Research ArticleModelling and Optimizing an Open-Pit TruckScheduling Problem

Yonggang Chang12 Huizhi Ren1 and Shijie Wang1

1School of Mechanical Engineering Shenyang University of Technology Shenyang 110870 China2ChinaCoal Pingshuo Group Co Ltd Shuozhou 036006 China

Correspondence should be addressed to Huizhi Ren renhuizhi126com

Received 18 December 2014 Accepted 14 February 2015

Academic Editor Purushothaman Damodaran

Copyright copy 2015 Yonggang Chang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper addresses a special truck scheduling problem in the open-pit mine with different transport revenue consideration Amixed integer programmingmodel is formulated to define the problem clearly and a few valid inequalities are deduced to strengthenthe model Some properties and two upper bounds of the problem are proposed Based on these inequalities properties and upperbounds a heuristic solution approach with two improvement strategies is proposed to resolve the problem and the numericalexperiment demonstrates that the proposed solution approach is effective and efficient

1 Introduction

In an open-pit mine trucks are a kind of crucial equipmentwhich undertakes nearly all ore andwaste rock transportation[1] Typically the truck is loaded a full load ofmaterial (maybeore gangue or other waste rocks that cover ores and thusneed to be moved away) under an electric shovel transportsit to a dumping depot dumps its loaded material and thenreturns back to the shovel to carry out the next round Figure 1illustrates an open-pit photo and a simplified map for theroad network in an open-pit mine All of thematerial loadingpoints each of which is equipped with an electric shoveland dumping depots are relatively constant while trucks cantransport material from varied loading points and unload atdifferent depots [2] The efficient truck scheduling can notonly reduce the truck transportation cost but also increase theshovel utilization ratio and the mine productivity [1]

There are plenty of researches published on the operationsresearch in the open-pit mine but most of them focus on thepit design and production planning [3] such as the strategicultimate pit limit design problem [4ndash7] and tactical block-sequencing problem [8ndash11] Among researches on the tacticaland operational equipment-allocation in the open-pit minethe stochastic feature of equipment has been emphasizedBoth queuing theory and simulation were used as stochastic

methodologies tomodel and analyse the open-pit productionscheduling problem especially in the early published liter-atures For example Manla and Ramani [12] simulated thetruck haulage process which initialized the truck schedulingresearch Kappas and Yegulalp [13] used the queuing theorycombined with extensions of Markovian principles to ana-lyze the steady-state performance of a typical truck-shovelsystem and estimate its operation parameters Oraee and Asi[14] simulated the production process of trucks and shovels inSongun Copper Mine in Iran considering fuzzy parameterson each shovel Najor and Hagan [15] focused on stochasticbehavior in the truck-shovel system and provided a modeland analysis based on queuing theory Their analysis showedthat productivity can be overestimated by about 8 percentbecause of overlooking queuing

In some of the open-pit equipment routing and selectionmodels optimization methodologies that is integer pro-gramming and dynamic programming were used to deter-mine fleet size and allocation [3] For example White andOlson [16] applied networkmodels linear programming anddynamic programming methods to the truck schedulingproblem and proposed a feasible truck-dispatching systemThey divided the problem into three stages The first one wasto decide the shortest paths between all locations in theminewhich is a shortest path problem actually In the next stage a

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 745378 8 pageshttpdxdoiorg1011552015745378

2 Discrete Dynamics in Nature and Society

(a) An open-pit mine

L1

L2

L3

L4

L5L6

D1

D2

D3

(b) A sketch map of the open-pit road networkwhere 1198711 1198713 1198716 represent the loading pointsand11986311198632 and1198633 represent the dumping depots

Figure 1 An open-pit mine and a part of its road network

linear program was used to determine material flows alongthese paths Finally the dynamic program assigned trucks tothe optimized paths between shovels and dumpsTheir truck-dispatching system not only yields significant (10ndash20 percentin most cases) productivity improvements but also producedan effect on truck scheduling theory for its three-stage process[17] And Yang et al [18] adopted the three-stage process toresolve the truck dispatch problem

In this paper we consider the varied transportationrevenue for different material loading point and address thejoint path planning and tuck dispatching problem

The outline of this paper is as follows Section 2 describesthe considered open-pit truck scheduling problem withvaried truck transporting revenue in detail and provides itsinteger programming model In Section 3 a few validinequalities two properties of optimal solution and twoupper bounds for the problem are presented In Section 4 aheuristic approach with two improvement strategies is pro-posed to resolve the problem Section 5 tests the proposedupper bounds and solution approach Section 6 concludes thepaper

2 Problem Description andMathematical Model

21 Problem Description In the open pit there are usuallyseveral loading points and dumping depots (to be calleddumps later) as shown in Figure 1 At each loading pointthere is always a pile of loosened ores gangue or waste rocks(to be referred to as material collectively) since the load-ing point needs a loosening blasting before the excavationoperation starting Each loading point is assumed to be

equipped with one and only one electric shovel (without lossof generality it will be viewed as twodifferent loading points iftwo shovels are equipped) The electric shovel excavates andloads the loosened material onto the trucks at each loadingpoint and the truck unloads itself with its self-dischargingdevice at a dump Similar to the loading point we assumethat one dump can accommodate only one truck at the sametime and the unloading area of more than one truck loadingpositions will be viewed as several different dumps

It is worth noting that an electric shovel (or loading point)can load only one truck at the same time and a dump allowsonly one truck simultaneously There is often a park for thewaiting trucks near an operating electric shovel or dump It isactually the key task to reduce the truck waiting times in thetraditional truck scheduling problem

In our considered problem scenario a truck of materialfrom different loading point is given different transportrevenue and our optimization objective will be equal tothe total transport revenue completed in the schedulinghorizonThe revenue per truck loading (transporting a truckof material) of different loading point is always decided bythe management and the decision-making basis includes twoconsiderations the possible transport distance and the trans-port priority The former is easy to be understood since thelonger distance always causes more transport times and thusshould be given more revenue The transport priority comesfrom the mining sequence which has never been modeled asa kind of constraints in the traditional research [1 3] andorthe varied market demand The transport revenue per truckloading can be viewed as the product of the possible transportdistance and the transport priority

From the viewpoint of a scheduler who is in charge ofmaking truck scheduling decisions in a shift (8 hours) ourobjective is to maximize the total transport revenues basedon the given revenue value per truck loading The decisionsinclude the dump selection as well as the truck routingThe dump selection is virtually the same as the first-stagedecisions in literature [16 18] deciding the shortest pathbetween all locations in the mine

The constraints to be considered include the following(1) loading capacity constraint that is each shovel (load-

ing point) can load only one truck at the same time(2) unloading capacity constraint that is each dump can

unload only one truck at the same time(3) dump selection constraint that is a set of candidate

dumps is given for each loading point and thematerialfrom the loading point must be dumped at one of thecandidate dumps

22 The Mixed Integer Programming Model To define andformulate the problemexplicitly the followingnotations needto be introduced to represent all loading points dumpsmaterial to be loaded in the loading points trucks pathsparameters and so on

119878 set of available electric shovels (loading points) 119878 =1 2 1198731198781015840 1198781015840 = 119878 cup 0 0 denoting dummy shovels

Discrete Dynamics in Nature and Society 3

119879 set of available trucks119886119894 revenue per truck of material from electric shovel119894 119894 isin 119878119863119894 set of candidate dumps corresponding to material

(waste rock or coal) loaded at electric shovel 119894 119894 isin 119878119863 = cup

119894isin119878119863119894 including all dumps

119867 the horizon considered in the problem119898119894 max number of truck loadings an electric shovel 119894

can complete at most during the horizon 119894 isin 119878119870119894 119870119894= 1 2 119898

119894 set of possible trucks of rock

(coal) that electric shovel 119894 loads 119894 isin 119878(119896 119894) a truck of material loaded by shovel 119894 119894 isin 119878119896 isin 119870

119894 hereafter referred to as truck loading (119896 119894)

11987001198700= 0 (0 0) denoting dummy truck loading

1199010

119894 loading times per truck at electric shovel 119894 119894 isin 1198781199011

119895 unloading times per truck at depot 119895 119895 isin 119863

1198750

119895119894 times for an empty truck to move from depot 119895 to

shovel 119894 119895 isin 119863 119894 isin 1198781198751

119894119895 times for a full-loaded truck to move form shovel119894 to depot 119895 119895 isin 119863 119894 isin 119878119902119905

119894 times for truck 119905 to move from its initial position

to shovel 119894 119894 isin 119878 119905 isin 119879119872 very big plus number

To represent the dump selection and truck dispatchdecisions we introduce the following decision variables

11990911990511989610158401198941015840119896119894=

1 if truck 119905 transports (119896 119894)

immediatly after (1198961015840 1198941015840)

0 otherwise

for 119905 isin 119879 119894 1198941015840 isin 1198781015840 119896 isin 119870119894 1198961015840isin 1198701198941015840

119911119896119894=

1 if truck loading (119896 119894) is loaded

and transported in the horizon

0 otherwise

for 119894 isin 119878 119896 isin 119870119894

11990611989511989610158401198941015840119896119894

=

1 if material (119896 119894) is dumped immediatly

after (1198961015840 1198941015840) at despot 119895

0 otherwise

for isin 119863119894cap 1198631198941015840 119894 1198941015840isin 119878 119896 isin 119870

119894 1198961015840isin 1198701198941015840

119910119896119894119895=

1 if material (119896 119894) is dumped at despot 119895

0 otherwise

for119895 isin 119863119894119894 isin 119878 119896 isin 119870

119894

(1)

1198880

119896119894is complete time of loadingmaterial (119896 119894) for 119894 isin 119878

119896 isin 1198701015840

119894

1198881

119896119894is complete time of unloading material (119896 119894) for

119894 isin 119878 119896 isin 1198701015840119894

Based on the above notations the problem can be formu-lated as the following integer programmingmodel (similar tomodel in Tang et al [19] which involved a truck schedulingproblem in container terminals)

Max sum

119894isin119878

sum

119896isin119870119894

119886119894119911119896119894 (2)

st 1198880

119896+1119894minus 1198880

119896119894ge 1199010

119894 119894 isin 119878 119896 isin 119870

119894(3)

1198881

119896119894minus 1198881

11989610158401198941015840 ge 1199011

119895minus119872(1 minus 119906

11989511989610158401198941015840119896119894)

119894 1198941015840isin 119878 119896 isin 119870

119894

1198961015840isin 1198701198941015840 119895 isin 119863

119894cap 1198631198941015840

(4)

1198881

119896119894minus 1198880

119896119894ge sum

119895isin119863

119910119896119894119895(119875119894119895+ 1199011

119895) 119894 isin 119878 119896 isin 119870

119894 (5)

1198880

119896119894minus 1198881

11989610158401198941015840 ge sum

119895isin119863

11991011989610158401198941015840119895119875119895119894+ 1199010

119894

minus119872(1 minus sum

119905isin119879

11990911990511989610158401198941015840119896119894)

119894 1198941015840isin 119878 119896 isin 119870

119894 1198961015840isin 1198701198941015840

(6)

1198880

119896119894ge sum

119905isin119879

11990911990500119896119894

119902119905

119894+ 1199010

119894minus119872(1 minus sum

119905isin119879

11990911990500119896119894

)

119894 isin 119878 119896 isin 119870119894

(7)

1198881

119896119894le 119867 +119872(1 minus 119911

119896119894) 119894 isin 119878 119896 isin 119870

1015840

119894(8)

sum

119894isin1198781015840

sum

119896isin119870119894

11990911990511989610158401198941015840119896119894= 11991111989610158401198941015840 119905 isin 119879 119894

1015840isin 119878 119896

1015840isin 1198701198941015840 (9)

sum

1198941015840isin1198781015840

sum

1198961015840isin1198701198941015840

11990911990511989610158401198941015840119896119894= 119911119896119894 119905 isin 119879 119894 isin 119878 119896 isin 119870

119894 (10)

119909119905119896119894119896119894= 0 119905 isin 119879 119894 isin 119878 119896 isin 119870

119894 (11)

sum

119894isin1198781015840

sum

119896isin119870119894

11990911990511989611989400

= sum

119894isin1198781015840

sum

119896isin119870119894

11990911990500119896119894

= 1 119905 isin 119879 (12)

sum

1198941015840isin1198781015840

sum

1198961015840isin1198701015840

119894

11990611989511989610158401198941015840119896119894= sum

1198941015840isin1198781015840

sum

119896isin1198701015840

119894

11990611989511989611989411989610158401198941015840 = 119911119896119894

119905 isin 119879 119894 isin 119878 119896 isin 1198701015840

119894

(13)

119906119895119896119894119896119894= 0 119895 isin 119863

119894 119894 isin 119878 119896 isin 119870

1015840

119894(14)

sum

119894isin119878

sum

119896isin119870119894

11990611989511989611989400

= sum

119894isin119878

sum

119896isin119870119894

11990611989500119896119894

= 1 119895 isin 119863 (15)

sum

119895isin119863119894

119910119896119894119895= 119911119896119894 119894 isin 119878 119896 isin 119870

119894 (16)

4 Discrete Dynamics in Nature and Society

The objective function (2) of the model maximizes thetotal transport value that is sum of transport revenue ofall truck loadings unloaded in horizon 119867 Constraints (3)can provide each truck with enough loading times betweenit and its adjacent trucks under the same shovel Similarlyconstraints (4) ensure enough time to unload materialbetween two adjacent trucks that continuously dump at thesame depot Constraints (5) guarantee that each truck hasenough traveling time to transportmaterial from the involvedshovel to depot while constraints (6) provide a truck withenough empty traveling time to return its next loading shovelConstraints (7) guarantee that each truck starts from its initialposition and reserves enough empty traveling time to reachits first loading spot before the involved shovel starts the truckloading operation Constraints (8) define the variable 119911

119896119894 that

is set 119911119896119894= 1 if truck loading (119896 119894) can be finished in horizon

119867 Constraints (9) and (10) guarantee that each valid truckloading (119911

119896119894= 1) is assigned to one and only one truck and

each truck can operate continuously Constraints (11) preventtruck loading self-loop Constraints (12) state that one andonly one truck loading serves as the head (last) truck loadingfor each available truck Similar to constraints (9) and (10)constraints (13) state that each valid truck loading is assignedto one and only one dumping depot and each dumpingdepot operates continuously Constraints (14) perform sameas constraints (11) prohibiting the self-loop in unloadingsequence Similar to constraints (12) constraints (15) statethat one and only one truck loading serves as the head (last)truck for each dumping depot Constraints (16) define therelationship between variables 119910

119896119894119895and 119911119896119894 that is each valid

truck loading has one and only one dumping depot

3 Valid Inequalities Property andUpper Bounds for the Problem

31 Valid Inequalities In this section a few valid inequalitiesare introduced To the problem definition the model inSection 22 is already adequate and all the valid inequalitiesare redundant but they can reduce the computing time ifwe attempt to directly solve the model through commercialoptimization software such as CPLEX In addition the validinequalities are also helpful for constructing the heuristicapproach

311 Electric Shovel Capacity Inequality Consider

sum

119896isin119870119894

1199111198961198941199010

119894le 119867 minusmin

119905isin119879

119902119905

119894minusmin119895isin119863119894

1199011

119894119895+ 1199011

119895 119894 isin 119878 (17)

Inequality (17) gives the maximum possible number oftrucks each electric shovel can finish in the horizon In theinequality 119867 minus min

119905isin119879119902119905

119894minus min

119895isin119863119894

1199011

119894119895is the practical time

limit for shovel 119894 where min119905isin119879119902119905

119894is the time for the nearest

empty truck to move from its initial position to shovel 119894 thatis the least possible times before shovel 119894 starting the firsttruck-loading and min

119895isin119863119894

1199011

119894119895+ 1199011

119895 is the permitted least

retained time to transport and unload the last truck loadedby shovel 119894

L1

L2

L3

L4

D1

D2

D3

Figure 2 Illustration of the truck round trip

312 Unloading Capacity Inequality Consider

sum

119894isin119878

sum

119896isin119870119894

1199101198961198941198951199011

119895le 119867 minus min

119894isin1198941015840|119895isin1198631198941015840

min119905isin119879

119902119905

119894+ 1199010

119894+ 1199011

119894119895

119895 isin 119863

(18)

Inequality (18) gives the maximum possible number oftrucks each unloading depot can unload in the horizon In theinequality119867minusmin

119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905

119894+1199010

119894+1199011

119894119895 is the practi-

cal time limit for depot 119895 wheremin119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905

119894+1199010

119894+

1199011

119894119895 is the earliest possible starting time of unloading the first

truck in depot 119895

313 Variables Relationship Inequality Consider

119911119896119894le 1199111198961015840119894 119896 le 119896

1015840 119894 isin 119878 (19)

Inequality (19) comes from the material transportationsequence as ensured by formula (3) where truck loading (119896 119894)is doomed to be transported before (1198961015840 119894)when 119896 le 1198961015840 stands

32 Property of the Optimal Solution For the above defineopen-pit truck scheduling problem we observe two proper-ties for the optimal solutions Some new concepts need to beintroduced to describe the properties

Definition 1 An empty truck starting from a depot 119895 119895 isin 119863119894

moves to a shovel 119894 119894 isin 119878 loads material (with shovel 119894)transports it to a depot 1198951015840 119895 isin 119863

119894 and unloads the material

(eg travelling from dump1198631to loading point 119871

1and to119863

3

in Figure 2) The whole movement process is called a truckround trip denoted by trip (119895 119894 1198951015840)

It is noteworthy that the end depot involved in a truckround trip can be the same as the starting depot

Definition 2 For any a truck round trip (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin119863119894 119894 isin 119878 119890

1198951198941198951015840 = 119886119894(1198750

119895119894+1198751

1198941198951015840 +1199010

119894+1199011

119895) is called trip transport

value

Discrete Dynamics in Nature and Society 5

Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value

Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times

(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel

(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot

(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon

It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011

119894119895and1199010

119895119894) through

the path

Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to

sum

119905isin119879

sum

(1198951198941198951015840)isinΨ

11989711990511989511989411989510158401198901198951198941198951015840 (20)

where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has

spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips

The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound

Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750

119895119894+ 1198751

1198941198951015840 + 1199010

119894+ 1199011

119895) sdot

min1198850119894 1198851

1198951015840 where1198850119894 and119885

1

1198951015840 denote themaximum loading

capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively

In Property 2 1198850119894and 1198851

1198951015840 can be obtained through

formulae (17) and (18) respectively

33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem

331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum

119894isin1198781198861198941198850

119894

In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873

119879denote the number of available trucks then

119862119879= 119867 sdot 119873

119879denotes the total available truck times involved

in the problem Let 119903119894denote the optimal truck round trip

involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires

the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast

119894and 119864lowast

119894denote the required valuable truck times and trip

transport value corresponding to trip 119903119894 respectively

It is assumed that 1198941 1198942 119894

119873is the shovel sequence in

the descending order of 119864lowast119894 that is 119864lowast

1198941

ge 119864lowast

1198942

ge sdot sdot sdot ge 119864lowast

119894119873

and1198941 1198942 119894

119873 = 119878

Upper bound 2 can be described and formulated clearlybased on the above notations and assumption

332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value

119891 (119862119879) =

119864lowast

1198941

119862119879

119862119879le 1198850

1198941

119875lowast

1198941

1198861198941

1198850

1198941

+ 119864lowast

1198942

(119862119879minus 1198850

1198941

119875lowast

1198941

)

1198850

1198941

119875lowast

1198941

le 119862119879

le 1198850

1198941

119875lowast

1198941

+ 1198850

1198942

119875lowast

1198942

119873minus2

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873minus1

(119862119879minus

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873minus1

119875lowast

119894119873minus1

119873minus1

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873

(119862119879minus

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873

119875lowast

119894119873

(21)

The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues

4 Heuristic Solution Method to the Problem

41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here

Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875

0

119895119894+

1198751

1198941198951015840 + 1199010

119894+ 1199011

119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863

119894 119894 isin 119878 is called

expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip

Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps

Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in

6 Discrete Dynamics in Nature and Society

the scheduling and set 119878idle to include all trucks in the initialstate

Build a truck set 119878running and let 119878running = 0 in the initialstate

Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0

Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4

Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip

Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running

Go to Step A2

Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop

The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process

42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution

Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894

1 1198942 119894

119873in the

descending order of 119864lowast119894(see Section 33) Because the initial

truck position can be far away from the shovel with higher119864lowast

119894 the strategy should reassign more trucks to these shovels

for the objective pursuitSteps of Strategy 1 are listed as follows

Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899

Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894

119899+1 119894119899+2 119894

119873

Table 1 Parameters of the problem instances

Path Problem Trucks Shovels Dumps

1

1 73 6 32 71 6 33 74 6 34 69 6 3

2

5 84 6 46 82 6 47 85 6 48 85 6 4

Step B3 If 119905idle1199010119894 ge 120583119867119875lowast

119894lowast in the beginning reassign

lfloor119905idle119875lowast

119894lowast1205831199010

119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule

the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned

Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step

B2 otherwise terminate

Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps

Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained

Step C2 Let 1199031 1199032 119903

119898 denote the set of truck round trips

executed by truck 119905Set 119896 = 1

Step C3 Attempt to reassign truck round trip 119903119896to other

shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration

Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3

Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate

5 Computational Experiments

To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM

The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances

Discrete Dynamics in Nature and Society 7

Table 2 Experiment result for the small sized problems

Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp

1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1

Table 3 Experiment result and comparison for the practical problem size

Problem Objective value Relative imp UB GAPHeuristic Improvement

1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446

including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances

The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances

The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively

The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative

imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table

From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations

6 Conclusion

In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient

8 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)

References

[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002

[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007

[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010

[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965

[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989

[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002

[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006

[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985

[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988

[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001

[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006

[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971

[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991

[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004

[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006

[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992

[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003

[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006

[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

(a) An open-pit mine

L1

L2

L3

L4

L5L6

D1

D2

D3

(b) A sketch map of the open-pit road networkwhere 1198711 1198713 1198716 represent the loading pointsand11986311198632 and1198633 represent the dumping depots

Figure 1 An open-pit mine and a part of its road network

linear program was used to determine material flows alongthese paths Finally the dynamic program assigned trucks tothe optimized paths between shovels and dumpsTheir truck-dispatching system not only yields significant (10ndash20 percentin most cases) productivity improvements but also producedan effect on truck scheduling theory for its three-stage process[17] And Yang et al [18] adopted the three-stage process toresolve the truck dispatch problem

In this paper we consider the varied transportationrevenue for different material loading point and address thejoint path planning and tuck dispatching problem

The outline of this paper is as follows Section 2 describesthe considered open-pit truck scheduling problem withvaried truck transporting revenue in detail and provides itsinteger programming model In Section 3 a few validinequalities two properties of optimal solution and twoupper bounds for the problem are presented In Section 4 aheuristic approach with two improvement strategies is pro-posed to resolve the problem Section 5 tests the proposedupper bounds and solution approach Section 6 concludes thepaper

2 Problem Description andMathematical Model

21 Problem Description In the open pit there are usuallyseveral loading points and dumping depots (to be calleddumps later) as shown in Figure 1 At each loading pointthere is always a pile of loosened ores gangue or waste rocks(to be referred to as material collectively) since the load-ing point needs a loosening blasting before the excavationoperation starting Each loading point is assumed to be

equipped with one and only one electric shovel (without lossof generality it will be viewed as twodifferent loading points iftwo shovels are equipped) The electric shovel excavates andloads the loosened material onto the trucks at each loadingpoint and the truck unloads itself with its self-dischargingdevice at a dump Similar to the loading point we assumethat one dump can accommodate only one truck at the sametime and the unloading area of more than one truck loadingpositions will be viewed as several different dumps

It is worth noting that an electric shovel (or loading point)can load only one truck at the same time and a dump allowsonly one truck simultaneously There is often a park for thewaiting trucks near an operating electric shovel or dump It isactually the key task to reduce the truck waiting times in thetraditional truck scheduling problem

In our considered problem scenario a truck of materialfrom different loading point is given different transportrevenue and our optimization objective will be equal tothe total transport revenue completed in the schedulinghorizonThe revenue per truck loading (transporting a truckof material) of different loading point is always decided bythe management and the decision-making basis includes twoconsiderations the possible transport distance and the trans-port priority The former is easy to be understood since thelonger distance always causes more transport times and thusshould be given more revenue The transport priority comesfrom the mining sequence which has never been modeled asa kind of constraints in the traditional research [1 3] andorthe varied market demand The transport revenue per truckloading can be viewed as the product of the possible transportdistance and the transport priority

From the viewpoint of a scheduler who is in charge ofmaking truck scheduling decisions in a shift (8 hours) ourobjective is to maximize the total transport revenues basedon the given revenue value per truck loading The decisionsinclude the dump selection as well as the truck routingThe dump selection is virtually the same as the first-stagedecisions in literature [16 18] deciding the shortest pathbetween all locations in the mine

The constraints to be considered include the following(1) loading capacity constraint that is each shovel (load-

ing point) can load only one truck at the same time(2) unloading capacity constraint that is each dump can

unload only one truck at the same time(3) dump selection constraint that is a set of candidate

dumps is given for each loading point and thematerialfrom the loading point must be dumped at one of thecandidate dumps

22 The Mixed Integer Programming Model To define andformulate the problemexplicitly the followingnotations needto be introduced to represent all loading points dumpsmaterial to be loaded in the loading points trucks pathsparameters and so on

119878 set of available electric shovels (loading points) 119878 =1 2 1198731198781015840 1198781015840 = 119878 cup 0 0 denoting dummy shovels

Discrete Dynamics in Nature and Society 3

119879 set of available trucks119886119894 revenue per truck of material from electric shovel119894 119894 isin 119878119863119894 set of candidate dumps corresponding to material

(waste rock or coal) loaded at electric shovel 119894 119894 isin 119878119863 = cup

119894isin119878119863119894 including all dumps

119867 the horizon considered in the problem119898119894 max number of truck loadings an electric shovel 119894

can complete at most during the horizon 119894 isin 119878119870119894 119870119894= 1 2 119898

119894 set of possible trucks of rock

(coal) that electric shovel 119894 loads 119894 isin 119878(119896 119894) a truck of material loaded by shovel 119894 119894 isin 119878119896 isin 119870

119894 hereafter referred to as truck loading (119896 119894)

11987001198700= 0 (0 0) denoting dummy truck loading

1199010

119894 loading times per truck at electric shovel 119894 119894 isin 1198781199011

119895 unloading times per truck at depot 119895 119895 isin 119863

1198750

119895119894 times for an empty truck to move from depot 119895 to

shovel 119894 119895 isin 119863 119894 isin 1198781198751

119894119895 times for a full-loaded truck to move form shovel119894 to depot 119895 119895 isin 119863 119894 isin 119878119902119905

119894 times for truck 119905 to move from its initial position

to shovel 119894 119894 isin 119878 119905 isin 119879119872 very big plus number

To represent the dump selection and truck dispatchdecisions we introduce the following decision variables

11990911990511989610158401198941015840119896119894=

1 if truck 119905 transports (119896 119894)

immediatly after (1198961015840 1198941015840)

0 otherwise

for 119905 isin 119879 119894 1198941015840 isin 1198781015840 119896 isin 119870119894 1198961015840isin 1198701198941015840

119911119896119894=

1 if truck loading (119896 119894) is loaded

and transported in the horizon

0 otherwise

for 119894 isin 119878 119896 isin 119870119894

11990611989511989610158401198941015840119896119894

=

1 if material (119896 119894) is dumped immediatly

after (1198961015840 1198941015840) at despot 119895

0 otherwise

for isin 119863119894cap 1198631198941015840 119894 1198941015840isin 119878 119896 isin 119870

119894 1198961015840isin 1198701198941015840

119910119896119894119895=

1 if material (119896 119894) is dumped at despot 119895

0 otherwise

for119895 isin 119863119894119894 isin 119878 119896 isin 119870

119894

(1)

1198880

119896119894is complete time of loadingmaterial (119896 119894) for 119894 isin 119878

119896 isin 1198701015840

119894

1198881

119896119894is complete time of unloading material (119896 119894) for

119894 isin 119878 119896 isin 1198701015840119894

Based on the above notations the problem can be formu-lated as the following integer programmingmodel (similar tomodel in Tang et al [19] which involved a truck schedulingproblem in container terminals)

Max sum

119894isin119878

sum

119896isin119870119894

119886119894119911119896119894 (2)

st 1198880

119896+1119894minus 1198880

119896119894ge 1199010

119894 119894 isin 119878 119896 isin 119870

119894(3)

1198881

119896119894minus 1198881

11989610158401198941015840 ge 1199011

119895minus119872(1 minus 119906

11989511989610158401198941015840119896119894)

119894 1198941015840isin 119878 119896 isin 119870

119894

1198961015840isin 1198701198941015840 119895 isin 119863

119894cap 1198631198941015840

(4)

1198881

119896119894minus 1198880

119896119894ge sum

119895isin119863

119910119896119894119895(119875119894119895+ 1199011

119895) 119894 isin 119878 119896 isin 119870

119894 (5)

1198880

119896119894minus 1198881

11989610158401198941015840 ge sum

119895isin119863

11991011989610158401198941015840119895119875119895119894+ 1199010

119894

minus119872(1 minus sum

119905isin119879

11990911990511989610158401198941015840119896119894)

119894 1198941015840isin 119878 119896 isin 119870

119894 1198961015840isin 1198701198941015840

(6)

1198880

119896119894ge sum

119905isin119879

11990911990500119896119894

119902119905

119894+ 1199010

119894minus119872(1 minus sum

119905isin119879

11990911990500119896119894

)

119894 isin 119878 119896 isin 119870119894

(7)

1198881

119896119894le 119867 +119872(1 minus 119911

119896119894) 119894 isin 119878 119896 isin 119870

1015840

119894(8)

sum

119894isin1198781015840

sum

119896isin119870119894

11990911990511989610158401198941015840119896119894= 11991111989610158401198941015840 119905 isin 119879 119894

1015840isin 119878 119896

1015840isin 1198701198941015840 (9)

sum

1198941015840isin1198781015840

sum

1198961015840isin1198701198941015840

11990911990511989610158401198941015840119896119894= 119911119896119894 119905 isin 119879 119894 isin 119878 119896 isin 119870

119894 (10)

119909119905119896119894119896119894= 0 119905 isin 119879 119894 isin 119878 119896 isin 119870

119894 (11)

sum

119894isin1198781015840

sum

119896isin119870119894

11990911990511989611989400

= sum

119894isin1198781015840

sum

119896isin119870119894

11990911990500119896119894

= 1 119905 isin 119879 (12)

sum

1198941015840isin1198781015840

sum

1198961015840isin1198701015840

119894

11990611989511989610158401198941015840119896119894= sum

1198941015840isin1198781015840

sum

119896isin1198701015840

119894

11990611989511989611989411989610158401198941015840 = 119911119896119894

119905 isin 119879 119894 isin 119878 119896 isin 1198701015840

119894

(13)

119906119895119896119894119896119894= 0 119895 isin 119863

119894 119894 isin 119878 119896 isin 119870

1015840

119894(14)

sum

119894isin119878

sum

119896isin119870119894

11990611989511989611989400

= sum

119894isin119878

sum

119896isin119870119894

11990611989500119896119894

= 1 119895 isin 119863 (15)

sum

119895isin119863119894

119910119896119894119895= 119911119896119894 119894 isin 119878 119896 isin 119870

119894 (16)

4 Discrete Dynamics in Nature and Society

The objective function (2) of the model maximizes thetotal transport value that is sum of transport revenue ofall truck loadings unloaded in horizon 119867 Constraints (3)can provide each truck with enough loading times betweenit and its adjacent trucks under the same shovel Similarlyconstraints (4) ensure enough time to unload materialbetween two adjacent trucks that continuously dump at thesame depot Constraints (5) guarantee that each truck hasenough traveling time to transportmaterial from the involvedshovel to depot while constraints (6) provide a truck withenough empty traveling time to return its next loading shovelConstraints (7) guarantee that each truck starts from its initialposition and reserves enough empty traveling time to reachits first loading spot before the involved shovel starts the truckloading operation Constraints (8) define the variable 119911

119896119894 that

is set 119911119896119894= 1 if truck loading (119896 119894) can be finished in horizon

119867 Constraints (9) and (10) guarantee that each valid truckloading (119911

119896119894= 1) is assigned to one and only one truck and

each truck can operate continuously Constraints (11) preventtruck loading self-loop Constraints (12) state that one andonly one truck loading serves as the head (last) truck loadingfor each available truck Similar to constraints (9) and (10)constraints (13) state that each valid truck loading is assignedto one and only one dumping depot and each dumpingdepot operates continuously Constraints (14) perform sameas constraints (11) prohibiting the self-loop in unloadingsequence Similar to constraints (12) constraints (15) statethat one and only one truck loading serves as the head (last)truck for each dumping depot Constraints (16) define therelationship between variables 119910

119896119894119895and 119911119896119894 that is each valid

truck loading has one and only one dumping depot

3 Valid Inequalities Property andUpper Bounds for the Problem

31 Valid Inequalities In this section a few valid inequalitiesare introduced To the problem definition the model inSection 22 is already adequate and all the valid inequalitiesare redundant but they can reduce the computing time ifwe attempt to directly solve the model through commercialoptimization software such as CPLEX In addition the validinequalities are also helpful for constructing the heuristicapproach

311 Electric Shovel Capacity Inequality Consider

sum

119896isin119870119894

1199111198961198941199010

119894le 119867 minusmin

119905isin119879

119902119905

119894minusmin119895isin119863119894

1199011

119894119895+ 1199011

119895 119894 isin 119878 (17)

Inequality (17) gives the maximum possible number oftrucks each electric shovel can finish in the horizon In theinequality 119867 minus min

119905isin119879119902119905

119894minus min

119895isin119863119894

1199011

119894119895is the practical time

limit for shovel 119894 where min119905isin119879119902119905

119894is the time for the nearest

empty truck to move from its initial position to shovel 119894 thatis the least possible times before shovel 119894 starting the firsttruck-loading and min

119895isin119863119894

1199011

119894119895+ 1199011

119895 is the permitted least

retained time to transport and unload the last truck loadedby shovel 119894

L1

L2

L3

L4

D1

D2

D3

Figure 2 Illustration of the truck round trip

312 Unloading Capacity Inequality Consider

sum

119894isin119878

sum

119896isin119870119894

1199101198961198941198951199011

119895le 119867 minus min

119894isin1198941015840|119895isin1198631198941015840

min119905isin119879

119902119905

119894+ 1199010

119894+ 1199011

119894119895

119895 isin 119863

(18)

Inequality (18) gives the maximum possible number oftrucks each unloading depot can unload in the horizon In theinequality119867minusmin

119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905

119894+1199010

119894+1199011

119894119895 is the practi-

cal time limit for depot 119895 wheremin119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905

119894+1199010

119894+

1199011

119894119895 is the earliest possible starting time of unloading the first

truck in depot 119895

313 Variables Relationship Inequality Consider

119911119896119894le 1199111198961015840119894 119896 le 119896

1015840 119894 isin 119878 (19)

Inequality (19) comes from the material transportationsequence as ensured by formula (3) where truck loading (119896 119894)is doomed to be transported before (1198961015840 119894)when 119896 le 1198961015840 stands

32 Property of the Optimal Solution For the above defineopen-pit truck scheduling problem we observe two proper-ties for the optimal solutions Some new concepts need to beintroduced to describe the properties

Definition 1 An empty truck starting from a depot 119895 119895 isin 119863119894

moves to a shovel 119894 119894 isin 119878 loads material (with shovel 119894)transports it to a depot 1198951015840 119895 isin 119863

119894 and unloads the material

(eg travelling from dump1198631to loading point 119871

1and to119863

3

in Figure 2) The whole movement process is called a truckround trip denoted by trip (119895 119894 1198951015840)

It is noteworthy that the end depot involved in a truckround trip can be the same as the starting depot

Definition 2 For any a truck round trip (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin119863119894 119894 isin 119878 119890

1198951198941198951015840 = 119886119894(1198750

119895119894+1198751

1198941198951015840 +1199010

119894+1199011

119895) is called trip transport

value

Discrete Dynamics in Nature and Society 5

Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value

Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times

(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel

(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot

(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon

It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011

119894119895and1199010

119895119894) through

the path

Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to

sum

119905isin119879

sum

(1198951198941198951015840)isinΨ

11989711990511989511989411989510158401198901198951198941198951015840 (20)

where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has

spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips

The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound

Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750

119895119894+ 1198751

1198941198951015840 + 1199010

119894+ 1199011

119895) sdot

min1198850119894 1198851

1198951015840 where1198850119894 and119885

1

1198951015840 denote themaximum loading

capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively

In Property 2 1198850119894and 1198851

1198951015840 can be obtained through

formulae (17) and (18) respectively

33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem

331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum

119894isin1198781198861198941198850

119894

In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873

119879denote the number of available trucks then

119862119879= 119867 sdot 119873

119879denotes the total available truck times involved

in the problem Let 119903119894denote the optimal truck round trip

involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires

the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast

119894and 119864lowast

119894denote the required valuable truck times and trip

transport value corresponding to trip 119903119894 respectively

It is assumed that 1198941 1198942 119894

119873is the shovel sequence in

the descending order of 119864lowast119894 that is 119864lowast

1198941

ge 119864lowast

1198942

ge sdot sdot sdot ge 119864lowast

119894119873

and1198941 1198942 119894

119873 = 119878

Upper bound 2 can be described and formulated clearlybased on the above notations and assumption

332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value

119891 (119862119879) =

119864lowast

1198941

119862119879

119862119879le 1198850

1198941

119875lowast

1198941

1198861198941

1198850

1198941

+ 119864lowast

1198942

(119862119879minus 1198850

1198941

119875lowast

1198941

)

1198850

1198941

119875lowast

1198941

le 119862119879

le 1198850

1198941

119875lowast

1198941

+ 1198850

1198942

119875lowast

1198942

119873minus2

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873minus1

(119862119879minus

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873minus1

119875lowast

119894119873minus1

119873minus1

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873

(119862119879minus

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873

119875lowast

119894119873

(21)

The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues

4 Heuristic Solution Method to the Problem

41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here

Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875

0

119895119894+

1198751

1198941198951015840 + 1199010

119894+ 1199011

119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863

119894 119894 isin 119878 is called

expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip

Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps

Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in

6 Discrete Dynamics in Nature and Society

the scheduling and set 119878idle to include all trucks in the initialstate

Build a truck set 119878running and let 119878running = 0 in the initialstate

Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0

Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4

Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip

Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running

Go to Step A2

Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop

The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process

42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution

Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894

1 1198942 119894

119873in the

descending order of 119864lowast119894(see Section 33) Because the initial

truck position can be far away from the shovel with higher119864lowast

119894 the strategy should reassign more trucks to these shovels

for the objective pursuitSteps of Strategy 1 are listed as follows

Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899

Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894

119899+1 119894119899+2 119894

119873

Table 1 Parameters of the problem instances

Path Problem Trucks Shovels Dumps

1

1 73 6 32 71 6 33 74 6 34 69 6 3

2

5 84 6 46 82 6 47 85 6 48 85 6 4

Step B3 If 119905idle1199010119894 ge 120583119867119875lowast

119894lowast in the beginning reassign

lfloor119905idle119875lowast

119894lowast1205831199010

119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule

the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned

Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step

B2 otherwise terminate

Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps

Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained

Step C2 Let 1199031 1199032 119903

119898 denote the set of truck round trips

executed by truck 119905Set 119896 = 1

Step C3 Attempt to reassign truck round trip 119903119896to other

shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration

Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3

Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate

5 Computational Experiments

To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM

The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances

Discrete Dynamics in Nature and Society 7

Table 2 Experiment result for the small sized problems

Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp

1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1

Table 3 Experiment result and comparison for the practical problem size

Problem Objective value Relative imp UB GAPHeuristic Improvement

1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446

including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances

The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances

The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively

The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative

imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table

From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations

6 Conclusion

In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient

8 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)

References

[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002

[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007

[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010

[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965

[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989

[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002

[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006

[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985

[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988

[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001

[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006

[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971

[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991

[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004

[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006

[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992

[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003

[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006

[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

119879 set of available trucks119886119894 revenue per truck of material from electric shovel119894 119894 isin 119878119863119894 set of candidate dumps corresponding to material

(waste rock or coal) loaded at electric shovel 119894 119894 isin 119878119863 = cup

119894isin119878119863119894 including all dumps

119867 the horizon considered in the problem119898119894 max number of truck loadings an electric shovel 119894

can complete at most during the horizon 119894 isin 119878119870119894 119870119894= 1 2 119898

119894 set of possible trucks of rock

(coal) that electric shovel 119894 loads 119894 isin 119878(119896 119894) a truck of material loaded by shovel 119894 119894 isin 119878119896 isin 119870

119894 hereafter referred to as truck loading (119896 119894)

11987001198700= 0 (0 0) denoting dummy truck loading

1199010

119894 loading times per truck at electric shovel 119894 119894 isin 1198781199011

119895 unloading times per truck at depot 119895 119895 isin 119863

1198750

119895119894 times for an empty truck to move from depot 119895 to

shovel 119894 119895 isin 119863 119894 isin 1198781198751

119894119895 times for a full-loaded truck to move form shovel119894 to depot 119895 119895 isin 119863 119894 isin 119878119902119905

119894 times for truck 119905 to move from its initial position

to shovel 119894 119894 isin 119878 119905 isin 119879119872 very big plus number

To represent the dump selection and truck dispatchdecisions we introduce the following decision variables

11990911990511989610158401198941015840119896119894=

1 if truck 119905 transports (119896 119894)

immediatly after (1198961015840 1198941015840)

0 otherwise

for 119905 isin 119879 119894 1198941015840 isin 1198781015840 119896 isin 119870119894 1198961015840isin 1198701198941015840

119911119896119894=

1 if truck loading (119896 119894) is loaded

and transported in the horizon

0 otherwise

for 119894 isin 119878 119896 isin 119870119894

11990611989511989610158401198941015840119896119894

=

1 if material (119896 119894) is dumped immediatly

after (1198961015840 1198941015840) at despot 119895

0 otherwise

for isin 119863119894cap 1198631198941015840 119894 1198941015840isin 119878 119896 isin 119870

119894 1198961015840isin 1198701198941015840

119910119896119894119895=

1 if material (119896 119894) is dumped at despot 119895

0 otherwise

for119895 isin 119863119894119894 isin 119878 119896 isin 119870

119894

(1)

1198880

119896119894is complete time of loadingmaterial (119896 119894) for 119894 isin 119878

119896 isin 1198701015840

119894

1198881

119896119894is complete time of unloading material (119896 119894) for

119894 isin 119878 119896 isin 1198701015840119894

Based on the above notations the problem can be formu-lated as the following integer programmingmodel (similar tomodel in Tang et al [19] which involved a truck schedulingproblem in container terminals)

Max sum

119894isin119878

sum

119896isin119870119894

119886119894119911119896119894 (2)

st 1198880

119896+1119894minus 1198880

119896119894ge 1199010

119894 119894 isin 119878 119896 isin 119870

119894(3)

1198881

119896119894minus 1198881

11989610158401198941015840 ge 1199011

119895minus119872(1 minus 119906

11989511989610158401198941015840119896119894)

119894 1198941015840isin 119878 119896 isin 119870

119894

1198961015840isin 1198701198941015840 119895 isin 119863

119894cap 1198631198941015840

(4)

1198881

119896119894minus 1198880

119896119894ge sum

119895isin119863

119910119896119894119895(119875119894119895+ 1199011

119895) 119894 isin 119878 119896 isin 119870

119894 (5)

1198880

119896119894minus 1198881

11989610158401198941015840 ge sum

119895isin119863

11991011989610158401198941015840119895119875119895119894+ 1199010

119894

minus119872(1 minus sum

119905isin119879

11990911990511989610158401198941015840119896119894)

119894 1198941015840isin 119878 119896 isin 119870

119894 1198961015840isin 1198701198941015840

(6)

1198880

119896119894ge sum

119905isin119879

11990911990500119896119894

119902119905

119894+ 1199010

119894minus119872(1 minus sum

119905isin119879

11990911990500119896119894

)

119894 isin 119878 119896 isin 119870119894

(7)

1198881

119896119894le 119867 +119872(1 minus 119911

119896119894) 119894 isin 119878 119896 isin 119870

1015840

119894(8)

sum

119894isin1198781015840

sum

119896isin119870119894

11990911990511989610158401198941015840119896119894= 11991111989610158401198941015840 119905 isin 119879 119894

1015840isin 119878 119896

1015840isin 1198701198941015840 (9)

sum

1198941015840isin1198781015840

sum

1198961015840isin1198701198941015840

11990911990511989610158401198941015840119896119894= 119911119896119894 119905 isin 119879 119894 isin 119878 119896 isin 119870

119894 (10)

119909119905119896119894119896119894= 0 119905 isin 119879 119894 isin 119878 119896 isin 119870

119894 (11)

sum

119894isin1198781015840

sum

119896isin119870119894

11990911990511989611989400

= sum

119894isin1198781015840

sum

119896isin119870119894

11990911990500119896119894

= 1 119905 isin 119879 (12)

sum

1198941015840isin1198781015840

sum

1198961015840isin1198701015840

119894

11990611989511989610158401198941015840119896119894= sum

1198941015840isin1198781015840

sum

119896isin1198701015840

119894

11990611989511989611989411989610158401198941015840 = 119911119896119894

119905 isin 119879 119894 isin 119878 119896 isin 1198701015840

119894

(13)

119906119895119896119894119896119894= 0 119895 isin 119863

119894 119894 isin 119878 119896 isin 119870

1015840

119894(14)

sum

119894isin119878

sum

119896isin119870119894

11990611989511989611989400

= sum

119894isin119878

sum

119896isin119870119894

11990611989500119896119894

= 1 119895 isin 119863 (15)

sum

119895isin119863119894

119910119896119894119895= 119911119896119894 119894 isin 119878 119896 isin 119870

119894 (16)

4 Discrete Dynamics in Nature and Society

The objective function (2) of the model maximizes thetotal transport value that is sum of transport revenue ofall truck loadings unloaded in horizon 119867 Constraints (3)can provide each truck with enough loading times betweenit and its adjacent trucks under the same shovel Similarlyconstraints (4) ensure enough time to unload materialbetween two adjacent trucks that continuously dump at thesame depot Constraints (5) guarantee that each truck hasenough traveling time to transportmaterial from the involvedshovel to depot while constraints (6) provide a truck withenough empty traveling time to return its next loading shovelConstraints (7) guarantee that each truck starts from its initialposition and reserves enough empty traveling time to reachits first loading spot before the involved shovel starts the truckloading operation Constraints (8) define the variable 119911

119896119894 that

is set 119911119896119894= 1 if truck loading (119896 119894) can be finished in horizon

119867 Constraints (9) and (10) guarantee that each valid truckloading (119911

119896119894= 1) is assigned to one and only one truck and

each truck can operate continuously Constraints (11) preventtruck loading self-loop Constraints (12) state that one andonly one truck loading serves as the head (last) truck loadingfor each available truck Similar to constraints (9) and (10)constraints (13) state that each valid truck loading is assignedto one and only one dumping depot and each dumpingdepot operates continuously Constraints (14) perform sameas constraints (11) prohibiting the self-loop in unloadingsequence Similar to constraints (12) constraints (15) statethat one and only one truck loading serves as the head (last)truck for each dumping depot Constraints (16) define therelationship between variables 119910

119896119894119895and 119911119896119894 that is each valid

truck loading has one and only one dumping depot

3 Valid Inequalities Property andUpper Bounds for the Problem

31 Valid Inequalities In this section a few valid inequalitiesare introduced To the problem definition the model inSection 22 is already adequate and all the valid inequalitiesare redundant but they can reduce the computing time ifwe attempt to directly solve the model through commercialoptimization software such as CPLEX In addition the validinequalities are also helpful for constructing the heuristicapproach

311 Electric Shovel Capacity Inequality Consider

sum

119896isin119870119894

1199111198961198941199010

119894le 119867 minusmin

119905isin119879

119902119905

119894minusmin119895isin119863119894

1199011

119894119895+ 1199011

119895 119894 isin 119878 (17)

Inequality (17) gives the maximum possible number oftrucks each electric shovel can finish in the horizon In theinequality 119867 minus min

119905isin119879119902119905

119894minus min

119895isin119863119894

1199011

119894119895is the practical time

limit for shovel 119894 where min119905isin119879119902119905

119894is the time for the nearest

empty truck to move from its initial position to shovel 119894 thatis the least possible times before shovel 119894 starting the firsttruck-loading and min

119895isin119863119894

1199011

119894119895+ 1199011

119895 is the permitted least

retained time to transport and unload the last truck loadedby shovel 119894

L1

L2

L3

L4

D1

D2

D3

Figure 2 Illustration of the truck round trip

312 Unloading Capacity Inequality Consider

sum

119894isin119878

sum

119896isin119870119894

1199101198961198941198951199011

119895le 119867 minus min

119894isin1198941015840|119895isin1198631198941015840

min119905isin119879

119902119905

119894+ 1199010

119894+ 1199011

119894119895

119895 isin 119863

(18)

Inequality (18) gives the maximum possible number oftrucks each unloading depot can unload in the horizon In theinequality119867minusmin

119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905

119894+1199010

119894+1199011

119894119895 is the practi-

cal time limit for depot 119895 wheremin119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905

119894+1199010

119894+

1199011

119894119895 is the earliest possible starting time of unloading the first

truck in depot 119895

313 Variables Relationship Inequality Consider

119911119896119894le 1199111198961015840119894 119896 le 119896

1015840 119894 isin 119878 (19)

Inequality (19) comes from the material transportationsequence as ensured by formula (3) where truck loading (119896 119894)is doomed to be transported before (1198961015840 119894)when 119896 le 1198961015840 stands

32 Property of the Optimal Solution For the above defineopen-pit truck scheduling problem we observe two proper-ties for the optimal solutions Some new concepts need to beintroduced to describe the properties

Definition 1 An empty truck starting from a depot 119895 119895 isin 119863119894

moves to a shovel 119894 119894 isin 119878 loads material (with shovel 119894)transports it to a depot 1198951015840 119895 isin 119863

119894 and unloads the material

(eg travelling from dump1198631to loading point 119871

1and to119863

3

in Figure 2) The whole movement process is called a truckround trip denoted by trip (119895 119894 1198951015840)

It is noteworthy that the end depot involved in a truckround trip can be the same as the starting depot

Definition 2 For any a truck round trip (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin119863119894 119894 isin 119878 119890

1198951198941198951015840 = 119886119894(1198750

119895119894+1198751

1198941198951015840 +1199010

119894+1199011

119895) is called trip transport

value

Discrete Dynamics in Nature and Society 5

Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value

Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times

(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel

(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot

(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon

It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011

119894119895and1199010

119895119894) through

the path

Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to

sum

119905isin119879

sum

(1198951198941198951015840)isinΨ

11989711990511989511989411989510158401198901198951198941198951015840 (20)

where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has

spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips

The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound

Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750

119895119894+ 1198751

1198941198951015840 + 1199010

119894+ 1199011

119895) sdot

min1198850119894 1198851

1198951015840 where1198850119894 and119885

1

1198951015840 denote themaximum loading

capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively

In Property 2 1198850119894and 1198851

1198951015840 can be obtained through

formulae (17) and (18) respectively

33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem

331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum

119894isin1198781198861198941198850

119894

In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873

119879denote the number of available trucks then

119862119879= 119867 sdot 119873

119879denotes the total available truck times involved

in the problem Let 119903119894denote the optimal truck round trip

involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires

the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast

119894and 119864lowast

119894denote the required valuable truck times and trip

transport value corresponding to trip 119903119894 respectively

It is assumed that 1198941 1198942 119894

119873is the shovel sequence in

the descending order of 119864lowast119894 that is 119864lowast

1198941

ge 119864lowast

1198942

ge sdot sdot sdot ge 119864lowast

119894119873

and1198941 1198942 119894

119873 = 119878

Upper bound 2 can be described and formulated clearlybased on the above notations and assumption

332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value

119891 (119862119879) =

119864lowast

1198941

119862119879

119862119879le 1198850

1198941

119875lowast

1198941

1198861198941

1198850

1198941

+ 119864lowast

1198942

(119862119879minus 1198850

1198941

119875lowast

1198941

)

1198850

1198941

119875lowast

1198941

le 119862119879

le 1198850

1198941

119875lowast

1198941

+ 1198850

1198942

119875lowast

1198942

119873minus2

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873minus1

(119862119879minus

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873minus1

119875lowast

119894119873minus1

119873minus1

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873

(119862119879minus

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873

119875lowast

119894119873

(21)

The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues

4 Heuristic Solution Method to the Problem

41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here

Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875

0

119895119894+

1198751

1198941198951015840 + 1199010

119894+ 1199011

119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863

119894 119894 isin 119878 is called

expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip

Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps

Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in

6 Discrete Dynamics in Nature and Society

the scheduling and set 119878idle to include all trucks in the initialstate

Build a truck set 119878running and let 119878running = 0 in the initialstate

Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0

Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4

Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip

Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running

Go to Step A2

Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop

The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process

42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution

Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894

1 1198942 119894

119873in the

descending order of 119864lowast119894(see Section 33) Because the initial

truck position can be far away from the shovel with higher119864lowast

119894 the strategy should reassign more trucks to these shovels

for the objective pursuitSteps of Strategy 1 are listed as follows

Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899

Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894

119899+1 119894119899+2 119894

119873

Table 1 Parameters of the problem instances

Path Problem Trucks Shovels Dumps

1

1 73 6 32 71 6 33 74 6 34 69 6 3

2

5 84 6 46 82 6 47 85 6 48 85 6 4

Step B3 If 119905idle1199010119894 ge 120583119867119875lowast

119894lowast in the beginning reassign

lfloor119905idle119875lowast

119894lowast1205831199010

119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule

the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned

Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step

B2 otherwise terminate

Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps

Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained

Step C2 Let 1199031 1199032 119903

119898 denote the set of truck round trips

executed by truck 119905Set 119896 = 1

Step C3 Attempt to reassign truck round trip 119903119896to other

shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration

Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3

Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate

5 Computational Experiments

To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM

The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances

Discrete Dynamics in Nature and Society 7

Table 2 Experiment result for the small sized problems

Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp

1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1

Table 3 Experiment result and comparison for the practical problem size

Problem Objective value Relative imp UB GAPHeuristic Improvement

1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446

including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances

The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances

The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively

The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative

imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table

From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations

6 Conclusion

In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient

8 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)

References

[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002

[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007

[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010

[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965

[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989

[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002

[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006

[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985

[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988

[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001

[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006

[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971

[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991

[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004

[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006

[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992

[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003

[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006

[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

The objective function (2) of the model maximizes thetotal transport value that is sum of transport revenue ofall truck loadings unloaded in horizon 119867 Constraints (3)can provide each truck with enough loading times betweenit and its adjacent trucks under the same shovel Similarlyconstraints (4) ensure enough time to unload materialbetween two adjacent trucks that continuously dump at thesame depot Constraints (5) guarantee that each truck hasenough traveling time to transportmaterial from the involvedshovel to depot while constraints (6) provide a truck withenough empty traveling time to return its next loading shovelConstraints (7) guarantee that each truck starts from its initialposition and reserves enough empty traveling time to reachits first loading spot before the involved shovel starts the truckloading operation Constraints (8) define the variable 119911

119896119894 that

is set 119911119896119894= 1 if truck loading (119896 119894) can be finished in horizon

119867 Constraints (9) and (10) guarantee that each valid truckloading (119911

119896119894= 1) is assigned to one and only one truck and

each truck can operate continuously Constraints (11) preventtruck loading self-loop Constraints (12) state that one andonly one truck loading serves as the head (last) truck loadingfor each available truck Similar to constraints (9) and (10)constraints (13) state that each valid truck loading is assignedto one and only one dumping depot and each dumpingdepot operates continuously Constraints (14) perform sameas constraints (11) prohibiting the self-loop in unloadingsequence Similar to constraints (12) constraints (15) statethat one and only one truck loading serves as the head (last)truck for each dumping depot Constraints (16) define therelationship between variables 119910

119896119894119895and 119911119896119894 that is each valid

truck loading has one and only one dumping depot

3 Valid Inequalities Property andUpper Bounds for the Problem

31 Valid Inequalities In this section a few valid inequalitiesare introduced To the problem definition the model inSection 22 is already adequate and all the valid inequalitiesare redundant but they can reduce the computing time ifwe attempt to directly solve the model through commercialoptimization software such as CPLEX In addition the validinequalities are also helpful for constructing the heuristicapproach

311 Electric Shovel Capacity Inequality Consider

sum

119896isin119870119894

1199111198961198941199010

119894le 119867 minusmin

119905isin119879

119902119905

119894minusmin119895isin119863119894

1199011

119894119895+ 1199011

119895 119894 isin 119878 (17)

Inequality (17) gives the maximum possible number oftrucks each electric shovel can finish in the horizon In theinequality 119867 minus min

119905isin119879119902119905

119894minus min

119895isin119863119894

1199011

119894119895is the practical time

limit for shovel 119894 where min119905isin119879119902119905

119894is the time for the nearest

empty truck to move from its initial position to shovel 119894 thatis the least possible times before shovel 119894 starting the firsttruck-loading and min

119895isin119863119894

1199011

119894119895+ 1199011

119895 is the permitted least

retained time to transport and unload the last truck loadedby shovel 119894

L1

L2

L3

L4

D1

D2

D3

Figure 2 Illustration of the truck round trip

312 Unloading Capacity Inequality Consider

sum

119894isin119878

sum

119896isin119870119894

1199101198961198941198951199011

119895le 119867 minus min

119894isin1198941015840|119895isin1198631198941015840

min119905isin119879

119902119905

119894+ 1199010

119894+ 1199011

119894119895

119895 isin 119863

(18)

Inequality (18) gives the maximum possible number oftrucks each unloading depot can unload in the horizon In theinequality119867minusmin

119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905

119894+1199010

119894+1199011

119894119895 is the practi-

cal time limit for depot 119895 wheremin119894isin1198941015840|119895isin1198631198941015840 min119905isin119879119902119905

119894+1199010

119894+

1199011

119894119895 is the earliest possible starting time of unloading the first

truck in depot 119895

313 Variables Relationship Inequality Consider

119911119896119894le 1199111198961015840119894 119896 le 119896

1015840 119894 isin 119878 (19)

Inequality (19) comes from the material transportationsequence as ensured by formula (3) where truck loading (119896 119894)is doomed to be transported before (1198961015840 119894)when 119896 le 1198961015840 stands

32 Property of the Optimal Solution For the above defineopen-pit truck scheduling problem we observe two proper-ties for the optimal solutions Some new concepts need to beintroduced to describe the properties

Definition 1 An empty truck starting from a depot 119895 119895 isin 119863119894

moves to a shovel 119894 119894 isin 119878 loads material (with shovel 119894)transports it to a depot 1198951015840 119895 isin 119863

119894 and unloads the material

(eg travelling from dump1198631to loading point 119871

1and to119863

3

in Figure 2) The whole movement process is called a truckround trip denoted by trip (119895 119894 1198951015840)

It is noteworthy that the end depot involved in a truckround trip can be the same as the starting depot

Definition 2 For any a truck round trip (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin119863119894 119894 isin 119878 119890

1198951198941198951015840 = 119886119894(1198750

119895119894+1198751

1198941198951015840 +1199010

119894+1199011

119895) is called trip transport

value

Discrete Dynamics in Nature and Society 5

Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value

Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times

(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel

(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot

(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon

It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011

119894119895and1199010

119895119894) through

the path

Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to

sum

119905isin119879

sum

(1198951198941198951015840)isinΨ

11989711990511989511989411989510158401198901198951198941198951015840 (20)

where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has

spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips

The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound

Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750

119895119894+ 1198751

1198941198951015840 + 1199010

119894+ 1199011

119895) sdot

min1198850119894 1198851

1198951015840 where1198850119894 and119885

1

1198951015840 denote themaximum loading

capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively

In Property 2 1198850119894and 1198851

1198951015840 can be obtained through

formulae (17) and (18) respectively

33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem

331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum

119894isin1198781198861198941198850

119894

In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873

119879denote the number of available trucks then

119862119879= 119867 sdot 119873

119879denotes the total available truck times involved

in the problem Let 119903119894denote the optimal truck round trip

involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires

the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast

119894and 119864lowast

119894denote the required valuable truck times and trip

transport value corresponding to trip 119903119894 respectively

It is assumed that 1198941 1198942 119894

119873is the shovel sequence in

the descending order of 119864lowast119894 that is 119864lowast

1198941

ge 119864lowast

1198942

ge sdot sdot sdot ge 119864lowast

119894119873

and1198941 1198942 119894

119873 = 119878

Upper bound 2 can be described and formulated clearlybased on the above notations and assumption

332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value

119891 (119862119879) =

119864lowast

1198941

119862119879

119862119879le 1198850

1198941

119875lowast

1198941

1198861198941

1198850

1198941

+ 119864lowast

1198942

(119862119879minus 1198850

1198941

119875lowast

1198941

)

1198850

1198941

119875lowast

1198941

le 119862119879

le 1198850

1198941

119875lowast

1198941

+ 1198850

1198942

119875lowast

1198942

119873minus2

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873minus1

(119862119879minus

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873minus1

119875lowast

119894119873minus1

119873minus1

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873

(119862119879minus

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873

119875lowast

119894119873

(21)

The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues

4 Heuristic Solution Method to the Problem

41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here

Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875

0

119895119894+

1198751

1198941198951015840 + 1199010

119894+ 1199011

119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863

119894 119894 isin 119878 is called

expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip

Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps

Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in

6 Discrete Dynamics in Nature and Society

the scheduling and set 119878idle to include all trucks in the initialstate

Build a truck set 119878running and let 119878running = 0 in the initialstate

Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0

Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4

Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip

Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running

Go to Step A2

Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop

The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process

42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution

Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894

1 1198942 119894

119873in the

descending order of 119864lowast119894(see Section 33) Because the initial

truck position can be far away from the shovel with higher119864lowast

119894 the strategy should reassign more trucks to these shovels

for the objective pursuitSteps of Strategy 1 are listed as follows

Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899

Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894

119899+1 119894119899+2 119894

119873

Table 1 Parameters of the problem instances

Path Problem Trucks Shovels Dumps

1

1 73 6 32 71 6 33 74 6 34 69 6 3

2

5 84 6 46 82 6 47 85 6 48 85 6 4

Step B3 If 119905idle1199010119894 ge 120583119867119875lowast

119894lowast in the beginning reassign

lfloor119905idle119875lowast

119894lowast1205831199010

119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule

the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned

Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step

B2 otherwise terminate

Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps

Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained

Step C2 Let 1199031 1199032 119903

119898 denote the set of truck round trips

executed by truck 119905Set 119896 = 1

Step C3 Attempt to reassign truck round trip 119903119896to other

shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration

Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3

Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate

5 Computational Experiments

To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM

The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances

Discrete Dynamics in Nature and Society 7

Table 2 Experiment result for the small sized problems

Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp

1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1

Table 3 Experiment result and comparison for the practical problem size

Problem Objective value Relative imp UB GAPHeuristic Improvement

1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446

including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances

The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances

The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively

The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative

imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table

From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations

6 Conclusion

In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient

8 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)

References

[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002

[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007

[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010

[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965

[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989

[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002

[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006

[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985

[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988

[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001

[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006

[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971

[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991

[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004

[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006

[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992

[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003

[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006

[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

Roughly speaking the trip transport value can be viewedas the truck transport revenue per unit time for some truckround trip There is an intuitionistic suggestion from thedefinition that trucks should be dispatched to the trip withhigher trip transportation value

Definition 3 The truck operation times excluding the belowthree classes of invaluable time are called valuable times

(1) times when truck waiting at an electronic shovelfor loading due to another truck occupies the sameshovel

(2) timeswhen truckwaiting at a depot for unloading dueto another truck occupies the depot

(3) times for the unfinished round trip if a truck cannotunload its last truck loading in the horizon

It is noteworthy that the trucks are assumed not to travelmore time than the given travelling time (1199011

119894119895and1199010

119895119894) through

the path

Property 1 If the initial position of each truck is one of thedepots the objective value of a feasible solution to the jointproblem is equal to

sum

119905isin119879

sum

(1198951198941198951015840)isinΨ

11989711990511989511989411989510158401198901198951198941198951015840 (20)

where 1198971199051198951198941198951015840 denotes the total valuable times that truck 119905 has

spent on trip (119895 119894 1198951015840) in the solution and Ψ denotes the set ofall truck round trips

The property provides another formulation for the prob-lem objective function and will play an important role inconstructing the upper bound

Property 2 In a feasible solution for the joint problem thetotal valuable time spent on truck round trip (119895 119894 1198951015840) 119895 isin 1198631198951015840isin 119863119894 119894 isin 119878 is not larger than (1198750

119895119894+ 1198751

1198941198951015840 + 1199010

119894+ 1199011

119895) sdot

min1198850119894 1198851

1198951015840 where1198850119894 and119885

1

1198951015840 denote themaximum loading

capacity of electronic shovel 119894 and maximum unloadingcapacity of dump 1198951015840 respectively

In Property 2 1198850119894and 1198851

1198951015840 can be obtained through

formulae (17) and (18) respectively

33 Upper Bounds Since the number of trucks loadings eachelectronic shovel can finish is limited in the horizon there isobviously the below upper bound for the problem

331 Upper Bound 1 The objective value of the optimalsolution cannot be larger than sum

119894isin1198781198861198941198850

119894

In view of both the truck transport capacity and shovelloading capacity anothermore tightened upper bound can beobtained Let119873

119879denote the number of available trucks then

119862119879= 119867 sdot 119873

119879denotes the total available truck times involved

in the problem Let 119903119894denote the optimal truck round trip

involving shovel 119894 119894 isin 119878 that is to say trip 119903119894requires

the shortest valuable truck times in the trips that transportmaterials from shovel 119894 to one of the available dumps and119875lowast

119894and 119864lowast

119894denote the required valuable truck times and trip

transport value corresponding to trip 119903119894 respectively

It is assumed that 1198941 1198942 119894

119873is the shovel sequence in

the descending order of 119864lowast119894 that is 119864lowast

1198941

ge 119864lowast

1198942

ge sdot sdot sdot ge 119864lowast

119894119873

and1198941 1198942 119894

119873 = 119878

Upper bound 2 can be described and formulated clearlybased on the above notations and assumption

332 Upper Bound 2 The objective value of the optimalsolution must not be larger than the following piecewisefunction value

119891 (119862119879) =

119864lowast

1198941

119862119879

119862119879le 1198850

1198941

119875lowast

1198941

1198861198941

1198850

1198941

+ 119864lowast

1198942

(119862119879minus 1198850

1198941

119875lowast

1198941

)

1198850

1198941

119875lowast

1198941

le 119862119879

le 1198850

1198941

119875lowast

1198941

+ 1198850

1198942

119875lowast

1198942

119873minus2

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873minus1

(119862119879minus

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus2

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873minus1

119875lowast

119894119873minus1

119873minus1

sum

119899=1

119886119894119899

1198850

119894119899

+ 119864lowast

119894119873

(119862119879minus

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

)

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

le 119862119879le

119873minus1

sum

119899=1

1198850

119894119899

119875lowast

119894119899

+ 1198850

119894119873

119875lowast

119894119873

(21)

The underlying idea of upper bound is that the truckshould be dispatched to the electronic shovel and round tripwith transport value as high as possible to maximize the totalrevenues

4 Heuristic Solution Method to the Problem

41 A Constructive Heuristic Method to Generate InitialSolution In the practical scheduling process truck waitingtimes are often inevitable and thus the conception of expectedreal-time transport value is introduced here

Definition 4 In the practical truck scheduling 119890lowast1198951198941198951015840 = 119886119894(119875

0

119895119894+

1198751

1198941198951015840 + 1199010

119894+ 1199011

119895+ 120575) (119895 119894 1198951015840) 119895 isin 119863 1198951015840 isin 119863

119894 119894 isin 119878 is called

expected real-time transport value (ERTV) for a possibletruck round trip selection where 120575 is the expected truckwaiting time (invaluable truck times) during the whole trip

Based on the above definition a constructive heuristicmethod can be introduced to generate the initial solution Itincludes the following steps

Step A1 Build a truck set 119878idle including all trucks that arenot assigned to a round trip or just finished a round trip in

6 Discrete Dynamics in Nature and Society

the scheduling and set 119878idle to include all trucks in the initialstate

Build a truck set 119878running and let 119878running = 0 in the initialstate

Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0

Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4

Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip

Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running

Go to Step A2

Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop

The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process

42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution

Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894

1 1198942 119894

119873in the

descending order of 119864lowast119894(see Section 33) Because the initial

truck position can be far away from the shovel with higher119864lowast

119894 the strategy should reassign more trucks to these shovels

for the objective pursuitSteps of Strategy 1 are listed as follows

Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899

Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894

119899+1 119894119899+2 119894

119873

Table 1 Parameters of the problem instances

Path Problem Trucks Shovels Dumps

1

1 73 6 32 71 6 33 74 6 34 69 6 3

2

5 84 6 46 82 6 47 85 6 48 85 6 4

Step B3 If 119905idle1199010119894 ge 120583119867119875lowast

119894lowast in the beginning reassign

lfloor119905idle119875lowast

119894lowast1205831199010

119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule

the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned

Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step

B2 otherwise terminate

Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps

Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained

Step C2 Let 1199031 1199032 119903

119898 denote the set of truck round trips

executed by truck 119905Set 119896 = 1

Step C3 Attempt to reassign truck round trip 119903119896to other

shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration

Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3

Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate

5 Computational Experiments

To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM

The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances

Discrete Dynamics in Nature and Society 7

Table 2 Experiment result for the small sized problems

Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp

1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1

Table 3 Experiment result and comparison for the practical problem size

Problem Objective value Relative imp UB GAPHeuristic Improvement

1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446

including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances

The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances

The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively

The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative

imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table

From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations

6 Conclusion

In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient

8 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)

References

[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002

[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007

[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010

[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965

[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989

[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002

[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006

[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985

[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988

[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001

[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006

[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971

[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991

[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004

[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006

[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992

[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003

[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006

[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

the scheduling and set 119878idle to include all trucks in the initialstate

Build a truck set 119878running and let 119878running = 0 in the initialstate

Build a truck scheduling information table119879truck a shoveldispatch information table 119879shovel and a dump schedulinginformation table 119879dump Initialize the three informationtables with 119879truck = 0 119879shovel = 0 and 119879dump = 0

Step A2 If 119878idle = 0 for each truck in 119878idle construct thecorresponding round trips compute their expected real-timetransport values in the current state (referring to119879truck119879shoveland119879dump) and record the trip with the highest ERTV and itsERTV otherwise go to Step A4

Step A3 Select the truck with the highest ERTV assign itto the recorded optimal trip once and decide the startingtime of the loading operation as early as possible according toboth the expected truck arriving time and the involved shovelavailable time Compute the complete times of both the truckand the shovel in the assigned round trip

Delete the truck from 119878idle If the obtained trip completetime is in the given horizon append the involved assignationand temporal information of truck shovel and dump to119879truck 119879shovel and 119879dump respectively and append the truckto 119878running

Go to Step A2

Step A4 If 119878running = 0 select a truck of the earliest tripcomplete time from 119878running delete the truck from 119878runningand append it to 119878idle otherwise output the current 119879truck119879shovel and 119879dump and stop

The above heuristic virtually simulates the equipmentrunning process especially the truck operation and thus canbe called a simulation-based solution construction approachIt is also appropriate to the online scheduling scene throughdisplacing the above simulation process with the real produc-tion process

42 Two Improvement Strategies Throughobservation on theproperties and upper bounds as well as analyzing the aboveheuristic method two improvement strategies are proposedto improve the initial solution

Strategy 1 (the shovel capacity rebalancing strategy) Strategy1 will refer to the defined shovel sequence 119894

1 1198942 119894

119873in the

descending order of 119864lowast119894(see Section 33) Because the initial

truck position can be far away from the shovel with higher119864lowast

119894 the strategy should reassign more trucks to these shovels

for the objective pursuitSteps of Strategy 1 are listed as follows

Step B1 Let 119899 = 1 and current shovel 119894lowast = 119894119899

Step B2 Check the shovel dispatch information table119879shovel inthe current solution and compute the total idle times of shovel119894lowast after finishing its first truck loading Let 119905idle denote thecomputed idle times and let 119878remained denote the set of trucksassigned to shovel 119894

119899+1 119894119899+2 119894

119873

Table 1 Parameters of the problem instances

Path Problem Trucks Shovels Dumps

1

1 73 6 32 71 6 33 74 6 34 69 6 3

2

5 84 6 46 82 6 47 85 6 48 85 6 4

Step B3 If 119905idle1199010119894 ge 120583119867119875lowast

119894lowast in the beginning reassign

lfloor119905idle119875lowast

119894lowast1205831199010

119894119867rfloor trucks to shovel 119894lowast from 119878remained reschedule

the scheduled total truck routine through repeating Steps(A2ndashA4) in the initial solution approach and delete thesereassigned trucks from truck set 119878remained Note that truckswith lower ERTV should be given priority to be reassigned

Step B4 If 119878remained = 0 set 119899 = 119899 + 1 119894lowast= 119894119899 and go to Step

B2 otherwise terminate

Strategy 2 (neighborhood swapping strategy) Strategy 2 isa simple neighborhood swapping strategy It includes thefollowing steps

Step C1 Set truck set 119878remained = 119879 Let 119905 denote any truck in119878remained and delete 119905 from 119878remained

Step C2 Let 1199031 1199032 119903

119898 denote the set of truck round trips

executed by truck 119905Set 119896 = 1

Step C3 Attempt to reassign truck round trip 119903119896to other

shovels and revise the involved truck schedule If there isa reassignment that can bring an improvement accept thealteration

Step C4 If 119896 lt 119898 set 119896 = 119896 + 1 and go to Step C3

Step C5 If 119878remained = 0 take a new truck 119905 from 119878remained andreturn to Step C2 otherwise terminate

5 Computational Experiments

To test the performance of the formulatedmathematic modeland the proposed heuristic approach with improvementstrategies we complemented all the involved programs underthe development environment of VC++ 2010 and solved theinteger programmingmodel in Section 2 with CPLEX a kindof optimization software The experiments are all performedon a computer withWin 7 operating system and 28GHz Intel2 Core CPU and 4GBRAM

The data in the numerical experiments comes froma practical open-pit mine The data includes 8 probleminstances but involves only 2 different mine path configu-rations due to the relative constancy of the open-pit roadnetwork Table 1 shows the details of the problem instances

Discrete Dynamics in Nature and Society 7

Table 2 Experiment result for the small sized problems

Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp

1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1

Table 3 Experiment result and comparison for the practical problem size

Problem Objective value Relative imp UB GAPHeuristic Improvement

1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446

including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances

The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances

The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively

The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative

imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table

From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations

6 Conclusion

In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient

8 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)

References

[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002

[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007

[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010

[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965

[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989

[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002

[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006

[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985

[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988

[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001

[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006

[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971

[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991

[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004

[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006

[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992

[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003

[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006

[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

Table 2 Experiment result for the small sized problems

Path Size Objective value CPU time (seconds)Model Heu Model Model InEq Heu Imp

1 3 54 54 437 446 lt12 3 58 58 961 784 lt11 6 111 111 3714 2483 lt12 6 118 118 4190 3409 lt11 9 169 169 17348 11387 lt12 9 177 177 14334 8617 lt11 12 226 220 47955 45484 lt12 12 234 234 51363 49456 lt1

Table 3 Experiment result and comparison for the practical problem size

Problem Objective value Relative imp UB GAPHeuristic Improvement

1 543 551 158 585 5702 536 543 126 559 2963 569 578 161 601 3804 509 534 468 558 4365 637 659 334 697 5466 588 619 505 650 4797 652 667 234 705 5408 661 691 434 714 319Average mdash mdash 302 mdash 446

including path configuration (Path) problem instance ID(Problem) number of trucks (Trucks) number of shovels(Shovels) and number of dumps (Dumps) The schedulinghorizon is a shift 8 hours for all instances

The transport revenue per truck of material is product ofthe expected shortest distance and the priority factor of theloading point The priority factor ranges in [09 11] in theproduction scheduling In the truck-shovel system the truckcapacity is usually scarce for the loading capacity of shovelsAll these features are also reflected in the collected probleminstances

The small sized problems are first solved using the for-mulatedmathematicmodels (with CPLEX) and the proposedheuristic approachThe small sized problem instances are cutout from instances of path configurations 1 and 2 in Table 1and the scheduling horizon is decreased to 2 hours Each ofthe small sized problem instances involves 2 shovels 2 dumpsand different number of available trucks Table 2 shows theexperiment results with the different problem sizes (numberof available trucks) In the table columns Path and Sizeshow the path configuration and number of available trucksrespectively Models Model Ineq and Heu Imp represent themathematical model model with inequalities (17)ndash(19) andthe heuristic with improvement strategy respectively

The experiment results of the practical problem instancesare listed in Table 3 with problem ID (Problem) objectivevalues of the initial solution (Heuristic) and improved solu-tion (Improvement) which are improved by the developedtwo improvement strategies relative improvement (Relative

imp) best upper bound (UB) which adopts the maximumof two upper bounds in Section 33 and the relative deviationof the obtained optimized solution and the upper boundThecomputation times for all instances are not longer than 5CPUseconds and thus neglected in the table

From Table 2 we can see that the mathematical model isvalid but not up to the practical problem size From Table 3we can observe that the average relative deviation is 446and the two improvement strategies perform 302 increaseon average The experiments show that the proposed upperbound is good and the developed heuristic solution approachwith improvement strategies is effective and efficient Andthe solution performance is relatively stable against differentproblem configurations

6 Conclusion

In this paper a special open-pit truck scheduling problemwas studied A mixed integer programming model wasfirst formulated considering transport revenue varied withdifferent loading points And a few valid inequalities twoproperties and two upper bounds of the problem werededuced Based on them a heuristic solution approach withtwo improvement strategies was proposed to resolve theproblem At last the numerical experiments were carried outwhich demonstrated that the proposed solution approachwaseffective and efficient

8 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)

References

[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002

[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007

[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010

[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965

[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989

[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002

[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006

[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985

[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988

[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001

[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006

[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971

[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991

[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004

[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006

[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992

[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003

[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006

[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is partly supported by National Natural ScienceFoundation of China (71201105) and China PostdoctoralScience Foundation (2013M530947)

References

[1] S Alarie and M Gamache ldquoOverview of solution strategiesused in truck dispatching systems for open pit minesrdquo Interna-tional Journal of Surface Mining Reclamation and Environmentvol 16 no 1 pp 59ndash76 2002

[2] H Askari-Nasab S Frimpong and J Szymanski ldquoModellingopen pit dynamics using discrete simulationrdquo InternationalJournal of Mining Reclamation and Environment vol 21 no 1pp 35ndash49 2007

[3] A M Newman E Rubio R Caro A Weintraub and K EurekldquoA review of operations research in mine planningrdquo Interfacesvol 40 no 3 pp 222ndash245 2010

[4] H Lerchs and I Grossmann ldquoOptimum design of open-pitminesrdquo CanadianMining andMetallurgical Bulletin vol 58 pp17ndash24 1965

[5] E Wright ldquoDynamic programming in open pit miningsequence planning a case studyrdquo in Proceedings of the 21stInternational Application of Computers and Operations ResearchSymposium (APCOM rsquo89) A Weiss Ed pp 415ndash422 SMELittleton Colo USA 1989

[6] S Frimpong E Asa and J Szymanski ldquoIntelligent modelingadvances in open pit mine design and optimization researchrdquoInternational Journal of Surface Mining Reclamation and Envi-ronment vol 16 no 2 pp 134ndash143 2002

[7] S E Jalali M Ataee-Pour and K Shahriar ldquoPit limits optimiza-tion using a stochastic processrdquo CIM Magazine vol 1 no 6 p90 2006

[8] E Busnach A Mehrez and Z Sinuany-Stern ldquoA productionproblem in phosphate miningrdquo Journal of the OperationalResearch Society vol 36 no 4 pp 285ndash288 1985

[9] D Klingman and N Phillips ldquoInteger programming for opti-mal phosphate-mining strategiesrdquo Journal of the OperationalResearch Society vol 39 no 9 pp 805ndash810 1988

[10] W Cai ldquoDesign of open-pit phases with consideration of sched-ule constraintsrdquo in 29th International Symposium onApplicationof computers and Operations research in the mineral industry(APCOM rsquo01) H Xie Y Wang and Y Jiang Eds pp 217ndash221China University of Mining and Technology Beijing China2001

[11] K Kawahata A new algorithm to solve large scale mine pro-duction scheduling problems by using the Lagrangian relaxationmethod [Doctoral thesis] Colorado School of Mines GoldenColo USA 2006

[12] C B Manla and R V Ramani Simulation of an Open-Pit TruckHaulage System New York Press New York NY USA 1971

[13] G Kappas and T M Yegulalp ldquoAn application of closed queue-ing networks theory in truck-shovel systemsrdquo InternationalJournal of Surface Mining amp Reclamation vol 5 no 1 pp 45ndash53 1991

[14] KOraee andBAsi ldquoFuzzymodel for truck allocation in surfaceminesrdquo in Proceedings of the 13th International Symposium onMine Planning Equipment Selection (MPES rsquo04) M HardygoraG Paszkowska and M Sikora Eds pp 585ndash591 WroclawPoland 2004

[15] J Najor and P Hagan ldquoCapacity constrained productionschedulingrdquo in Proceedings of the 15th International Symposiumon Mine Planning and Equipment Selection (MPES rsquo06) MCardu R Ciccu E Lovera and E Michelotti Eds pp 1173ndash1178 Fiordo Srl Torino Italy 2006

[16] J White and J Olson ldquoOn improving truckshovel productivityin open pit minesrdquo in Proceedings of the 23rd International Sym-posium on Application of Computers and Operations Researchin the Minerals Industries (APCOM rsquo92) pp 739ndash746 SMELittleton Colo USA 1992

[17] P Fang and L Li ldquoThe model and commentary of truckplanning for an opencast iron minerdquo Jounal of EngineeringMathematics vol 20 no 7 pp 91ndash100 2003

[18] L Yang BWang and K Li ldquoResearch on theory andmethod oftruck dispatching in surface minerdquo Opencast Mining Technol-ogy no 4 pp 12ndash14 2006

[19] L Tang J Zhao and J Liu ldquoModeling and solution of the jointquay crane and truck scheduling problemrdquo European Journal ofOperational Research vol 236 no 3 pp 978ndash990 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of