Advanced OR and AI Methods in Transportation

VEHICLE REPLACEMENT PLANNING IN FREIGHT TRANSPORTATION COMPANIES

Adam REDMER1

Abstract. The paper presents the method that allows defining the optimal replacement policy for vehicles utilized in a freight transportation company. The mathematical model of the problem as well as the solution procedure are described. The problem has been formulated in terms of the singlecriterion, linear, deterministic, static and discrete mathematical programming. The exact solution procedure has been proposed. The problem has been solved as a real life case study.

1. Introduction

Replacement theory, that is a part of a maintenance theory, has grown up out of an industrial environment [5], [6], where an optimal length of the operational life of production machinery or its components has been determined. In general, there are two fundamental replacement strategies (cases): replacement on failure and preventive replacement [3]. The first category of replacement strategies can be applied to the equipment that deteriorates with time or not, assuming that in both cases we can finally expect a failure. The second strategy can be applied when the following conditions are met [5]:

• the total cost associated with a failure replacement is grater than the total cost associated with a preventive maintenance,

• the hazard rate r(t) of the equipment is increasing with the time of utilization. In case of both strategies the general problem is how to determine the optimal time of utilisation to replacement or optimal cumulative usage of equipment to replacement [4]. Determining an optimal utilisation period two common replacement policies can by applied [4], [7]:

• the age replacement policy assuming that equipment is immediately replaced at a specified age (specified number of cycles) or at failure, whichever occurs first,

• the block replacement policy (the constant interval policy) assuming that each unit of equipment belonging to a given group is replaced by a new one periodically after specified and the same time period or a number of cycles independently of its age or immediately at failure.

1 Poznan University of Technology, 3 Piotrowo street, 60-965 Poznan, Poland, adam.redmer@put.poznan.pl

142 A. Redmer

There are many other replacement policies too [1], [7], for example: replacement after N failures, maintenances, repairs or shocks; an idle time policy; cost or repair limit policies and so on. One of the most popular replacement policies is the policy balancing exploitation (operation and maintenance) costs and ownership costs (policy so called minimal average costs during lifecycle) [1], [5]. This policy is known since the time of World War II [8], and has many modifications. In general, it assumes that exploitation costs should increase with time faster than ownership costs decreases giving the classic “bathtub” curve where the minimum indicates the optimum economic operational life of equipment (see figure 1).

Time

Costs

[mon

etary

unit /

time u

nit]

Optimum (tp) Total costs Exploitation costs Ownership costs

C(tp )k(t)

P - S(t p )

tp

Figure 1: Total average unit (per year or other time period) costs of exploitation and ownership of equipment replaced at age (time) t.

An example of such a policy is the method proposed by S. Eilon, J. R. King and D. E. Hutchinson in the year 1966 based on the following mathematical model [3]:

( ) ( ) ( ) ( )[ ]

⋅−−+⋅= ∫ SPtAtSdttkPt

tC pp

t

pp

p

0

1

(1)

where: C(tp) – total average annual (or per other time period) costs of exploitation and

ownership of equipment assuming its replacement at age (time) tp, P – total acquisition costs of „new” (including second-hand) equipment, k(t) – unit costs of exploitation of equipment at age (time) t, S(tp) – resale (market) value of equipment at age (time) tp, A(tp) – accumulated depreciation costs borne to the time tp, SP – the rate of taxation. The method has been applied to find an optimum replacement policy for a fleet of fork lifts giving results similar to those presented in figure 1. Aforementioned replacement policies and methods represent only small part of all attempts that have been done to solve the equipment replacement problem in general [7], [9], and the vehicle replacement problem particularly [1], [3]. Even though, the vehicle replacement policy has a crucial impact on different effectiveness parameters of transportation companies, there are a lot of difficulties with applying existing methods in the area of road, freight transportation. Such difficulties arise from the following features of the existing replacement methods that are:

Vehicle replacement planning in freight transportation companies 143

• dedicated to simple (e.g. electronic parts, bearings) or more complex (gear boxes, starters, alternators) elements, but not to complete vehicles,

• dedicated to production lines and other manufacturing equipment operating in more stable environment than vehicle’s operational conditions affected by the way the vehicle is used, the load carried, the type of journeys, the climate, and the random stresses inducted from road surfaces,

• focused on a given group (type) of vehicles instead of a single vehicle, • assuming the constant utilisation rate of the equipment during its operational life.

All the mentioned reasons induce the author to make some efforts, presented in the paper, to adapt one of the existing methods to the conditions of the freight transportation companies.

2. The replacement model Presented below a singlecriterion, linear, deterministic, static and discrete mathematical model of the vehicle replacement problem is based on the method proposed by S. Eilon, J. R. King and D. E. Hutchinson [3]. The carried out computational experiments revealed that the direct application of this method to the case of vehicles utilized by freight transportation companies is not possible. The main problem is the decreasing utilization intensity of such vehicles whereas the analyzed method does not take into account such a possibility. It is due the fact that all analyzed costs are given per time unit (usually one year). As a result the optimal operational life of vehicles bounds for infinity (see figure 2 – costs given per year). To apply the minimal average costs replacement policy to the vehicles which utilisation intensity decreases with time, the following changes in the model have to be done:

• the utilization intensity (annual mileage) of vehicles in consecutive years of their operational lives has to be taken into account,

• the total costs of exploitation and ownership have to be given per one kilometre, • the technical durability of vehicles (e.g. maximal mileage) has to be taken into account, • different forms of financing the fleet investments (buying for cash, renting, leasing

and hiring) have to be considered. As a result, the following mathematical model of the vehicle replacement problem is proposed:

( ) ( ) ( ) ( ) ( ) ( )[{

( )( ) ( )] ( ) } ( ) ( )( ) ]( ) ( ) ( ) ( )[{

( )( ) ( )] } ( )[ ( ) ] } ( ) ( )

+⋅⋅−⋅−+=−

+⋅−+=+⋅⋅⋅+++⋅⋅−+⋅++

+⋅−=−=+⋅⋅⋅++

++⋅⋅−+⋅+=

∑∑

∑

∑ ∑

⋅−+

+=

+

+=

−+⋅

⋅−+

+=+⋅

⋅−+⋅−

=

+

+=+⋅−

ppp

p

pp

p

ppp

ppp

p

pp

ppp

p pp

p

p

ttHHW

Wt

tW

WtpHppp

pppWtttHppZ

ttHHW

WtSttH

tnppWttnppZ

tH

n

tW

WtStn

pH

tIUtIUtHrKattHHWtS

ttHHWtPPrtIUctztKE

tKESPtSAPWKtPrKaWw

rKattSttPPrtIUctztKE

tKESPtSAPWKtPrKaWwtTUDC

11

1

1

1

1 1

11

(2)

where: TUDC – total unit discounted costs of exploitation and ownership of a vehicle [monetary

unit / kilometre], assuming its replacement at age (time) tp,

144 A. Redmer

H – length of the planning horizon [year], tp – age of a vehicle at the moment of replacement (tp = 1, 2, 3, …) – DECISION

VARIABLE, t – consecutive years of exploitation of a vehicle (t = 1, 2, 3, …, tp), n – number of consecutive repetitions of tp-year long operational life of a vehicle

within the planning horizon H [-], r – the discount rate [-], Ww – advance payment [monetary unit], if leasing of a vehicle is considered

(e.g. handling charges), Ka – deposit [monetary unit], if hiring of a vehicle is considered, Wp – age of a vehicle when newly acquired [year] (Wp = 0 for a new vehicle and 1, 2,

3, … for a second-hand vehicle), P(t) – total ownership costs borne in consecutive years t of exploitation of a vehicle

[monetary unit / year], PP(t) – remaining ownership costs that have to be borne in the last year of exploitation t =

tp or ppp ttHHWt ⋅−+= [monetary unit], assuming that operational life of a vehicle can be shorter than period of repayments of liabilities connected with its acquisition / ownership,

PWK – gross book value of a vehicle when newly acquired [monetary unit], SA(t) – annual depreciation rate in consecutive years t of exploitation of a vehicle [-], SP – the annual rate of taxation [-], KES(t) – annual fixed costs of exploitation of a vehicle in consecutive years t of its

operational life [monetary unit / year], KEZ(t) – unit variable costs of exploitation of a vehicle in consecutive years t of its

operational life [monetary unit / kilometre], excluding fuel costs, zp(t) – average fuel consumption of a vehicle in consecutive years t of its exploitation

[litre / kilometre], cp – average fuel price [monetary unit / litre], IU(t) – annual utilization intensity (mileage) of a vehicle in consecutive years t of its

exploitation [kilometre / year], S(t) – resale (market) value of a vehicle in the last year of its exploitation t = tp or

ppp ttHHWt ⋅−+= [monetary unit]. The following constraints have to be taken into account: 1. Technical durability of a vehicle, e.g. maximal mileage to overhaul or cumulated

utilisation intensity SIUmax could not be exceeded: ( ) max

1SIUtIUP

pp

p

tW

Wtp ≤+ ∑+

+=

(3) where: Pp – a total mileage of a vehicle when newly acquired [kilometre]. 2. Decision variable, i.e. age of a vehicle at the moment of replacement can equal to

positive, integer values only: tp ∈ ℜ\{0} (4)

3. Maximal age of a vehicle at the moment of replacement tp max cannot be grater than the length of the planning horizon H:

tp max ≤ H (5)

Vehicle replacement planning in freight transportation companies 145

3. Solution procedure The proposed mathematical model of the vehicle replacement problem is relatively easy to solve, so the exact solution can be obtained. It can be done due the fact that the decision variable can equal to positive, integer values only and that a reasonable, maximum age of a vehicle to replacement roughly equals to 20 years (even if it is 100 years, it is still easy to solve). The most difficult task when solving the proposed model seems to be determination of a time-dependency of some parameters of the model, such as: P(t), PP(t), and so on. It seems to be difficult especially because the model is dedicated to particular vehicles, however it can be applied to a fleet of vehicles too. The best way to determine the mentioned relationships is to analyze the data for a whole or a homogeneous part of a company’s fleet, assuming that the age of vehicles varies between 1 and 20 years, or covers a part of this time period at least. Based on such data it is possible to determine the mentioned relationships and their trends. The determined this way relationships for a certain fleet or a group of vehicles can be applied when calculating age to replacement for particular vehicles. In such a case the present data for particular vehicles are utilised to calibrate generic relationships characteristic for a whole fleet. And finally, before starting computational experiment the length of the planning horizon has to be chosen (skipping other, relatively simple to determine the input data of the model). To do this the author utilized method proposed by J. F. Sousa and R. C. Guimarães in the year 1997 [10]. They suggest that to choose the length H of the planning horizon, each H-optimal policy (i.e. an optimal operational life of vehicle under the given length H of the planning horizon) should be repeated indefinitely over time giving the expected annual, discounted costs of vehicle’s ownership and exploitation.

4. Results To validate the proposed model and the solution procedure (implemented with the help of MS Excel) some computational experiments have been carried out based on a real life data from a freight transportation company operating fleet composed of 66 vehicles (including ordinary and specialized trucks, tractors, trailers and semitrailers), vehicles that differ by: age, load capacity, technical condition, average ton-kilometer cost, fuel consumption etc. Figure 2 (costs given per kilometre) presents general results of carried out computational experiments for 5-year old, leased tractors such makes as: Scania, Iveco, MAN, Renault and Volvo.

0,00

0,20

0,40

0,60

0,80

1,00

1,20

1,40

1,60

1,80

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Age of vehicle to replacement [years]

Cost

per k

ilome

tre [E

uro /

km]

0

20 000

40 000

60 000

80 000

100 000

120 000

140 000

160 000

Costs

per y

ear [E

uro /

year]

Ownership costs [Euro/km] Exploitation costs [Euro/km] Total costs [Euro/km]Ownership costs [Euro/year] Exploitation costs [Euro/year] Total costs [Euro/year]

Figure 2: Relationship between costs and age of vehicle to replacement.

146 A. Redmer

An average optimal age of vehicles (tractors) at the moment of replacement equals to 5 years, but as figure 2 shows replacement at age of 6 or 7 years results in a little bit higher total unit discounted costs – TUDC (up to 0.5% higher). The replacement of vehicles at the age of 8 gives TUDC almost 2% higher, while the replacement at age of 9 gives TUDC 4% higher.

5. Conclusions The presented vehicle replacement method lets the management of the freight transportation company to define an optimal replacement policy for utilized fleet of vehicles resulting in the costs reduction and the improvement of the fleet technical and economical condition. The method solves not only replacement problem but two other fleet management problems, too. They are, the selection of „new” vehicles and the form of financing the fleet investments. The method requires a certain number of historical and forecasted data describing fleet operations and characteristics. This data should be collected, updated and processed with the application of a modern database. This database combined with the proposed model should constitute a Decision Support System (DSS) for a fleet management in a transportation company. Construction of such a DSS is a further step of this research.

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Government Operational Research Unit of the Royal Institute of Public Administration, United Kingdom, 1971.

[2] P. Dejax and T. Crainic. A Review of Empty Flows and Fleet Management Models in Freight Transportation. Transportation Science, 21: 227-247, 1987.

[3] S. Eilon, J. R. King and D. E. Hutchinson. A study in equipment replacement. Operational Research Quarterly, 17(1): 59-71, 1966.

[4] G. J. Glasser. Planned replacement: Some theory and its application. Journal of Quality Technology, 1(2): 110-119, 1969.

[5] A. K. S. Jardine and J. A. Buzacott. Equipment reliability and maintenance. European Journal of Operational Research, 19: 285-296, 1985.

[6] A. Kelly and M. Harris. Management of industrial maintenance. Butterworths, London, 1987. [7] T. Nakagawa. A summary of discrete replacement policies. European Journal of

Operational Research, 17: 382-392, 1984. [8] G. A. D. Preinreich. Economic life of industrial equipment. Econometrica, 8: 12-44, 1940. [9] P. Ritchken and J. Wilson. (m, T) Group Maintenance Policies. Management Science,

36: 632-639, 1990. [10] J. F. Sousa and R. C. Guimarães. Setting the length of the planning horizon in the

vehicle replacement problem. European Journal of Operational Research, 101: 550-559, 1997.