international express courier routing and scheduling under uncertain demands

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International express courier routingand scheduling under uncertaindemandsJ.-R. Lin a , S. Yan b & C. W. Lai ba Department of Transportation Science , National Taiwan OceanUniversity , Keelung , Taiwanb Department of Civil Engineering , National Central University ,Jhongli , TaiwanPublished online: 22 Aug 2012.

To cite this article: J.-R. Lin , S. Yan & C. W. Lai (2013) International express courier routingand scheduling under uncertain demands, Engineering Optimization, 45:7, 881-897, DOI:10.1080/0305215X.2012.709511

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Engineering Optimization, 2013Vol. 45, No. 7, 881–897, http://dx.doi.org/10.1080/0305215X.2012.709511

International express courier routing and scheduling underuncertain demands

J.-R. Lin1, S. Yan2* and C.W. Lai2

aDepartment of Transportation Science, National Taiwan Ocean University, Keelung, Taiwan;bDepartment of Civil Engineering, National Central University, Jhongli, Taiwan

(Received 26 November 2011; final version received 17 June 2012)

The purpose of this study is to develop a planning model and a real-time adjustment model to help aninternational express company facing uncertain demands to plan courier routes and schedules and adjust theplanned routes in actual operations. A solution procedure is then developed to efficiently solve the real-timeadjustment model. A simulation-based evaluation method is also developed to compare the performanceof the proposed models. The test results, related to an international express company’s operations, showthe good performance of the proposed models. Sensitivity analysis is also performed to gain better insightsinto knowing how several important parameters affect the solutions.

Keywords: courier scheduling; uncertain demand; time–space network; heuristic

1. Introduction

The purpose of this research is to develop a courier routing and scheduling model to help aninternational express company that faces uncertain demands to plan courier routes and sched-ules. Using the planned routes and schedules as a basis, a real-time adjustment model is thendeveloped to adjust the actual daily courier routes and schedules dynamically to respond to thecustomer requests that suddenly pop up. The distinctive features of this problem are the inherentdynamic and stochastic nature of customer demands. This problem is motivated by the opera-tions of an international express company that serves an urban area. The international expresscompany provides pick-up and delivery requests for the express transportation of letters andsmall parcels from one country to another country in timely fashion. An international shipmentorder consists of a pick-up request in one country and a delivery request in another country.In general, customer service requests can be classified according to three types: delivery ser-vice, pick-up service with a day’s advance notice, and same-day pop-up pick-up service. Thedelivery services are those shipments collected from other countries on the previous day andrequested to be delivered to a customer located in the city by the service day. Most of themrequest delivery by noon. The pick-up services with a day’s notice are those shipments for whichnotice is received on the previous day, where a pick-up service is requested on the service day,

*Corresponding author. Email: [email protected]

© 2013 Taylor & Francis

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882 J.-R. Lin et al.

for delivery to another country by the next day. The same-day pop-up pick-up services are thoseshipment notices received on the service day, whereby a pick-up service is requested on the serviceday for delivery to another country by the next day. Most of them occur during the afternoon.The delivery services and pick-up services with a day’s notice in advance can be consideredto be certain demands since they are known a priori. However, the same-day pick-up servicerequests, accounting for a large portion of total customer requests, may not be completely knownahead of time. The same-day pick-up orders come up continuously over time during a serviceday and usually vary on a daily basis. However, in actual planning practice, the controllers firstallocate a number of zones that are defined by natural boundaries such as rivers, highways, rail-ways and hills. Then the controllers use the average demands to calculate how many customerstops should be included in a courier route to ensure that a predefined productivity target isachieved. In general, the productivity target is defined as the number of stops that a courier canservice in an hour (stops per hour). Once the number of customer stops that should be includedin a route has been determined, the controllers plan courier routes and schedules by followinga trial-and-error process based on their experiences. In accordance with the average demandsand the manual process, the planned courier routes and schedules may not be efficient, or maynot be at all feasible to satisfy all pop-up customer demands. Courier routing and schedulingare essential to courier service, since the setting up of good courier routes and schedules cannot only enhance operating efficiency, but also improve the customer service level. Therefore,there is a need to develop a courier route planning model that takes into consideration uncertaincustomer demands.

Moreover, in courier routing and scheduling problems, the disturbances created by those suddenpop-up requests that occur in actual operations are difficult to estimate at the planning stageusing projected (average) demands. Supposing that a courier trip is to be planned based on itsprojected customer demands, if the courier trip is detoured owing to servicing a pop-up request, thesubsequent downstream trips of the courier may not be able to be executed on time. Internationalexpress companies thus need to apply a suitable real-time adjustment rule to adjust the downstreamservice trips. Such adjustments could propagate disturbances or adversely affect either this or othercouriers. Unfortunately, it is very difficult to accurately pre-estimate the disturbances propagatedor adverse impacts because they are related to the real-time adjustment, including the adjustmentrules used by the staff, those pop-up requests, the real-time adjustment results and other constraints,all of which are difficult to predict ahead of time. In actual practice, the controllers may adjustthe planned routes and schedules periodically (e.g. every half hour or every hour), immediatelywhen a new request occurs, or in a mixed way involving both. In general, the controllers assignthe new request to the most appropriate courier based on the courier locations. The request isinserted in a least-cost fashion into the planned route of the closest courier. If there is no courierthat can respond to the request, a back-up courier will be assigned to service the request. However,such an adjustment rule may not be feasible or effective when it comes to responding to all thepop-up requests. The planned courier route schedule may not be able to be adjusted to satisfy all ofthose demands that suddenly pop up, because a large portion of customer demands are same-dayrequests. To satisfy those same-day request demands, the controllers need a better adjustmentmethod to dynamically adjust the planned courier routes and schedules several times during theactual daily operations. On the other hand, an adjusted courier route that is as similar to theplanned route as possible is preferable from the perspective of actual operations. The controllersalso need to make the route adjustment as small as possible while satisfying all customer demands.For these reasons, a real-time adjustment model is developed to systematically and dynamicallyadjust the planned courier routes to respond to the disturbances of pop-up requests that occur atthe real-time operation stage.

A fixed size of fleet is usually available to service the requests. Some pop-up requestsmay remain unserviced, with a fixed-size fleet. These unserviced requests may usually be

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Engineering Optimization 883

postponed until the next day. However, in the actual operations of the company, there aresome back-up contract couriers available. If some requests cannot be served by regular couri-ers, the controller may dispatch the back-up contract couriers to service the requests, insteadof leaving the requests unserviced. A higher operating cost may thus be incurred if a back-up contract courier is dispatched. Mitrovic-Minic and Laporte (2004) solved an urban courierservice problem where the number of couriers was used as a variable to avoid unservicedrequests.

Researchers have studied the stochastic vehicle routing problem (SVRP) with uncertaindemands in various ways, and there is an extensive literature on this topic. For a detailed reviewarticle on the SVRP with uncertain demands see Gendreau et al. (1996). With regard to the gen-eral vehicle routing problem, another distinctive characteristic of the problem is the absence ofcapacity constraints owing to the small sizes of the letters and parcels. Therefore, only somekey works of particular relevance to this research are briefly reviewed, rather than the exten-sive works related to general vehicle routing problems. Several past studies have focused onthe improvement of courier routing and scheduling by addressing the issue of these uncertaincustomer demands, e.g. Angelelli et al. (2004), Sungur et al. (2010) and Ghiani et al. (2010).The dynamic occurrence of new requests has been addressed in several past studies, most ofwhich used heuristics to solve the problem (e.g. Horn 2002, Angelelli et al. 2004, Bent andHentenryck 2004, Diana and Dessouky 2004, Gendreau et al. 2006, Ghiani et al. 2009). Toreduce the search space of the problem, heuristic variable fixing has been applied to a capac-itated network design problem (Holmberg and Yuan 2000) and a time–space network-basedvehicle and crew scheduling problem (Steinzen et al. 2010). This encouraged the present authorsto develop a solution procedure for the problem, based on the concept of heuristic variablefixing.

The courier routing and scheduling problem and the real-time adjustment of the courier routingand scheduling problem, with their multiple customer service types, involve complicated analysisamong numerous time-window and space constraints which are highly correlated with each other.It is difficult to use the traditional integer programming techniques (e.g. traditional vehicle routingmodels) to formulate and efficiently solve this type of problem. On the other hand, the time–spacenetwork method has been popularly employed to solve conveyance scheduling/routing problems,because it provides a natural and efficient way to represent multiple conveyance routings withmultiple courier trips in the dimensions of time and space. Although the resulting model scaleis generally enlarged owing to extension in the dimension of time, complicated time-relatedconstraints can normally be easily modelled for realistic problems, particularly in comparisonwith the space network models. Coupled with the development of efficient algorithms, the time–space models (usually formulated as multiple commodity network flow problems or network flowproblems with side constraints) can be effectively and efficiently solved. For examples, see Wangand Lo (2008), Yan and Tang (2008), Yan and Shih (2009), Steinzen et al. (2010) and Yan et al.(2012). Based on these characteristics, the time–space network technique could be a suitableway of solving the courier routing and scheduling problem as well as the real-time adjustmentof the courier routing and scheduling problem, although, to the authors’ knowledge, so far nomodel has been formulated using this technique to solve this type of problem. Therefore, in thisstudy the time–space network flow technique is employed to develop a model designed to helpan international express company operating in an urban area to plan courier routes and schedulesand adjust its routes in actual operations. The development of other models using other methodsfor solving this type of problem and comparing the results with those of the model presented herecould be a direction for future research.

The rest of this article is organized as follows. In Section 2, the models are introduced. InSection 3, the solution algorithms and evaluation methods are developed. In Section 4, numericaltests are performed. Finally, some conclusions are offered in Section 5.

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2. Modelling approach

In this section, the network structure and mathematical formulation for the planning and thereal-time adjustment of courier routing and scheduling under uncertain demands are discussed insequence. A time–space network model is employed to denote the courier routing and scheduling.Using the projected demands, a planning model (PM) is developed to address the issue of not know-ing the same-day request services in advance. A real-time adjustment model (RAM) is then devel-oped to dynamically adjust the optimal planned courier routes and schedules once some requestsof the same-day pick-up service are known during the service day. In what follows, the time–spacenetwork that serves as the basis for the mathematical formulations will be first discussed.

2.1. Network structure

As shown in Figure 1, the time–space network of courier flows denotes the potential movementsof the couriers within certain time periods and space locations. The horizontal axis represents thedepot and the customer locations, while the vertical axis stands for the duration of courier servicetimes. Note that, in practice, the couriers need to return to the depot site by a deadline in orderto enable the outbound parcels to be loaded onto the outbound cargos in time. The end of theduration denotes the latest time that the couriers need to return to the depot. Each node denotes thedepot or a customer location at a specific time. A short time span is taken as the interval of a node.The node interval affects the number of nodes in the analysis period. Obviously, the more nodes(the shorter the intervals) the network contains, the more accurate the decision, but the larger theproblem, the more difficult it is to solve. Each arc represents an activity for the courier movements.The arc flows express the flow of couriers in the network. Two types of arcs are defined below.

(1) Service arc

A service arc [see (1), (2) and (3) in Figure 1] represents a courier’s travel from the depot toa customer location, a customer location to another customer location, or a customer locationto the depot. All possible trips between two locations in the network, within a reasonable blockof time, are considered, as long as time slots at the depot or customer locations are available.Each service arc contains information about the dispatch time, the dispatch location, the arrivaltime, the arrival location and the arc cost. For simplicity, the travel time between two locationsis equal to the actual travel time required between two locations plus the service time required atthe arrival customer location. The arrival time at the arrival customer location is then equal to thetravel time between two locations plus the departure time at the previous location. The arc costis the operating cost of dispatching a courier to service the customer request. For an internationalexpress company, the goal is to provide the service on time without delay. The arc cost can thus besimplified as the time duration of the associated service. In other applications, the arc cost can bea composite operating cost of dispatching a courier to service the customer request. The arc flow,which is a binary variable, indicates whether or not a courier moves from the dispatch location tothe arrival location for the associated block of time. The arc flow’s upper bound is one, indicatingthat a customer can be served at most once using this trip. The arc flow’s lower bound is zero,indicating that no courier serves the customer using this trip.

(2) Holding arc

A holding arc, shown as (4) in Figure 1, represents the holding of couriers at the depot or acustomer location in a time window. The arc cost denotes the cost incurred for holding a courier

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Time windows (the latest time)

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From the depot to a customer locationFrom a customer location to anotherFrom a customer location to the depot

Figure 1. The time–space network of courier flows.

at the depot or a customer location. Since holding a courier at a customer location implies wastingtime waiting for the next service duty, the arc cost is set as being larger than that of a service arc.Holding a courier at the depot indicates that there is no need to dispatch the courier, and thus thearc cost is set with a relatively low value. For holding arcs connecting the depot, the arc flow’supper bound is set at the fleet size, and denotes the largest number of couriers that can be held atthe depot during a specific time window. The arc flow’s lower bound is set at zero, meaning thatno courier is held at the depot. For holding arcs connecting a customer location, the arc flow’s

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upper bound is set at one, meaning that at most one courier can be held at a customer location.The lower bound is set at zero, meaning that no courier may be held at the customer location.

2.2. Mathematical formulation of the planning model (PM)

Before introducing the model’s formulation, the notation and symbols are listed in Table 1.

Table 1. List of variables and parameters.

Name/description

Variablesxij arc (i,j) flow in the time–space networkParametersA the set of all arcs in the time–space networkN the set of all nodes in the time–space networkUt

d the set of all service arcs and holding arcs connecting the customer location d within time window tDN the set of all customer locationsTd the set of time slots within the request service time windows of customer dR the set of all service arcscij arc (i,j) cost, if the arc is a service arc, the arc cost is the travel time between node iand j; if the arc is a holding

arc, the arc cost is the waiting time plus a penalty time incurred for holding the courier for the next serviceduty

uij arc (i,j) flow’s upper boundai supply or projected demand for couriers at node i, if ai > 0: the supply of couriers; ai < 0: the projected

demand for couriers; at the time when the service day begins the supply at the depot node is the fleet size,and the demand at the depot node is the negative fleet size at the time when the service day ends

Based on the notation, the PM can be formulated as follows:Min

Z =∑

ij∈A

cij xij (1)

subject to

∑

j∈N

xij −∑

k∈N

xki = ai, ∀i ∈ N (2)

∑

ij∈Utd

xij − 1, ∀t ∈ Td , ∀d ∈ DN (3)

0 ≤ xij ≤ uij, ∀ij ∈ A (4)

xij ∈ INT , ∀ij ∈ A (5)

The objective function (1) minimizes the sum of the operating times of all courier services.Constraints (2) are the courier flow conservation constraints for each node. Constraints (3) ensurethat each customer location is served once within the time window. Constraints (4) guarantee thatall arc flows are within their bounds. Constraints (5) ensure that all flow variables are integers.

Note that the courier flows obtained from the above model cannot yet be put to practicaluse without identifying each planned courier route in the time–space network. Using the flowdecomposition method (Yan and Young 1996), embedded in a simple graph search, the courierflows in the network are decomposed into several arc chains, each representing a courier route forone day and the input of the real-time adjustment model. The courier-flow time–space network is

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Customer locationsThe depot

lthcourier

(1)the planned service requests

(2)the pop-up service requests

(2)

Figure 2. The courier-flow time–space network of courier flows.

shown in Figure 2 and the procedure of the flow decomposition method is described as follows:

Step 1: Use a graph search to find a courier route and schedule (an arc chain), that was dispatchedfrom the depot, serviced some customers (with locations and times) and returned to thedepot, by tracing the arcs with positive flows.

Step 2: Deduct one unit from those arc flows on the arc chain.Step 3: If there is no arc with a positive flow, then go to Step 4; otherwise, return to Step 1.Step 4: Record all the planned courier routes and schedules (arc chains).

Note that although the courier routes (the arc chains) may not be unique, the objective valuesfor different patterns of courier routes are the same since the courier fleet routes are the same.Moreover, practical operational considerations such as the working load balance between eachcourier route can be easily implemented while constructing the courier routes. Step 1 has incor-porated this concern into the graph search as follows. Given the projected demands, the servicetarget that is the average number of customers that a courier needs to service in a day can bedetermined in advance. While constructing a courier route (an arc chain) in the graph search, at

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every node it can be checked whether the current number of customers that a courier has servicedper day (or per hour) has exceeded the average one. If yes, a holding arc is chosen to proceedto the search, otherwise a service arc is chosen. The search is repeated until the courier route isfinished.

2.3. The real-time adjustment model (RAM)

In actual practice, when a new request occurs, the controllers assign the request to the mostappropriate courier based on the courier locations. The request is inserted in a least-cost fashioninto the planned route of the closest courier. If there is no courier that can respond to the request, aback-up courier will be assigned to service the request. However, such a real-time adjustment maybe neither efficient nor feasible in responding to all the sudden pop-up requests. Because a largepart of the customer demands comprises same-day requests, based on actual practice, the plannedcourier route and schedule may not be adjusted to satisfy all those demands that suddenly popup. To satisfy those same-day request demands, a systematic method is required to dynamicallyadjust the planned courier route and schedule several times in a day. On the other hand, an adjustedcourier route that is as close to the planned route as possible is preferred. Therefore, there is a needto make the route adjustment as small as possible. To address this issue, similar arcs are defined.If the allowable disturbance time is , all service arcs within the allowable disturbance time areconsidered to be similar arcs. For example, as shown in Figure 3, if the allowable disturbance isone time unit and the planned service arc is the service arc from spot 2 to spot 9, then the servicearcs from spot 1 to spot 8 and from spot 3 to spot 10 are similar arcs, as are the service arcs fromspot 1 to spot 9, from time spot 2 to spot 8, from time spot 2 to spot 10 and from spot 3 to spot 9.

To formulate the RAM, some more notation and symbols as listed in Table 2 are introduced,in addition to the notation and symbols already introduced.

Based on the notation and symbols, the RAM can be formulated as follows:Min

Z =∑

n∈B

∑

ij∈APn

cnijy

nij (6)

subject to

∑

j∈NPn

ynij −

∑

k∈NPn

ynki = an

i , ∀i ∈ NPn, ∀n ∈ B (7)

∑ ∑

n∈Bij∈Ut,nd

ynij = 1, ∀t ∈ Td , ∀d ∈ DN (8)

gn ≤ (1 + δ)T , ∀n ∈ PB (9)

gn =∑

i′j′∈En yni′j′∑

ij∈ASn xnij

, ∀n ∈ PB (10)

0 ≤ ynij ≤ un

ij, ∀ij ∈ APn, ∀n ∈ B (11)

ynij ∈ INT , ∀ij ∈ APn, ∀n ∈ B (12)

The objective function (6) minimizes the sum of the operating times of all the courier services.Constraints (7) are the courier flow conservation constraints for each node. Constraints (8) ensurethat each customer location is served once within the time window. Constraints (9) and (10) ensurethat the courier route adjustments are within the limit. Constraints (9) ensure that the courier routeadjustment ratios are within the limit which is the minimal allowable adjustment ratio plus the

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Figure 3. Similar arcs.

Table 2. List of additional variables and parameters.

Name/description

Variablesyn

ij arc (i, j) flow in the nth courier-flow time–space network

gn the route adjustment ratio of the nth courierParametersAPn the set of all arcs in the nth courier-flow time–space networkNPn the set of all nodes in the nth courier-flow time–space networkASn the set of all planned service arcs of the nth courier

Ut,nd the set of all service arcs and holding arcs connecting the customer d within time window t in the nth

courier-flow time–space networkB the set of all time–space network layers (a total of l + 1 layers)PB the set of time–space network layers for planned couriers (a total of l layers)En the set of unsimilar arcs to the planned arcs in the nth courier-flow time–space networkxn

ij the planned route arcs of the nth courier, if arc (i, j) is on the route xnij = 1; otherwise 0

cnij arc (i, j) cost in the nth courier-flow time–space network;

l the number of planned dispatched couriers;un

ij arc (i, j) flow’s upper bound, if n ≤ l, unij = 1; if n = l + 1, un

ij is equal to the total back-up fleet size;

ani the supply or demand for couriers at node i in the nth courier-flow time–space network;

T the lower bound of the maximal allowable adjustment ratio of a planned courier route;δ the allowable tolerance of route adjustment;

allowable tolerance. Constraints (10) determine the route adjustment ratio of each planned courierroute. Constraints (11) guarantee that all arc flows are within their bounds. Constraints (12) ensurethat all flow variables are integers.

Note that the back-up courier flows in the l + 1th layer obtained from the RAM cannot yet beput to practical use without identifying each back-up courier route and schedule in the time–spacenetwork. Using the flow decomposition method mentioned in Section 2.2, the back-up courierflows in the network are decomposed into several arc chains, each representing a back-up courierroute and schedule for one day.

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The Minimax Theorem is applied to determine the lower bound of the maximal allowableadjustment ratio of a planned courier route T , which is an input of the RAM. The model (RM)can be formulated as follows:Min

Z = T (13)

subject to∑

j∈NPn

ynij −

∑

k∈NPn

ynki = an

i , ∀i ∈ NPn, ∀n ∈ B (14)

∑ ∑

n∈Bij∈Ut,nd

ynij = 1, ∀t ∈ Td , ∀d ∈ DN (15)

gn ≤ T , ∀n ∈ PB (16)

gn =∑

i′j′∈En yni′j′∑

ij∈ASn xnij

, ∀n ∈ PB (17)

0 ≤ ynij ≤ un

ij, ∀ij ∈ APn, ∀n ∈ B (18)

ynij ∈ INT , ∀ij ∈ APn, ∀n ∈ B (19)

The objective function (13) minimizes the lower bound of the maximal allowable adjustment ratioof a planned courier route T , which is the maximal route adjustment ratio of all couriers, withoutconsidering the operating times of all courier services. Therefore, T is the lower bound of themaximal allowable ratio of a planned courier route and schedule in the RAM, since the objectiveof the RAM is to minimize the total operating time of all courier services. Constraints (14) are thecourier flow conservation constraints for each node. Constraints (15) ensure that each customerlocation is served once within the time window. Constraints (16) ensure that the lower boundof the maximal allowable courier route adjustments ratio (an input in the RAM) is the maximalone across all planned courier routes. Constraints (17) determine the route adjustment ratio ofeach planned courier route. Constraints (18) guarantee that all arc flows are within their bounds.Constraints (19) ensure that all flow variables are integers.

3. Solution algorithms and evaluation methods

In this section, the solution procedures for the proposed models—RM, PM and RAM—are dis-cussed first. Based on the solution procedures for the proposed models, how to adjust the plannedcourier routes dynamically in a service day is then addressed. Finally, an evaluation method forthe proposed models is developed. The proposed models, RM, PM and RAM, are formulated asa mixed integer multiple network flow problem. For the problem of realistic cases, the RM andPM can be solved within a reasonable time, using the C++ computer language, coupled withCPLEX 11. However, it is almost impossible to optimally solve a realistically large problem ofRAM within a limited time. Therefore, a solution method is developed, coupled with CPLEXMIP, to solve the RAM. In what follows, the solution method for RAM will be first described.

3.1. Solution procedure for RAM

In actual practice, when a new request occurs, the controllers assign the request to the closestcourier in a least-cost fashion. The solution can be found very quickly but the solution may be faraway from the optimal solution, since the number of couriers that are allowed to adjust routes and

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schedules (for simplicity, called the number of adjusted couriers) only equals one. By contrast,the RAM can be solved optimally by CPLEX 11.0 where the number of adjusted couriers equalsthe courier fleet size. However, it is almost impossible to optimally solve a realistically largeproblem of RAM within a limited time for the controllers to respond to the pop-up request. Thelarger the number of adjusted couriers, the better the solution that can be found, although theproblem will take longer to solve. Therefore, to trade off the solution effectiveness and efficiencyan appropriate number (or several appropriate numbers) of adjusted couriers should be determinedwith tests by the carrier in advance to allow a better solution to be found than the solution foundby the actual practice and to ensure that the solution is found within a reasonable time, therebyallowing the controllers to respond to the pop-up request. The number of adjusted couriers Kis first determined and the flow variables associated with other couriers are fixed. Then, how toadjust the planned routes and schedules of the K couriers and back-up couriers to respond to thepop-up requests can be formulated. The reduced problem can easily be solved using CPLEX MIP.If those non-adjusted courier flow variables are fixed, the feasible solution set is reduced and thusthe problem can be solved much more quickly. The solution procedure is described as follows:

Step 1: Initialization: set K (the number of adjusted couriers).Step 2: Identify the new pop-up request customer demands with their locations and time windows.Step 3: Find K couriers who are closest to the new request demands.Step 4: Fix other courier service routes and schedules.Step 5: Solve the reduced problem (with CPLEX MIP) to identify the new K courier and the

back-up courier service routes and schedules, given that other courier service routes andschedules are fixed.

Step 6: Record the adjusted courier service routes and schedules and the actual operating time.

3.2. Dynamic adjustment of planned routes and schedules

In actual operations, the same-day pick-up service requests, which account for a large portionof total customer requests, may not be known a priori. The same-day pick-up orders come upcontinuously over time during a service day. It is almost impossible and may not be efficient toadjust the planned route and schedule to respond to every pop-up service request immediately. Inpractice, the control staff may adjust the planned routes and schedules every half hour or everyhour (this means the control staff adjust the planned routes and schedules at certain times in aservice day). In accordance with actual practice, a dynamic procedure is developed to adjust theplanned routes and schedules in a service day. The procedure is described as follows:

Step 1: Initialization: set P (the number of adjustment times) and divide the total service timeinto P time slots. Let the adjustment time at p be tp.

Step 2: Set p = 1.Step 3: Identify the new pop-up request customer demand within the time tp and the time when

the service day ends. The demand includes the pop-up requests that arise during timetp−1 and tp and the planned demand, with all of their service windows falling between tpand the time when the service day ends. Note that if some predicted uncertain demandsarise during time tp−1 and tp, they then become certain demands. If they are modifiedwith service time windows from the predicted ones, then the predicted demands shouldbe replaced by the updated ones.

Step 4: Use the RAM with the new K adjusted couriers closest to the pop-up requests (or thecurrent practice in real-world operations) to adjust the courier routes and schedules.

Step 5: If p = P, then go to Step 6; otherwise p = p + 1, and return to Step 3.Step 6: Record the adjusted courier service routes and schedules and the actual operations cost.

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3.3. Evaluation method

The performances of the PM and RAM, as well as the current practices in actual operations, areevaluated via a simulation. The planned courier routes and schedules are first identified, usingthe planning models based on the projected pop-up demands and certain demands. The pop-up requested service demand data for W evaluation days are then randomly generated. Giventhe planned courier routes and schedules and pop-up customer request demands, the real-timeadjustment method is used to adjust the planned routes and schedules dynamically and then toidentify the real courier routes and schedules and the actual operating time for each evaluationday. Finally, the PM, the RAM and the current practice in actual operations are compared withthe statistical results. The steps of the evaluation method are listed as follows:

Step 1: Use the planning model to identify the planned courier routes and schedules considered,based on the projected pop-up demands.

Step 2: Randomly generate the pop-up customer demand data of W evaluation days.Step 3: Set evaluation day w = 1.Step 4: Use the RAM (or the current practice in real-world operations) to adjust the courier routes

and schedules dynamically until all the requested customer services have been completedand the evaluation day ends.

Step 5: Record the courier routes and schedules and the actual operating time.Step 6: If w = W , then go to Step 7; otherwise w = w + 1 and return to Step 4.Step 7: Calculate the statistical results for W evaluation days.

4. Numerical tests

Numerical tests are performed, using operating data from a Taiwanese international express deliv-ery company with reasonable simplifications to test how well the models may be applied in thereal world. The C++ computer language, coupled with CPLEX 11, was used to build the modeland to solve the problems. The tests were performed on an Intel Core2 Quad Q9500 2.66 GHzdesktop computer with 4.0 GB RAM in a Microsoft Windows 2000 environment.

4.1. Data analysis

This numerical example focuses on the courier route scheduling of a depot of an internationalexpress delivery company (located in Taipei, Taiwan). There are 22 customer service areas inthe service area of the depot. The courier fleet size is 10. In practice, the couriers work in twonon-overlapping shifts. The couriers start work at 09:00 h and provide service until 12:30 h (i.e.the total service time is three and a half hours in the morning). After one and a half hours’ rest,they return to work and continue until 20:30 h. The total service time of a courier is 10 hours.Considering the operating constraints in practice, 5 minutes are taken as an interval of a node(i.e. 12 nodes per hour and 121 nodes in total for a location) in the network. The travel timebetween two locations in the network (the depot and a customer location, a customer location toanother customer location, or a customer location to the depot) is estimated based on a governmentsurvey study. The arrival time at a customer location less the departure time of the courier at theprevious location is equal to the actual travel time required between two locations plus the servicetime required at the arrival customer location, which is assumed to be 5 minutes. The arc cost ofholding back-up couriers is assumed to be zero, while the service arc costs of back-up couriersare assumed to be one and a half times those of regular couriers.

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A customer service request contains the information on customer service type, customer locationand time window. In general, the customer service time windows range from an hour to six anda half hours. Customer service demand can be classified according to the following three types:delivery service, pick-up service with notice in advance, and the same-day pop-up pick-up service,where they account for 44%, 16% and 40% of total demand, respectively. Delivery services andpick-up services with a day’s notice in advance are considered to be certain demands and thesame-day pick-up services are regarded as uncertain demands. The occurrence probabilities ofcertain demands and uncertain demands vary during certain periods in a day. The largest certaindemand appears from 17:01 to 18:30 h and the other peak appears around 12:01–12:30 h, whilemost of the stochastic demands appear in the afternoon.

4.2. Test results

The proposed evaluation method is used to calculate the actual operating time spent at the operationstage, in order to compare the performance of the PM, the current planning practice for real-worldoperations. The proposed models and solution methods described in Sections 2 and 3 provideseveral key parameters that may affect the solution: the number of adjusted couriers K , the numberof real-time adjustment times in a day P, the allowable tolerance of route adjustment δ, and theevaluation days W . The values of K , P, δ and W are first set up as 3, 20, 0.5 and 80, respectively.Then a sensitivity analysis of these parameters is performed. Note that the number of non-adjustedcouriers is 7, which is equal to the courier fleet size (i.e. 10) minus the number of adjusted couriers(i.e. 3). The planning results are shown in Table 3. Also note that the time measurement of ‘actualoperating time’ or ‘projected operating time’ used in Tables 3–7 (including all actual travel timesbetween locations, service times at all customer locations, and waiting times with penalty times atall locations) is 5 minutes, which is the time interval of a node. The objective value of the PM (witha value of 1452.9) is smaller than that of the current practice, although the number of couriers usedof the PM (with a value of 10) is the same as that of the current practice. The actual performanceof these two planning methods can be further evaluated after their application to actual operations.The evaluation results, using the actual practice as the real-time adjustment method, are also shownin Table 3. While using the actual current practice as the real-time adjustment method, the PMyields a better solution with an average daily actual operating time of 1854.6. The current practiceyields a solution with an average daily actual operating time of 1901.8, which is 2.5% higher thanthat of the PM. The evaluation results, using the RAM as the real-time adjustment method, areshown in Table 4. Similarly, while using the RAM as the real-time adjustment method, the PMalso yields a better solution with an average daily actual operating time of 1756.6. The current

Table 3. Comparison of planning methods, using the actual practices as the real-time adjustment method.

Current PlanningPlanning methods practice model

Planning stageObjective value (projected operating time) 1697.3 1452.9Solution time (sec) N/A 96.3Evaluation stage (current practice)Average daily actual operating time 1901.8 1854.6Solution time (sec) N/A N/AGap (%)Difference between projected and actual operating times 12.0 27.6Difference in actual operating time between two planning methods 2.5 0.0

Note: the average daily actual operating time is calculated over 80 evaluation days.

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Table 4. Comparison of planning methods, using the RAM as the real-time adjustment method.

Current PlanningPlanning methods practice model

Planning stageObjective value (projected operating time) 1697.3 1452.9Solution time (sec) N/A 96.3Evaluation stage (RAM)Average daily actual operating time 1802.7 1756.6Total solution time (sec) 330 335Gap (%)Difference between projected and actual operating times 6.2 20.9Difference in actual operating time between two planning methods 2.6 0.0

Note: The uncertain demand ratio is 40%.RAM = real-time adjustment model.

Table 5. Sensitivity analysis of the number of adjusted couriers.

Allowable adjusted couriers (k) 2 3 4 5

Evaluation stageAverage daily Actual operating time 1794.9 1756.6 1721.0 1672.3Solution time (sec) 289 335 483 672Gap between the lower bound (%) 14.3 11.9 9.6 6.5

Table 6. Sensitivity analysis of the allowable tolerance of route adjustment.

δ 0 0.5 1 ∞Evaluation stageAverage daily Actual operating time 1808.1 1756.6 1719.5 1650.3Solution time (sec) 264 335 553 883

Table 7. Sensitivity analysis of the ratio of stochastic demands to total demands.

Ratios (%) 20 40 60 80

Planning modelAverage daily Actual operating time (evaluation stage) 1690.9 1756.6 1814.8 1899.7Solution time (sec) 228 335 435 583

practice still performs worse, with an average daily actual operating time of 1802.7, which is 2.6%higher than that of the PM. The PM yields a better solution at the actual operation stage regardlessof which real-time adjustment methods are used. As shown in Table 4, the computation time forsolving the PM optimally is 96.3 seconds, which is within a reasonable time for the controllersto plan courier routes and schedules. Besides, the total computation time to solve the RAMs inthe proposed solution procedure is around 335 seconds, which is within a reasonable time for thecontrollers to adjust courier routes and schedules dynamically.

The planned routes of the two planning methods are used as a basis and the two real-timeadjustment methods are then used to adjust the planned route at the evaluation stage, to compare theperformance of the RAM and the actual real-time adjustment practice for real-world operations.The results show that the RAM outperforms the current practice in terms of the gap in actualoperating time between the two adjustment methods, regardless of which planned route is used

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as a basis. When the planned routes of the current practice are used as a basis, the average dailyactual operating time obtained from the actual practice (with a value of 1901.8) is roughly 5.5%higher than that obtained from the RAM (with a value of 1802.7). When the planned routes of theplanning method are used as a basis, the average daily actual operating time obtained from theactual practice (with a value of 1854.6) is roughly 5.6% higher than that obtained from the RAM(with a value of 1756.6).

If the uncertain demands are known in advance, the optimal courier routes and schedules andthe corresponding objective value (this is the lower bound of the actual operating time) can bedetermined. The average daily actual operating time obtained from the RAM (with a value of1756.6) is 11.9% higher than the lower bound (with a value of 1569.7).

4.3. Sensitivity analyses

The proposed models and solution methods described in Sections 2 and 3 provide several keyparameters that may affect the solution: the number of adjusted couriers K , the number of real-time adjustment times in a day P, and the allowable tolerance of route adjustment. A sensitivityanalysis of these parameters is first performed. To further understand the influence of uncertaincustomer demands on the solution, a sensitivity analysis of the ratio of the uncertain demand tothe total demand is then performed.

Four values of the number of allowable adjusted courier routes K are tested to investigate theinfluence of the number of adjusted couriers K on the solution quality and solution performance.As the number of adjusted couriers increases, the more flexibly they are able to respond to thestochastic demand. The actual operating time and the gap to the lower bound decrease as thenumber of adjusted couriers increases, as shown in Table 5. However, more time is required tosolve the problem as the number of adjusted couriers increases. Note that if the value of the numberof adjusted couriers K is set up as the fleet size, the original problem is solved and the problemsize is not reduced. If the value of the number of adjusted couriers K is set up as 1, the real-timeadjustment model is solved using the current practice. While determining the number of adjustedcouriers, there is a trade-off between the solution quality and the required computation times. Inthe operation stage, it is for the controller staff (the decision makers) to determine the numberof adjusted couriers. However, in order to respond to the pop-up requests quickly, the requiredcomputation times should not be too long.

How often the routes are adjusted may affect how well the real-time adjustment method per-forms. Thus, the number of adjustment times in a day is changed from 20 (which means that theroutes are dynamically adjusted every half hour) to 10 (which means that the routes are dynami-cally adjusted every hour). The solution time required decreases from 335 seconds to 151 seconds.The actual operating time (with a value of 1723.3) decreases slightly. This suggests that adjustingthe route more often does not necessarily lead to a lower operating cost. A further study is requiredto determine the optimal adjustment frequency and the trade-off between the solution quality andthe solution time required.

Four values of the allowable tolerance of the route adjustment are tested to investigate theinfluence of the allowable tolerance of the route adjustment on the performance of the real-timeadjustment method. As shown in Table 6, the more flexibly the routes can be adjusted, the betterthe actual operating time that can be obtained. However, the actual routes will be more dissimilarto the planned routes.

The ratio of the uncertain demands to the total demands is set up with four different values (20%,40%, 60% and 80%) to determine the influence of stochastic demand on the solutions. As shownin Table 7, the actual operating time increases as the value of the ratio increases, regardless ofwhich planned routes are used as a basis. This suggests that the higher the portion of the uncertain

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demands, the higher the actual operating time. If a way can be found to reduce the portion ofuncertain demand, the actual operating time may be reduced.

5. Conclusions

In this study, a courier route and schedule planning model is first developed to find a plannedcourier route and schedule for daily operations. A real-time adjustment model is then developedto adjust the actual daily courier routes and schedules dynamically to respond to the customerrequests that suddenly pop up, using the planned routes and schedules as a basis. A solutionprocedure, based on heuristic variable fixing, is then developed to efficiently solve the real-timeadjustment problems, and a simulation-based evaluation method is also developed to compare theperformance based on actual practice, the planning models and the real-time adjustment models.

Numerical tests, related to an international express company’s operation, are performed toevaluate the proposed models and the actual practice. The test results show that the PM performsbetter than the actual practice in terms of the actual operating time obtained at the operation stage.The test results also show that the RAM performs better than the actual practice in terms of theactual operating time obtained at the operation stage. Sensitivity analysis is conducted to test howseveral important parameters affect the performance of the proposed models.

Future research would be useful in at least the following directions. First, the stochastic traveltimes of courier trips are not addressed here. In actual operations, stochastic travel times oftenoccur in courier trips, thereby hampering the performance of planned courier routes and schedules.It would be useful to develop a courier route and schedule planning model with stochastic traveltimes. Second, the service requests for same-day pick-up and delivery are not considered.Althoughsuch a service request seldom occurs in international express courier operations, it often occursin local courier operations. It would therefore be useful in a local courier routing application toinclude the service requests for same-day pick-up and delivery.

Acknowledgements

This research was supported by a grant (NSC-96-2628-E-008-057-MY2) from the National Science Council of Taiwan.The authors would like to thank the unnamed international express company for providing the test data and their valuableopinions. The authors would also like to thank two anonymous referees for their helpful comments and suggestions, whichhave substantially improved the presentation of this paper.

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international express courier routing and scheduling under uncertain demands

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