A cooperative strategy for a vehicle routing problem with pickup and delivery time windows

Computers & Industrial Engineering 55 (2008) 766–782

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A cooperative strategy for a vehicle routing problem with pickup and delivery time windows q C.K.Y. Lin * Department of Management Sciences, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China

a r t i c l e

i n f o

Article history: Received 28 January 2007 Received in revised form 1 February 2008 Accepted 6 March 2008 Available online 13 March 2008 Keywords: Vehicle routing Pickup and delivery time windows Cooperative strategy Integer programming

a b s t r a c t A vehicle routing problem with pickup and delivery time windows is studied with the objective of determining resource requirements and daily routing by minimizing the sum of vehicle fixed costs and travelling costs. The multiple use of vehicles can reduce costs. Such a cooperative strategy, which has received little attention in past literature but which occurs in practice, may generate further savings. The strategy is studied here, with the single or multiple use of vehicles. Vehicles are allowed to travel to transfer items to another vehicle returning to the depot, provided no time window constraint is violated. This is modelled by an exact integer programming formulation which includes the solutions of the independent strategy. The proposed models are compared with a construction heuristic [Lu, Q., & Dessouky, M. M. (2006). A new insertion-based construction heuristic for solving pickup and delivery problem with time windows. European Journal of Operational Research, 175 (2) 672–687] which was applied to this problem. Experiments with instances generated from real-life data and simulated data show (i) significant savings over the construction heuristic as problem size grows; and (ii) multiple use of vehicles with a cooperative strategy may achieve cost savings over the independent strategy.  2008 Elsevier Ltd. All rights reserved.

1. Introduction This study was motivated by the daily operations of a local courier service of a multi-national logistics company. With the recent growth in demand for express delivery services, studying the resources required is crucial in setting contract prices for client projects in which the demand data can be assumed to be static during the contract period. With good cost estimation techniques, costs could be managed more easily. On the daily operational level, courier or vehicle routes need to be determined. Daily route sheets are issued to staff and signed by customers to acknowledge receipt. Each customer here is associated with two time window requirements: one for document pickup at the customer site and another for delivery to the mail centre to meet the designated flight departure time. The current problem belongs to the class of static-deterministic pickup and delivery problem with time windows (PDPTW) where there are many pickup points and only a single, common delivery point at the mail centre (or depot). Pickups occur before delivery to the mail centre. For urban deliveries, the vehicles could also represent couriers and letters/small parcels, which usually are not constrained by vehicle capacity. The objective is to find the minimum cost solution to servicing all customers and satisfying the time window constraints in this special case of PDPTW – the many-to-one pickup and delivery problem with time window constraints and without capacity constraints. This includes determining the number of vehicles required and the set of vehicle routes servicing the customers. According to a PDP survey paper by Savelsbergh and Sol (1995), ‘‘In the presence of time constraints the problem of finding a feasible pickup and delivery plan with time constraints is NP-hard.” Hence, the current problem is also NP-hard. q

The work described in this paper was fully supported by the Strategic Research Grant from City University of Hong Kong (Project No. 7001736). * Tel.: +852 2788 9485; fax: +852 2788 8560. E-mail address: [email protected]

0360-8352/$ - see front matter  2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2008.03.001

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This paper contributes by modelling the cooperating strategy in vehicle routing problem (VRP) with single or multiple use of vehicles. Independent vehicle operation is a common assumption in the literature; however, the multiple use of vehicles with a cooperative strategy can reduce cost. The cooperative strategy occurs in practice in parcel delivery services with multiple delivery resources, but is rarely explored and formulated for PDPTW. Studying the impact of cooperative strategy would provide new insights, but this has not received much attention in VRPs as compared with cooperative control systems for robots operating in manufacturing cells or unmanned aircraft in military operations. This study analyzes a VRP with PDPTW in four operational modes: independent single-route vehicle operations; the cooperative strategy applied on single-route vehicles; the multiple use of independent vehicles (or multi-route strategy); and the cooperative strategy to examine the cost savings and difference in route design. A route is defined as a trip starting from the depot and ending at the same depot after picking up customers’ documents along the route. Multiple use of vehicles allows each vehicle to be assigned to multiple routes (or a sequence of single routes) scheduled with no violation of constraints. The cooperative strategy here allows vehicles (or couriers) to travel to another customer location to transfer documents to a vehicle heading back to the mail centre, provided the delivery time constraints are satisfied. An analogy could be drawn with the savings algorithm (Clarke and Wright, 1964) where routes are merged to save travel distances and vehicles. The difference here is that not all vehicles operate independently. Some may cooperate, depending on problem data. In this paper, insights are generated on a favorable operating strategy under a given spatial distribution for customers and depot. The paper is organized as follows. Section 2 reviews the literature on pickup and delivery problems with time window constraints, related problems and the methodologies. Section 3 describes the model assumptions. Section 4 presents an exact model for the independent strategy for single- and multi-route vehicles. Section 5 proposes a model for the cooperative strategy. Section 6 reports on the computational experiments, followed by the managerial implications in Section 7. Finally, the conclusion and future research are discussed in Section 8. 2. Literature review A survey of pickup and delivery problems (PDP) was conducted by Savelsbergh and Sol (1995). The vehicle routing problem (VRP) is a special case of PDP in which either all pickup points or all delivery points are located at the depot. The current problem belongs to the static m-PDP with time windows in which each customer is characterized by a pair of pickup and delivery requests to be serviced by the same vehicle within its respective time window. Waiting is allowed when a vehicle is carrying items on board. The fleet consists of m identical vehicles all starting out from the depot, and all delivery points occur at the depot. Hence, the problem can be regarded as an m-VRP with pickup and delivery time windows. Depending on the operating constraints, pickup and delivery problems can have various classifications. Table 1 gives the classification with respect to problem structure and solution approach. This paper introduces a new category of cooperative vehicle operations (Table 1a: Type VIII). The present problem belongs to the class with static-deterministic demand where the problem is solved Table 1 Classification of VRP with pickup and delivery time windows (a) With respect to problem structure I. Availability of transportation requests

II. Number of depots

 Static-deterministic  Dynamic  Stochastic

 Single  Multiple

III. Vehicle fleet size

IV. Vehicle capacity

 Single  Multiple (identical/non-identical)

 Capacitated  Uncapacitated

V. Relationship between pickups and deliveries

VI. Objective function

   

Pickup before delivery Delivery-first, pickup-second Simultaneous pickups and deliveries Mixed pickups and deliveries

      

Minimize Minimize Minimize Minimize Minimize Minimize Minimize

duration completion time travel time route length client inconvenience number of vehicles cost (or Maximize profit)

VII. Wait allowed (when carrying items)

VIII. Vehicle operations

 Yes  No

 Independent  Cooperative

(b) With respect to solution approach Exact  Integer programming  Relaxation of set partitioning (or set covering) model and column generation  Dynamic programming  Branch-and-bound with strong lower bound  Lagrangian relaxation

Heuristic  Construction and insertion  Metaheuristic (tabu search, genetic algorithm)  Decomposition  Lagrangian relaxation  Hybrid

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once, ahead of the actual delivery time. There is only a single depot with multiple identical uncapacitated vehicles. For each customer, pickup occurs before delivery and the objective is to find a set of independent and cooperative vehicle routes, respectively, in order to minimize the total cost (comprising vehicle fixed cost and travelling cost). Exact and heuristic methods and the literature on PDP or VRP with time windows related to the present problem structure are discussed in the paper. The construction heuristic is a class of deterministic heuristic for solving scheduling problems. Lu and Dessouky (2006) applied an insertion heuristic for a PDP with time window and vehicle capacity constraints by systematically constructing the solution sequence. Unassigned customers are inserted into feasible positions incurring the minimum incremental cost, including both the incremental distance and the cost of reducing the time window slack. The visual attractiveness of solutions is also quantified and incorporated into the heuristic. An exact approach for a PDP with time windows and vehicle capacity constraints includes the set partitioning formulation in Dumas et al. (1991). Each column in the formulation represents a feasible route. A column generation method is applied to solve the linear relaxation of the set partitioning model, where new admissible columns are generated by solving a constrained shortest path problem. This exact algorithm works well when the vehicle capacity constraints are restrictive. The branch and price approach works in a similar manner. The problem is reformulated as a set covering problem. Starting with a feasible solution, the pricing strategy finds new columns with negative reduced costs to be re-optimized in another iteration of a set covering problem of reduced size. (If no such column exists, the current solution is optimal.) If the solution values are non-integers, the branching strategy will be applied to obtain integer solutions. Pricing algorithms are also developed in Angelelli and Mansini (2002) for a capacitated VRP with simultaneous pickup and delivery at each customer location with a time window constraint. Irnich (2000) focused on a special class of PDP where all possible routes can be easily enumerated either due to relatively narrow time windows or large demand quantities. Problem characteristics include (independent) heterogeneous capacitated vehicles (at multiple depots), time windows at customer sites and at the single hub. The heuristic approach involves a two-stage procedure. Relevant route/vehicle combinations are first enumerated, followed by relevant trips for each route/vehicle combination. With the resulting trips as input, the second stage solves a set covering model (by heuristic) to cover each request at least once. Requests that are being covered multiple times in the output will be assigned to exactly one trip. Exact algorithms solving the set covering problem mostly involve the use of Lagrangian heuristics and subgradient optimization embedded in a branch and bound framework as in Beasley and Jørnsten (1992). The Lagrangian relaxation approach can be both an exact or a heuristic approach, depending on the ability of the algorithm to find good Lagrangian multipliers. Kohl and Madsen (1997) applied Lagrangian relaxation to the customer assignment constraints of a VRP with a single time window per customer. The resulting model is decomposed into a set of simpler sub-problems: a shortest path problem with time windows and capacity constraints for each vehicle. Imai et al. (2007) relaxed the truck capacity constraints for a PDP where trucks transport full container loads. The relaxed problem is decomposed into an assignment problem forming delivery-pickup location pairs. The pairs are then assigned to trucks by solving a generalized assignment problem. A metaheuristic is an alternative approach to solving VRPs. Nanry and Barnes (2000) developed a reactive tabu search algorithm for a PDP with vehicle capacity and time window constraints. Data sets are modified from benchmark problems of up to 100 customers. Other applications of the tabu search algorithm include the capacitated VRP with backhauls and time windows (Currie and Salhi, 2004; Duhamel et al., 1997) where each demand point is classified as either linehaul or backhaul, requiring only delivery or pickup service, respectively, during its specified time window. The hybrid approach involves combining some of the above methods. Bent and Van Hentenryck (2006) applied simulated annealing and a large neighbourhood search to a PDP with time windows and two prioritized objectives. The simulated annealing is applied to minimize the number of routes and the large neighbourhood search is used to minimize the total travel cost. Zhong and Cole (2005) adopted a sweep algorithm to form customer clusters, followed by a guided local search to determine routes for a VRP with backhauls and time windows. There is little VRP literature on cooperative strategies. Shang and Cuff (1996) examined a multi-criteria pickup and delivery problem allowing transfer of hospital documents between vehicles provided no additional travel time is involved. They developed a look-ahead heuristic allowing users to input the number of delivery items to look-ahead, the number of time windows to look-ahead, and the minimum number of new routes to be constructed. Each new item is inserted into the existing schedule or into the best new route. New routes are evaluated according to a multi-criteria weighted score method considering four performance measures (number of items that can be transported, amount of travel time saved, number of unscheduled items that can use the route as origin or destination, and the number of items that could be transferred between the new route and existing routes). The route with the largest score is selected into the existing set. Shang and Cuff (1996) reported that there are no exact algorithms for similar problems. Here, however, this paper proposes an exact formulation supported by computational results. The problem studied in this paper is related to the location-routing problem (LRP), as both the vehicle routes and the location of transfer points need to be found. A review on the location-routing problem can be found in Nagy and Salhi (2007). Lin et al. (2002) and Lin and Kwok (2006) described applications of LRP in mail delivery services and applied metaheuristics. Jacobsen and Madsen (1980) examined a newspaper distribution problem where the transfer points needed to be determined for primary vehicles to transfer newspapers to secondary vehicles. As there was no optimal solution or a good lower bound, the existing operating system was simulated as a benchmark for comparison with the heuristics developed. The problem in this paper allows vehicles to pick up items from customers (within the time window) and transfer them to another

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vehicle returning to the depot, if cost is justified. The exact formulation can serve as a benchmark for evaluating heuristic performance for current and related problems. A related problem is the truck and trailer routing problem (TTRP), an extension of the vehicle routing problem, where customers are served by one (or more) of the following three routes: (i) a pure truck route, (ii) a complete vehicle route containing a truck and a trailer as one unit, or (iii) a complete vehicle route where the trailer is parked somewhere so that the truck can visit customer locations that are less accessible by a complete vehicle. Semet (1995) proposed a twophase heuristic procedure after presenting the integer programming formulation. The first phase assigns trailers to trucks and determines customers to be served by each truck or complete vehicle (truck plus trailer), followed by the second phase of route generation. Construction and improvement heuristics are used in Gerdessen (1996), and a tabu search algorithm combined with deterministic annealing in Chao (2002). Gerdessen (1996) assumes each trailer is parked exactly once to reduce the complexity of modelling the problem. This assumption is relaxed in Chao (2002). Both articles assume that each customer site can be a candidate parking place. This is similar to the current problem where each customer site can be a candidate location for document transfer. The differences are that pickup and delivery time windows are considered here and the number of instances of cooperation among vehicles (for document transfer) is determined by the model. Apart from the cooperative strategy, the multiple use of vehicles can also achieve cost savings. Taillard et al. (1996) applied the tabu search algorithm in a three-step procedure: the first two steps generate good vehicle routes as feasible VRP solutions and store them in sets in a search tree; the third step solves a bin packing problem on each set of stored VRP solution, to convert them into a solution with the multiple use of vehicles. The best overall solution is selected. Brandâo and Mercer (1997) designed a constructive and improvement heuristic for a similar multi-trip vehicle routing and scheduling problem with delivery time window, vehicle capacity and driver’s maximum working time constraints. Petch and Salhi (2004) examined the objective of reducing the maximum overtime for a prescribed vehicle fleet size. Special data structure and reduction techniques were applied to speed up the search for VRP solutions. Multi-trip VRP solutions will be considered in the present problem where vehicle capacity constraints can be ignored since letters/small parcels are usually not constrained by vehicle capacity. PDP or VRP with pickup and delivery time windows but without vehicle capacity constraints have been less studied. Van der Bruggen et al. (1993) used a two-phase approach with variable-depth exchange (similar to k-opt for the travelling salesman problem) and simulated annealing for solving a static single-vehicle PDPTW of up to 38 customers, giving nearly exact results. In the dynamic problem, only heuristics were developed due to the need to reschedule quickly. A constructive heuristic and a tabu search improvement procedure were applied to courier services by Mitrovic´-Minic´ et al. (2004) for the short-term and long-term horizons, respectively. The work in the present study contributes to modelling and provides exact methods for a cooperative strategy for a VRPTW without capacity constraints. 3. Model assumptions The assumptions in the models are mainly based on the current practice of a courier service. A day’s work is typically divided into two half-day service sessions: morning and afternoon. Each session represents an independent problem where customer pickup and delivery time windows occur in the same session. Problem size reduction can be achieved through dividing work period and locations into independent sub-problems. The capacity constraint is not considered as express documents are not very heavy and vehicles or couriers can usually pick up all their assigned documents in a service session. This assumption was adopted for the pickup and delivery of parcels or medical records in Langevin and Soumis (1989), Mitrovic´Minic´ et al. (2004), Mitrovic´-Minic´ and Laporte (2004) and Shang and Cuff (1996). For a set of N customers given in a service session, the pickup time window for customer i (=1, . . . , N) with pickup time specified at ti is [ti  dP, ti], where a given early allowance dP (>0) is common to all customers. An amount of on-site service time u is expected at each customer site. The delivery location for all customers is the mail centre at which parcels are processed before outbound delivery. For customer i, the time window for delivery time specified at si is [0, si + dD], where a given lateness allowance dD (>0) is common to all customers. To summarize, the assumptions underlying the models in this study include the following: (i) The service session is of duration T. (ii) Vehicle capacity is not constrained due to the small size of express documents. (iii) The pickup request at each customer location occurs before the delivery request to the common delivery location (mail centre). Both must be serviced by the same vehicle, unless an item is transferred to another vehicle after pickup. (iv) Travel time between a pair of locations could be asymmetrical or symmetrical. (v) When only distances between pairs of locations are available, vehicle speed is assumed to be an average traffic speed of V km/h. (This converts the collected travel distance into estimated travel time.) (vi) Pickup at the customer site should be no earlier than dP minutes before the specified pickup time. (vii) Waiting time is allowed if a vehicle arrives before the earliest pickup time at the customer site. (viii) On-site service time is assumed to be u minutes. (ix) Delivery time at the delivery location (mail centre) should be no later than dD minutes after the specified delivery time.

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4. Independent strategy Planning is based on the assumption that vehicles operate independently on their assigned route(s), which is common in the VRP literature. The advantage of independent operations is the flexibility offered to vehicles to react to real-time changes without affecting other vehicles. The disadvantage could be higher cost when vehicles do not share resources (time and travel distance) to achieve possible savings when demand is regular and stable (as in contract customers). Results from the independent strategy serve as a basis for comparison with the cooperative strategy (Section 5). Each route represents a feasible customer group (sequence of customers visited), originating from and returning to the mail centre exactly once. The entire set of distinct routes (or customer groups) is identified by a tree search Procedure S (Appendix 1). As the delivery time constraints may force a vehicle to return to the depot before the session ends, it is possible to assign a vehicle to multiple routes, one scheduled after another, if there is no time conflict. A multi-route group is defined as a sequence of single routes satisfying all pickup and delivery time windows in the group. The entire set of multi-route groups is determined by a tree search Procedure M (Appendix 2) applied to the single-route groups, similar to Procedure S (Appendix 1) in finding all feasible routes for the N customers. (The feasibility check for joining single routes into a feasible multi-route group is outlined in Step (5) of Procedure M.) The pickup and delivery time windows are checked whenever a single-route is appended to the end of a multi-route group. Serving a multi-route group is equivalent to the multiple use of vehicles where each vehicle starts out from the depot at least once to service a customer group during the service session. Planning based on multi-route groups will result in cost savings (at least) over the single-route groups since each single-route group is also counted as one multi-route group. The least-cost solution for single-route or multi-route groups can be determined exactly by solving a classical set partitioning problem (SPP): Model S : Set partitioning model Parameters: N number of customers in a service session n number of (single- or multi-route) groups formed (by Procedure S or Procedure M) cost of servicing group g (=sum of vehicle fixed cost and travelling cost), g = 1, . . . , n Cg set of groups that can service customer i, i = 1, . . . , N Si

Decisions : X g ¼ Min Z ¼

n X



1

if group g is selected ðto be assigned to a vehicleÞ;

0

otherwise

g ¼ 1; . . . ; n

Cg  Xg

g¼1

subject to :

X

X g ¼ 1;

i ¼ 1; . . . ; N

g2Si

X g ¼ 0; 1;

g ¼ 1; . . . ; n

The objective is to select a set of groups to service all customers at minimum cost. The single set of constraints in Model S requires each customer to be visited exactly once in some feasible group. When the problem size is large, the number of single- or multi-route customer groups formed (from Procedure S or Procedure M) could be huge. There are however many ways to form good independent customer groups from the past literature. For example, heuristics on VRPs with PDTW and single/multiple use of vehicles can be used to generate customer groups (vehicle routes) for large problems. As the focus of this paper is on modelling a cooperative strategy and applying it to the independent groups to examine the extent of cost savings, exact methods will be used to avoid approximation errors from any heuristic employed. 5. Cooperative strategy The multiple use of independent vehicles can reduce costs. This section examines how cooperation among vehicles (i.e., transferring goods) can achieve a similar purpose. (The goods here are express documents, hence the capacity constraint does not need not to be considered.) With today’s advances in communications technology, cooperation and change updates can be made easily. At the planning level, a simple type of cooperation is formulated in which a vehicle (or courier), on collecting all customer items on its assigned route(s), travels to a transfer point to transfer the collected items to a vehicle returning to the mail centre, provided that the delivery time constraints can all be satisfied. The transfer point chosen is the last customer pickup location of the returning vehicle in order not to incur delays for customer documents picked up earlier on the route. (Other transfer points are possible but at the increased risk of delay and additional computational time and memory.) The cooperation is analogous to the concept of merging feasible trips to find better trips in Clarke and Wright savings algorithm (1964) for VRPs and in the heuristic phase MERGE in Xu et al. (2003) for PDP. The difference here is to merge feasible routes at the transfer point to release vehicle(s) to visit other

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customers. On the other hand, when a set of candidate independent routes has been formed (say by heuristics), this model can be applied to the independent routes to check if further savings (in cost and vehicles) are possible through cooperation. First, the cooperative strategy is modeled on vehicles servicing single-route groups. The same model could also be applied to the multi-route groups by replacing the single-route groups with multi-route groups. A specially designed network P = (W, A) is defined for the cooperative strategy, where W and A denote the node set and arc set, respectively. The node set W in this network consists of two nodes for each customer group and the depot (node 0). Given n is the number of feasible customer groups, let node Pg denote the selection decision of group g and node Rg denote the return decision to the depot from the last customer location in group g (g = 1, . . . , n). Hence if node Rg is visited, so should node Pg. The reverse may not be true as documents picked up in group g could be transferred to another vehicle (at another location), instead of being returned directly to depot. More precisely, each node in W is defined with a set of inflow and outflow arcs, representing feasible vehicle movements. Two types of outflow arcs are defined for node Pg (g = 1, . . . , n): (i) (Pg, Rg): Group g is selected. Documents collected will be returned directly to the depot from the last customer location in g. (ii) (Pg, Rh), where h(–g) 2 W: Group g is selected. Documents collected in group g will be carried to the last customer location in group h and transferred to a vehicle returning directly to the depot (carrying documents of both groups). Two time constraints are checked to ensure that the departure time is not delayed at the last customer in h, and that the delivery time constraints of either group at the depot are not affected. Let the departure time from the last location of group g and h be denoted by d(g) and d(h), respectively, when they are serviced independently, and their specified delivery time at the depot by s(g) and s(h), respectively. Check:  Departure time constraint:d(g) + travel time(last customer of g ? last customer of h) 6 d(h)  Delivery time constraint:d(h) + travel time(last customer of h ? depot) 6 min{s(g), s(h)} + dD  No common customer exists in group g and h.(In the solution, the number of inflow arcs to node Rh represents the number of groups whose documents will be transferred/carried by a vehicle returning to the depot from the last customer location in h.) Two types of inflow arcs are defined for node Pg (g = 1, . . . , n): (i) (0, Pg): a vehicle starts out from the depot to collect all documents in group g. (ii) (Rf, Pg), where f(–g) 2 W: Group g is selected, where its documents are collected by a free vehicle released from the last customer location in group f. The pickup time constraint for group g documents is checked. Let the departure time from the last location of group f be denoted by d(f) and the latest pickup time of the first customer, g(1), in group g by t(g(1)). Check:  Pickup time constraint:d(f) + travel time(last customer of f ? customer g(1)) 6 t(g(1)).  No common customer exists in group f and g.

Two types of outflow arcs are defined for node Rg (g = 1, . . . , n): (i) (Rg, 0): a vehicle carrying group g documents returns to the depot from the last customer location in group g. (Depending on the number of inflow arcs to node Rg, the returning vehicle could carry other documents transferred to it at the last customer location in g.) (ii) (Rg, Ph), where h(– g) 2 W: (Analogous with type (ii) inflow arcs defined for node Pg above.) Two types of inflow arcs are defined for node Rg (g = 1, . . . , n): (i) (Pg, Rg): (Same interpretation as type (i) outflow arcs defined for node Pg above.) (ii) (Pf, Rg), where f (–g) 2 W: (Analogous with type (ii) outflow arcs defined for node Pg above.) On defining the elements in the network P, the uncapacitated VRP with pickup and delivery time windows and a cooperative strategy can be formulated by an exact integer programming model: Model C : Integer programming model for cooperative strategy Parameters: n number of customer groups formed W vehicle fixed cost travelling cost within group g (starting from the first customer and up to the last customer), g = 1, . . . , n vg travelling cost along arc (i, j), " (i, j) 2 A (xij = vg for (i, j) = (Pg, Rg), g = 1, . . . , n) xij set of groups that can service customer i, i = 1, . . . , N Si

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Decisions: flow (or connection) along arc (i, j), " (i, j) 2 A yij

Min Z ¼

n X

W  y0;Pg þ

g¼1

subject to: X X

X

xij  yij

yj ;Pg ¼ 1;

i ¼ 1; . . . ; N

g2S i ðj;P g Þ2A

X

y0;Pg þ

yRf ;Pg ¼ yPg ;Rg þ

ðRf ;Pg Þ2A

X

ð1Þ

ði;jÞ2A

yPf ;Rg ¼

ðPf ;Rg Þ2A

X

X

ð2Þ yPg ;Rh ;

g ¼ 1; . . . ; n

ð3Þ

ðP g ;Rh Þ2A h–g

yRg ;Ph þ yRg ;0 ;

g ¼ 1; . . . ; n

ð4Þ

ðRg ;P h Þ2A h–g

yP g ;Rg ¼ yRg ;0 ; g ¼ 1; . . . ; n X yPf ;Rg 6 N  yRg ;0 ; g ¼ 1; . . . ; n

ð5Þ ð6Þ

ðP f ;Rg Þ2A

yij ¼ 0; 1;

8 ði; jÞ 2 A

ð7Þ

The objective function in constraint (1) describes the total cost, comprising the vehicle fixed cost and the variable travelling cost. The set partitioning constraint for each customer (i) is formulated in constraint (2). The flow balance constraints for each node, Pg and Rg, associated with group g in the network are formulated in constraints (3) and (4), respectively. Constraint (5) requires that if group g is selected, all its documents collected should travel along arc (Rg, 0), if returned to the depot directly. Conversely, if node Rg is not visited (i.e., no vehicle returns to depot from this node), group g will not be selected ðyRg ;0 ¼ 0 ¼ yPg ;Rg Þ. Constraint (6) ensures that if document(s) including group g have been collected/transferred at the last customer location in g (inflow to node Rg P 1), node Rg must be visited (outflow = 1), a vehicle will return to depot along arc (Rg, 0). Constraint (5) needs to be formulated together with (6). If constraint (5) is omitted, it is observed from some computational experiments that yRg ;0 ¼ 1; yPg ;Rg ¼ 0; i.e., the last customer node in group g is only used as a transfer node for some vehicles carrying other customer documents, but not group g. Though this is possible for other extended cooperative models, the one here simply makes the best use of a returning vehicle carrying documents in its assigned group. In the computational experiments, the integer constraint (7) for decision variables yij, except yRg ;0 , is relaxed when running Model C. The solution values turn out to be 0 or 1 in most instances. As every solution representing independent operations is also included in Model C (with only positive flow on arcs (0, Pg), (Pg, Rg) and (Rg, 0), g 2 W), Model C incorporates both cooperative and independent solutions. Hence, the optimal solution for Model C is no worse than the independent strategy (Model S). In the case when only a set of acceptable independent routes is available, Model C can be applied to it to explore possible cost savings from cooperation. When considering the multiple use of vehicles, a group could be defined as a multi-route group (formed by Procedure M in Appendix 2) and Model C could be applied similarly. The candidate transfer points now become the last customer location visited in each multi-route customer group. As each single-route group is also counted as one multi-route group, the optimal solution for the latter is no worse than the former. Other characteristics of pickup and delivery problems could also be accounted for as follows. 5.1. Limited fleet size When the number of available vehicles is V, add an additional constraint to Model C: n X y0;Pg 6 V

ð8Þ

g¼1

5.2. Capacitated vehicles The many-to-one pickup and delivery problem requires all customer documents in a group to be picked up before the vehicle returns to the depot. Hence, by adding a load checking constraint in (1) and (6) of Procedure S (Appendix 1), all feasible single-route customer groups satisfying the vehicle capacity constraint can be formed. (As before, these will serve as input to find all multi-route customer groups by Procedure M (Appendix 2), where each single-route component satisfies the vehicle capacity constraint.) Let qf denote the total loading of customer group f (=1, . . . , n) and Q the vehicle capacity, then constraint (6) in Model C could be modified as follows: X qf  yPf ;Rg 6 Q  yRg ;0 ; g ¼ 1; . . . ; n ð60 Þ ðPf ;Rg Þ2A

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5.3. Restriction on number of cooperative vehicles As transfer of documents from one vehicle to another vehicle will possibly incur delay, one can restrict the number of vehicles involved in transfer at any location to some limit R by modifying constraint (6) in Model C as follows: X yPf ;Rg 6 R  yRg ;0 ; g ¼ 1; . . . ; n ð600 Þ ðPf ;Rg Þ2A

Figs. 1 and 2 illustrate the optimal solution for the independent and cooperative strategy, respectively, for a small instance of six customers. (Details are shown in Appendix 3.) The optimal solution for independent single-route vehicles is $21,539, requiring three vehicles. Routes 1 and 3 in Fig. 1 can be assigned to one vehicle with no constraints violation. Hence for the multiple use of independent vehicles, the optimal solution requires only two vehicles at a total cost of $14,539 (given vehicle fixed cost = $7000). The optimal cooperative solution (the same for single-route or multi-route vehicles for this instance) involves a transfer at location of customer 5 and requires two vehicles (Fig. 2). The total cost is $14,455, which is less than that of the independent single-route or multi-route solutions. Total travelling time is saved (390 min vs 462 min) at the expense of longer waiting time (163 min vs 86 min and 91 min). When fuel cost is high, waiting is preferred to driving to another location. (During the wait or slack time, dynamic requests can also be entertained.) 6. Computational experiments The proposed models were compared with an insertion-based construction heuristic by Lu and Dessouky (2006) applied to the present problem (without vehicle capacity constraints). Let H_construct denote this heuristic in which the next insertion task and position in each construction step is the one incurring the least insertion cost, composed of both the increase in distance and slack reduction in the time windows. (The visual attractiveness of the solution was not considered, as the proposed models aim at finding the minimum total cost. Hence, the cost of the solution is the main focus.) The resulting solution could consist of both single-route and/or muli-route vehicles. The experiments were designed based on two types of data. The first set (10 instances) was based on real-life data obtained from a local delivery service and the operating characteristics of a courier service. (Unknown parameters are simulated.) The second set (30 instances) contained simulated test problems adopting known operating parameters from the two local services. 6.1. Local instances The parameters of the first set of real-life data were estimated as follows: (i) The location data in instances 1–10 were adopted from a local delivery service. Each instance contained one single depot and 27–30 customers. Travel distances in instances 1–4 were provided by drivers (Lin et al., 2002) and by a geographical information system, and by Network Analyst software in instances 5–10 (Lin and Kwok, 2006). (ii) Cost components: the fixed cost of vehicle and staff (=HK$7000) was adopted from the courier service and the variable operating cost of vehicles (=HK$70 per hour) from the delivery service.

120 Route 1 2

100

Route 2 Route 3 3

80

1 (customer)

60

40

6 4

5

20 P(depot)

0 50

60

70

80

90

Fig. 1. Optimal independent strategy.

100

110

774

C.K.Y. Lin / Computers & Industrial Engineering 55 (2008) 766–782

120 Route 1

2

100

Route 2 3

80

1 (customer)

60

40

6 4

5

20 P (depot)

0 50

60

70

80

90

100

110

Fig. 2. Optimal cooperative strategy. Node 5: both a customer node and a transfer node.

(iii) Sizes of pickup, delivery time windows (dP = 5 min, dD = 10 min) and duration of service session (T = 300 min) were adopted from the courier service. For each instance, pickup times (ti) and delivery times (si) were simulated within ½0; T, accounting for sufficient travel time back to the depot. Average traffic speed V = 20.8 km/h was adopted (Transport Department, Hong Kong, 2001). In the real-life data, customers are located in densely populated districts, which are typical characteristics of the urban area. As dP is small compared with the travel times, one can assume that customers in a feasible route are served in the earliest pickup time order. Hence, customers were first sorted in increasing order of pickup time {ti, i = 1, . . . , N} before being formed into feasible single-route groups by Procedure S (Appendix 1) and multi-route groups by Procedure M (Appendix 2). 6.2. Simulated instances All locations were simulated on a rectangular service area expressed by travel time in a 200  200 min2 grid. (In urban pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi areas, a maximum 4-h ðffi 2002 þ 2002 Þ one-way travel between locations is assumed to be reasonable.) The simulated data sets were characterized by two factors: number of customers and depot location. The customer size was 50 and 100, respectively, and their locations were randomly generated in the grid. The depot locations were set at the edge, the centre and half-way between in order to examine the change in problem complexity and the best operating strategy. Five instances were generated for each combination of customer size and depot location, resulting in a total of 30 instances. Other parameters were adopted as in the local instances (Section 6.1). 6.3. Comparison of strategies All models were coded in Visual Basic.NET and all experiments were performed on a Pentium 4, 2.5 GHz processor. The optimization software CPLEX 9.1 was used in solving all (mixed) integer programming problems. Five different operational strategies were tested: (i) independent multi-route vehicles from H_construct; (ii) independent single-route vehicles from Model S; (iii) the cooperative strategy on single-route vehicles from Model C (iv) independent multi-route vehicles (or multiple use of vehicles) from Model S and (v) a cooperative strategy on multi-route vehicles from Model C. Performance was measured in total cost (fixed vehicle cost and travelling cost) and number of vehicles required. The savings of the proposed models (ii)–(v) over the construction heuristic (i) was evaluated in turn. (It was also possible to make relative comparisons among (ii)–(v) from the results in Tables 2–4.) The comparison was based on the best (or optimal) solution obtained, subject to computer memory requirement and running time. For the ten local instances (Table 2), it was possible to obtain exact results for nearly all instances (except one) for (ii)–(v). For the 30 simulated instances (Table 3), a maximum time limit of 10,800 CPU seconds was allowed for each instance. Computational time used is recorded in Table 4. Results from the local instances (Table 2) show that the average daily cost of the cooperative strategy on single-route vehicles is superior to the constructive heuristic H_construct, as are both multi-route strategies (Fig. 3). When vehicles operate on single routes, the cooperative strategy can achieve savings over its independent counterpart by as much as 20 per cent (instances 5 and 9). For the multiple use of vehicles, the maximum saving from the cooperative strategy is 12 per cent (instance 9). In three out of the ten instances (7, 9 and 10), the cooperative single-route solution may even outperform the independent multi-route solution. The tradeoff is the additional computational time and memory required.

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C.K.Y. Lin / Computers & Industrial Engineering 55 (2008) 766–782 Table 2 Computational results for the local instances 1–10 Instance

No. of customers

Daily cost in HK$ (number of vehicles) Construction heuristic (H_construct)

Single-route vehicles Independent strategy (Model S)

Cooperative strategy (Model C)

Independent strategy (Model S)

Multi-route vehicles Cooperative strategy (Model C)

1 2 3 4

27

115,141.83 (16) 107,889.83 (15) 93,853.67 (13) 107,969.17 (15)

100,716 (14) 114,640.17 (16) 107,675.17 (15) 114,654.17 (16)

100,716 (14) 114,640.17 (16) 93,646 (13) 114,654.17 (16)

100,716 (14) 93,640.17 (13) 86,675.17 (12) 100,655.33 (14)

100,716 (14) 93,640.17 (13) 86,646 (12) 100,655.33 (14)

5 6 7 8

29

50,373.17 (7) 79,292.5 (11) 101,722.83 (14) 71,755.83 (10)

64,229.67 (9) 78,887.67 (11) 108,599.17 (15) 78,557.5 (11)

50,145.67 (7) 71,769.83 (10) 101,413.67 (14) 78,543.5 (11)

43,268.17 (6) 58,091.83 (8) 101,599.17 (14) 57,709.17 (8)

43,268.17 57,769.83 94,413.67 57,709.17

9 10

30

79,675.17 (11) 79,487.33 (11)

64,824.67 (9) 79,034.67 (11)

50,606.5 (7) 71,872.5 (10)

57,824.67 (8) 72,034.67 (10)

50,606.5 (7)a 71,872.5 (10)

–

3.87% (4.38%)

4.71% (4.37%)

13.75% (13.81%)

15.43% (15.43%)

Average savings (%) over H_construct a

(6) (8) (13) (8)

Best upper bound obtained.

Table 3 Computational results for the simulated instances 11–40 Instance

No. of customers

Depot location

Daily cost in HK$ (number of vehicles) Construction heuristic (H_construct)

Single-route vehicles Independent strategy (Model S)

Cooperative strategy (Model C)

Independent strategy (Model S)

Cooperative strategy (Model C)

Edge (100, 0)

176,264.5 (24) 169,164.33 (23) 189,918.17 (26) 213,291.17 (29) 169,356.83 (23)

168,890.17 (23) 154,320.83 (21) 182,110.83 (25) 190,676.5 (26) 168,864.5 (23)

147,264.83 (20)a 147,234.5 (20) 160,667.5 (22) 190,676.5 (26) 161,641.67 (22)

161,885.5 (22) 140,320.83 (19) 161,154 (22) 190,676.5 (26) 161,819 (22)

147,264.83 (20) 140,234.5 (19) 160,667.5 (22) 190,676.5 (26) 161,641.67 (22)

16 17 18 19 20

Half-way (50, 0)

188,984.83 (26) 182,248.5 (25) 159,743.5 (22) 174,263.83 (24) 174,162.33 (24)

188,316.33 174,189.17 145,058.67 159,500.83 180,622.17

166,919.67 167,074.83 144,967.67 152,310.67 159,489.17

167,316.33 153,189.17 138,037.67 159,500.83 152,622.17

153,029.33 153,074.83 137,967.67 152,297.83 145,436.67

21 22 23 24 25

Centre (0, 0)

173,792.5 (24) 188,237 (26) 195,450.5 (27) 181,304.67 (25) 166,682.83 (23)

173,386.5 (24) 187,567.33 (26) 180,514.83 (25) 180,556.83 (25) 158,632.83 (22)

159,190.5 (22) 159,509 (22) 173,472.83 (24) 166,618.67 (23) 158,632.83 (22)

152,419.17 (21) 152,631.5 (21) 159,691 (22) 152,673.5 (21) 158,632.83 (22)

145,257 (20) 145,519.5 (20) 159,495 (22) 152,582.5 (21) 158,632.83 (22)

Average savings (%) over H_construct 26 100 Edge 27 (100, 0) 28 29 30

–

4.01% (3.74%)

10.54% (10.43%)

12.49% (12.56%)

14.67% (14.72%)

316,911 (43) 316,123.5 (43) 330,472.33 (45) 344,573.83 (47) 315,303.33 (43)

293,741 (40) 271,682.83 (37) 285,901 (39) 286,185.67 (39) 270,911.67 (37)

257,748.17 (35)a 264,629.17 (36) 278,630.33 (38) 271,610.5 (37) 263,931.5 (36)

279,766.67 (38) 271,682.83 (37) 285,901 (39) 272,203.17 (37) 270,911.67 (37)

250,717.83 (34) 264,625.67 (36) 278,630.33 (38) 264,610.50 (36) 263,931.5 (36)

31 32 33 34 35

Half-way (50, 0)

290,598 (40) 305,840.5 (42) 342,315.17 (47) 319,723.83 (44) 269,175.67 (37)

268,202.67 (37) 283,176.83 (39) 305,102 (42) 275,622.67 (38) 239,402.33 (33)

254,248.17 (35) 254,894.5 (35)a 290,888.5 (40)a 261,674 (36)a 232,612.33 (32)a

254,216.67 255,235.17 291,113.67 254,707.83 218,327.67

(35) (35) (40) (35) (30)

247,329.83 (34)a 247,929.5 (34)a (Out of memory) (Out of memory) (Out of memory)

36 37 38 39 40

Centre (0, 0)

304,628.33 (42) 297,465 (41) 282,684.5 (39) 304,612 (42) 333,208.17 (46)

260,384.83 (36) 281,922.67 (39) 260,772.17 (36) 317,215.5 (44) 310,166.5 (43)

239,290.33 (33)a 260,775.67 (36)a 232,411.67 (32)a 289,098.83 (40)a 267,939 (37)

225,726.67 (31) 239,979.83 (33) 225,639.17 (31) 261,233 (36) 261,277.33 (36)

(Out of memory) 232,893.5 (32)a 218,692.83 (30)a 261,266.83 (36)a 247,259.83 (34)

–

9.80% (9.57%)

16.02% (15.98%)

17.26% (17.32%)

19.14% (19.26%)

11 12 13 14 15

50

Average savings (%) over H_construct a

Best upper bound obtained.

(26) (24) (20) (22) (25)

Multi-route vehicles

(23) (23) (20) (21) (22)

(23) (21) (19) (22) (21)

(21) (21) (19) (21) (20)

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Table 4 Computational time taken Instance

No. of customers

CPU seconds Construction heuristic (H_construct)

Single-route vehicles Independent strategy (Model S)

Cooperative strategy (Model C)

Independent strategy (Model S)

Cooperative strategy (Model C)

(a) Local instances 1–10 1 27 2 3 4

5.04 5.83 5.27 5.73

2.62 2.56 2.71 2.66

2.67 3.9 3.9 3.03

2.5 2.48 4.46 4.31

7.3 5.91 4.53 6.23

5 6 7 8

29

6.8 6.22 10.92 6.57

6.62 9.8 5.81 10.43

22,741.75 1908.31 146.04 1096.56

2.99 6.71 11.32 5.33

4660.86 79.49 18.06 2,211.91

9 10

30

7.41 6.69

3.45 11.08

32,629.15 29,723.66

7.63 6.41

a

Independent strategy (Model S)

Cooperative strategy (Model C)

Independent strategy (Model S)

Cooperative strategy (Model C)

(b) Simulated instances 11–40 11 50 Edge 12.83 12 12.67 (100, 0) 13 17.72 14 12.71 15 13.91

7.67 9.29 8.12 8.21 8.37

b

11.88 15.15 12.14 11.18

27.39 7.27 11.68 10.44 11.43

87.66 116.96 49.97 13.95 10.94

16 17 18 19 20

Half-way (50, 0)

14.24 22.01 12.8 12.88 12.37

7.54 8.22 8.31 8.59 8.32

1871.46 16.79 11.07 10.59 16.12

7.6 8.31 9.64 15.24 8.26

29.88 16.02 120.97 13.92 15.08

21 22 23 24 25

Centre (0, 0)

15.47 11.81 11.84 11.66 12.9

12.31 8.3 8.18 8.21 8.42

10.93 18.57 13.07 15.15 11.9

7.51 11.12 10.82 8.39 8.81

18.77 103.75 12.24 11.58 14.74

Edge (100, 0)

46.89 44.64 44.89 46.9 45.28 41.9 43.44 52.94 41.15 44.79 38.37 46 44.12 45.22 46.05

28.7 34.97 30.14 30.94 30.14 30.75 31.11 31.59 30.95 31.65 31.28 31.12 31.24 29.9 30.83

b

32.95 28.06 27.69 110.07 28.33 29.86 249.29 69.55 39.09 122.27 39.59 63.02 32.92 32.12 32.42

1,251.02 77.05 74.21 543.26 2,083.24

Depot location

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 a b

100

Half-way (50, 0)

Centre (0, 0)

Multi-route vehicles

47,106.12

CPU seconds Construction heuristic (H_construct)

Single-route vehicles

Multi-route vehicles

49.66 49.66 666.89 804.55 5,428.42 b b b b b b b b

146.7

a b a a a a a b a

526.32

Computer memory exceeded. Maximum time limit (10,800 CPU s) reached.

The simulated instances (generated on a 200  200 min2 rectangular grid) carry more randomized geographical characteristics. Results for the constructive heuristic (H_construct) and exact results for the independent strategy (Model S) were obtained more easily (Table 4b) than for the cooperative strategy (Model C). More difficult problems are those with large problem size and when the depot location is located near the centre of the customer group. This creates many possible customer groups as the problem size increases. In some instances, either only an upper bound was obtained or the computer ran out of memory. (Many heuristics in the literature can create a smaller set of good independent routes for Model C to be applied for further improvement. The focus of this paper is however on modelling a cooperative strategy and comparing it with its independent counterparts through exact methods.) Results in average daily cost from all the optimization models outperformed the constructive heuristic, and the savings increased with the problem size (Figs. 3 and 4). In twelve out of the thirty instances (11, 13, 15–16, 19, 26–30 and 32–33), the cooperative strategy on single-route vehicles

C.K.Y. Lin / Computers & Industrial Engineering 55 (2008) 766–782

20%

30 or fewer customers

50 customers

15%

% cost savings

777

100 customers

10%

5%

0% Single-r_indep Single-r_coop Multi-r_indep Multi-r_coop

-5% Fig. 3. Average savings (%) in total cost over construction heuristic.

% reduction in vehicles

20%

30 or fewer customers 50 customers

15%

100 customers

10%

5%

0% Single-r_indep

Single-r_coop

Multi-r_indep

Multi-r_coop

-5% Fig. 4. Average reduction (%) in vehicles over construction heuristic.

outperformed even the independent multi-route vehicles. The observations from the experiments are summarized as follows:  The construction heuristic H_construct, based on Lu and Dessouky (2006), is of computational complexity O(n4), and has tractable running time (Table 4). The average solution quality was inferior to three optimization models: cooperative single-route vehicles; independent multi-route vehicles; and cooperative multi-route vehicles, in increasing order of dominance.  The cooperative strategy was able to achieve further savings over the independent strategy at the expense of more computer memory and running time.  The multiple use of vehicles in a PDPTW reduced total cost and number of vehicles.  The percentage savings in total cost is nearly the same as the percentage reduction in number of vehicles in each instance (Tables 2 and 3). This implies that the cooperative strategy is helpful in reducing the number of vehicles used and hence the fixed cost. The saving in travelling cost is relatively small.  Difficult problems carry one or more of the following characteristics: (i) large number of customers; (ii) depot located near the centre of customer area; and (iii) cluster of customers with small travel distances in between. (This can be handled by aggregating compatible customers into a single node through heuristic approaches.)

7. Managerial implications This study proposes a scientific approach to scheduling the daily routes and estimating the resources required for pickup and delivery problems with time windows. This allows resource utilization and slack times to be estimated during the planning stage and compared after implementation. The decision maker can also perform sensitivity analysis with respect to changes in available resource (vehicles), customer pickup times, or return times. These can generate helpful insights in negotiating contracts with clients. From the experiments, planning based on the multiple use of vehicles and the cooperative strategy is the most cost-effective for VRPs with pickup and delivery time windows that often force vehicles to return to the depot before the end of a

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session to meet delivery time constraints. Subject to computer memory or running time limits, the better solution between independent multi-route vehicles and cooperative (single- or multi-route) vehicles is suggested. When the customer characteristics (location, pickup and delivery times) are stable and occur regularly – e.g., corporate customers sign a contract – the savings from multiple use of vehicles and the cooperative strategy are significant and will increase with the length of the contract. The proposed methodologies also contribute to capacity planning of the depot (mail centre). The routing and scheduling results can indicate whether too many items, exceeding the service capacity, will be returning to the depot near the end of the service session. (If this occurs, it will cause delay in the processing of items and breaching of the service pledge for time-sensitive express services.) The study by Hall (1996) for overnight carriers adopting separate pickup and delivery sessions also suggests using models to adjust delivery and cutoff times to maintain a balanced workload at the terminal (mail centre). Hence, a scheduling tool for the upstream activities in a supply chain serves as a demand planning tool for downstream activities. This study provides a foundation to compare performances of the independent vs the cooperative strategy, and the single vs the multiple use of vehicles in any given instance. It also provides benchmarks for heuristics developed. As the geographical characteristics for each urban area could be quite different, the approximation formula in the literature for resource estimation needs to be tested. The use of information systems (such as geographical information systems and geographical positioning systems) with the models developed in this work may provide an objective estimate of the resource requirements during the planning stage without the assumption of spatial characteristics or customer distribution. The methods may also apply to services which are operating on defined service zones or routes constructed by the cluster-first, route-second approach. Cooperative strategies could take various formats. The challenge lies in defining reasonable ones for implementation, modelling, solving such problems and selecting appropriate cooperative solutions. The expected benefit is cost and vehicle savings, but close coordination is required among the cooperating vehicles. Hence, it is reasonable to involve only a small number of customer groups and vehicles in cooperation (by adding constraint (600 ) in Model C) as delay in one group will affect the servicing of other group(s). 8. Conclusions and future research This study presents exact approaches to VRPs with pickup and delivery time windows for the purpose of resource estimation and the scheduling of daily routes. The objective is to service all customers at minimum cost, which comprises vehicle fixed cost and variable travelling cost. Due to the problem structure of many pickup-to-one delivery points, the solution of independent vehicle operations could be represented by multi-route customer groups, made up of a sequence of singleroute groups (or the multiple use of vehicles). The cooperative strategy and its modelling by exact methods has received little attention in the vehicle routing literature or in commercial software, which usually assume independent vehicle operations. For VRPs with pickup and delivery time windows and without capacity constraints (e.g., courier services), the cooperative strategy here considers document transfer at each customer location where a vehicle would be returning to the depot. Cost saving over the independent (single- or multi-route) strategy is achieved for problem size of up to 100 customers. Compared with an insertion-based construction heuristic (Lu and Dessouky, 2006), the cooperative strategy and the independent multiroute strategy achieve savings in both total cost and vehicles in each tested instance. The savings also increase with problem size. Other types of cooperative strategies are worth investigating depending on the characteristics of the delivery service. An exact model for PDPTWs without capacity constraints is rare in literature and only a few heuristics have been developed (Mitrovic´-Minic´ et al., 2004), not to mention the cooperative version. Planning based on the cooperative strategy and the multiple use of vehicles is more cost-effective than the independent strategy. The exact performance of both strategies is constrained by problem size, computer memory and running time limits. Existing approaches in solving set partitioning problems could handle large problems for the independent strategy (Model S). Alternatively, given a candidate set of independent solutions, the cooperative model (Model C) could be applied to examine cost and vehicle savings. Future research will examine the case for multiple (non-identical) delivery resources. Acknowledgments The author is grateful to Dr. Gábor Nagy and the anonymous reviewer for their support and constructive comments on improving the presentation of this paper. Appendix 1. Procedure S: Depth-first search for feasible single routes

(1) Initially check feasibility of each customer node (i = 1,. . ., N):  Pickup time constraint: travel time (depot ? i) 6 ti  Delivery time constraint: Max{ti  dP, travel time (depot ? i)} + u + travel time (i ? depot) 6 si + dD (discard customer node(s) which cannot satisfy either of the above. Revise the value of N for the number of feasible customer nodes.)

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779

(2) Let ordered list L = (L1, L2, . . . , LN), where t L1 6 tL2 6    6 t LN . (3) The maximum number of levels in the search tree (or nodes in a route) is N + 2, where the start and end are the depot and each intermediate level consists of a customer node. (4) An arc (i, j) in the search tree corresponds to travelling from customer node i (after pickup) to node j, if such a move is feasible by Step (6) below. (5) Branching strategy: suppose the partial tree T consists of an ordered set of nodes (depot, i1, i2, . . . , ik). Denote the service end time of T by tðTÞ. Select the earliest node in list L, say j, which has not occurred in T and such that ti1 6 ti2 6    6 t ik 6 tj . (6) Feasibility check at each level: when considering appending a node j to a partial tree T = (depot, i1, i2, . . . , ik) when service at ik ends at time tðTÞ, check the following two conditions.  Pickup time constraint: tðTÞ þ travel time ðik ! jÞ 6 t j  Delivery time constraint:Maxftj  dP ; tðTÞ þ travel time ðik ! jÞg þ u þ travel timeðj ! depotÞ 6 minfsi1 ; si2 ; . . . ; sik ; sj g þ dD If both constraints hold, update partial tree T ! ðT; jÞ and service end time tðTÞ ! Maxft j  dP ; tðTÞþ travel timeðik ! jÞg þ u. (7) Backtracking strategy: when no node in list L can be appended to the partial tree T = (depot, i1, i2, . . ., ik) through Steps (5) and (6), this implies T ! depot forms a complete distinct route. Then backtrack to the previous level (k-1) in T and branch off to a new node after ik in list L (Step (5)). (8) Termination condition: this occurs when the algorithm backtracks to the first level of the partial tree T which contains node LN. Hence, the last route found is (depot ? LN ? depot).

Appendix 2. Procedure M: Depth-first search for feasible multi-route groups (1) Let n be the total number of single routes identified by Procedure S and R = (R1, . . . , Rn) be the ordered list of single routes. (Each route is feasible on its own with the start and end nodes at the depot.) Let Rj(i) denote the ith customer on route Rj, i = 1, . . . , |Rj|, where |Rj| is the total number of customers on route Rj ,j = 1, . . . , n. The earliest delivery deadline at the depot for customers on route Rj is denoted by sðRj Þ ¼ min fsRj ðiÞ g þ dD . i¼1;...;jRj j

(2) The maximum number of levels in the search tree is n. Each level consists of a single-route originating at and returning to a depot. (3) An arc (Ri, Rj) in the search tree corresponds to a vehicle servicing route Rj after it returns to the depot when route Ri is completed, if such a move is feasible by Step (5) below. (4) Branching strategy: suppose the partial tree T consists of a feasible sequence of routes (r1, r2, . . . , rk). Select the earliest element in list R, say Rj, currently not in T and whose position occurs after rk in list R. (The latter condition preserves the ascending order of pickup time and is checked in detail in Step (5).) (5) Feasibility check at each level: consider the partial tree T ¼ ðr1 ; r2 ; . . . ; r k Þ has its return time to depot denoted by tðTÞ, the following three conditions must hold if some route Rj is to be appended to the end of T.  T and Rj have no common customer 0  Pickup time constraints: let t denote a temporary time variable (initially set at tðT0Þ) representing the updated end time of the partial tree T. Check the pickup time constraint for each customer node Rj(i) on route Rj (i = 1, . . . , |Rj|): t0 þ travel timeðRj ði  1Þ ! Rj ðiÞÞ 6 tRj ðiÞ : If the above holds, t0 ! maxftRj ðiÞ  dP ; t0 þ travel timeðRj ði  1Þ ! Rj ðiÞÞg þ u. Next i. Otherwise, route Rj is not feasible to be appended to the end of T. Backtrack (Step (6)).  Delivery time constraint: if the pickup time constraints are satisfied for customers in route Rj, t0 represents the service end time at the last customer node Rj(|Rj|). Check t0 þ travel timeðRj ðj Rj jÞ ! depotÞ 6 sðRj Þ þ dD If all three above sets of constraints hold, update partial tree T ! ðT; Rj Þ and return time to depot tðTÞ ! t0 þ travel timeðRj ðj Rj jÞ ! depotÞ

(6) Backtracking strategy: when no node in list R can be appended to the partial tree T ¼ ðr1 ; r2 ; . . . ; rk Þ through Steps (4) and (5), this implies T forms a distinct multi-route group. Then backtrack to the previous level (k-1) in T and branch off to a new element after rk in list R (Step (4)).

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(7) Termination condition: this occurs when the algorithm backtracks to the first level of the partial tree T which contains node Rn. Hence, the last multi-route group found is Rn.

Appendix 3. Data and calculations for Figs. 1 and 2 Customer characteristics Node

P (depot)

1

2

3

4

5

6

Time window (min) Pickup Delivery (at depot)

– –

[117, 122] [0, 281]

[102, 107] [0, 280]

[137, 142] [0, 318]

[191, 196] [0, 360]

[228, 233] [0, 379]

[305, 310] [0, 393]

Pairwise travel time (min)

P 1 2 3 4 5 6

1

2

3

4

5

6

72 –

102 31 –

89 26 26 –

59 38 61 38 –

40 35 65 49 24 –

45 35 63 45 16 8 –

Symmetrical travel times are assumed above. Duration of service session ðTÞ ¼ 400 min. On-site service time (u) = 5 min. Vehicle fixed cost (W) = $7000. Unit travelling cost = $70 per hour = $1 16 per min.

Set of independent single-route customer groups and characteristics Singleroute group

Customer sequence

Finish time at depot (min)

Earliest delivery deadline at depot (min)

Singleroute group

Customer sequence

Finish time at depot (min)

Earliest delivery deadline at depot (min)

1 2 3 4 5 6 7 8 9 10 11 12

2–3–4–5 2–3–4 2–3–5 2–3 2–4–5 2–4 2–5 2 1–4–5 1–4 1–5 1

273 255 273 231 273 255 273 209 273 255 273 194

280 280 280 280 280 280 280 280 281 281 281 281

13 14 15 16 17 18 19 20 21 22 23

3–4–5 3–4 3–5 3 4–5–6 4–5 4–6 4 5–6 5 6

273 255 273 231 355 273 355 255 355 273 355

318 318 318 318 360 360 360 360 379 379 393

Set of independent multi-route customer groups (including the 23 single-route groups) Multi-route group

Customer sequence [single-route groups joined]

Multi-route group

Customer sequence [single-route groups joined]

24 25 26 27 28

2–3–4–5–P–6 [2 + 23] 2–3–P–6 [4 + 23] 2–4–P–6 [6 + 23] 2–P–6 [8 + 23] 1–4–P–6 [10 + 23]

29 30 31 32

1–P–6 [12 + 23] 3–4–P–6 [14 + 23] 3–P–6 [16 + 23] 4–P–6 [20 + 23]

For each multi-route group above, finish time at depot = 355 min and delivery deadline = 393 min. (Same values as for the last route (P–6–P) in each group.)

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Optimal schedule for independent single-route vehicles (Fig. 1) Route 1

Nodes visited P

2

3

4

P

Arrival time (min) vs pickup time window

–

102 2 [102, 107]

133 < 137

180 < 191

255

107 280

142 318

196 360

Departure time (min) Delivery deadline (min) Route 2

P

1

5

P

Arrival time (min) vs pickup time window Departure time (min) Delivery deadline (min)

–

72 < 117 122 281

157 < 228 233 379

273

Route 3

P

6

P

Arrival time (min) vs pickup time window Departure time (min) Delivery deadline (min)

–

45 < 305 310 393

355

Total travelling time = (102 + 26 + 38 + 59) + (72 + 35 + 40) + (45 + 45) = 462 min. Total cost = $7000  3 (vehicles) + $1 16  462 = $21,539. (Assuming latest start from depot for the first trip of each vehichle). Total waiting time = (137  133) + (191  180) + (228  157) = 86 min.

Optimal schedule for independent multi-route vehicles (Fig. 1) Vehicle 1

Arrival time (min) vs pickup time window

Nodes visited P

2

3

4

P

6

P

–

102 2 [102, 107]

133 < 137

180 < 191

255

300 < 305

355

107 280

142 318

196 360

Departure time (min) Delivery deadline (min) Vehicle 2

P

1

5

P

Arrival time (min) vs pickup time window Departure time (min) Delivery deadline (min)

–

72 < 117 122 281

157 < 228 233 379

273

310 393

Total travelling time = 462 min (same as independent single-route vehicles). Total cost = $7000  2 (vehicles) + $1 16  462 = $14,539. (Assuming latest start from depot for the first trip of each vehicle). Total waiting time = (137  133) + (191  180) + (305  300) + (228  157) = 91 min.

Optimal schedule for cooperative single-route vehicles (Fig. 2) Route 1

Arrival time (min) vs pickup time window

Nodes visited P

2

3

4

5

P

–

102 2 [102, 107]

133 < 137

180 < 191

220 < 228

273

107 280

142 318

196 360

233 379

Departure time (min) Delivery deadline (min) Route 2

P

1

5 (transfer to route 1)

6

P

Arrival time (min) vs pickup time window Departure time (min) Delivery deadline (min)

–

72 < 117 122 281

157 (n.a.) 233 –

241 < 305 310 393

355

Total travelling time = (102 + 26 + 38 + 24 + 40) + (72 + 35 + 8 + 45) = 390 min. Total cost = $7000  2 (vehicles) + $1 16  390 = $14,455. (Assuming latest start from depot for the first trip of each vehicle). Total waiting time = (137  133) + (191  180) + (228  220) + (233  157) + (305  241) = 163 min.

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The optimal cooperative solution (Fig. 2) is represented by the following network solution in P

The above solution is also optimal for cooperative multi-route vehicles.

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