Intl. Trans. in Op. Res. 19 (2012) 613–629 DOI: 10.1111/j.1475-3995.2012.00852.x

INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH

Multiobjective multiproduct parcel distribution timetabling: a real-world application Omar Ben-Ayeda,c and Salem Hamzaouib,c a Optimal Solutions Center for Management Science, Wichita, Kansas, USA Institute of Computer Science and Business, University of Kairouan, Kairouan, Tunisia c College of Economics and Business, University of Sfax, Sfax, Tunisia E-mail: [email protected] [Ben-Ayed]; [email protected] [Hamzaoui] b

Received 3 January 2011; received in revised form 11 March 2012; accepted 21 March 2012

Abstract Multiobjective multiproduct parcel distribution timetabling problem is concerned with generating effective timetables for parcel distribution companies that provide interdependent services (products) and have more than one objective. A parcel distribution timetabling problem is inherently multiobjective because of the multitude of criteria that can measure the performance of a timetable. This paper provides the mathematical formulation of the problem and applies the model to a real-world case study. The application shows that without a common ground with the practitioners, it would be impossible to define the actual requirements and objectives of the company; problem definition is as important as model construction and solution method. Keywords: scheduling; multiple objective programming; integer programming; transportation; distribution; shipping industry; postal services; practice of OR

1. Introduction Parcel distribution companies are concerned with collecting parcels from origin customers and delivering them to destination customers. The number of customers and their geographic location determine the size of the distribution company; some domestic companies may be servicing tens or hundreds of customers every day, while international companies such as FedEx and UPS are servicing millions of customers all over the world (Reuters, 2011a, 2011b). Since the volume from one origin customer to one destination customer is typically too small to justify a direct transport between the two customers, the shipments of the same neighborhood (e.g., city) are instead consolidated in a facility called station. In most cases, the number of stations is also too high to allow a direct transport between each pair. The shipments of each group of neighbor stations are usually consolidated in a higher level facility called hub.  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research  Published by Blackwell Publishing, 9600 Garsington Road, Oxford, OX4 2DQ, UK and 350 Main St, Malden, MA 02148, USA.

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The problem of locating parcel distribution facilities is referred to as a parcel distribution network design problem (PDNDP). Most of the works on PDNDP are confined to one level of facilities (Iyer and Ratliff, 1990; Min, 1996; Grunert and Sebastian, 2000; Zapfel and Wasner, 2002; Gunnarsson et al., 2006; Yaman et al., 2007; Lin and Chen, 2008). The few works dealing with the case of multilevel facilities include Wasner and Z¨apfel (2004), Yaman (2009), and Ben-Ayed (2010, 2011 and 2012a). PDNDP is mainly concerned with minimizing the shipping cost, i.e., the cost of sending all shipments from their origin customers to their destination customers. However, in addition to shipping cost, parcel distribution companies are usually facing other major concerns, such as the time of pickup, the time of delivery, and the time elapsing from the pickup of the parcel until its delivery. Since customers are sensitive to both cost and time, the success of a distribution company highly depends on its performance on these two competitive factors. Once the network is designed, the next step is to decide the timing of the movements of vehicles connecting the nodes in order to optimize the pickup and delivery times of the parcels. The problem, referred to as a parcel distribution timetabling problem (PDTP), is inherently a service network design problem (SNDP), but it is not distinctly classified among the components of SNDP (Crainic and Rousseau, 1986; Crainic and Laporte, 1997; Grunert and Sebastian, 2000). Survey on SNDP can be found in Crainic (2000) and Wieberneit (2008). Until 2011, there were no works in the literature on PDTP, except those of Kara and Tansel (2001), Yaman et al. (2007), and Tan and Kara (2007) that explicitly include the decisions concerning the arrival and departure times at each node, in addition to minimizing the time rather than the cost. However, their model has some limitations when applied to parcel distribution because it does not use clock times. Other models, such as parcel hub scheduling problem (McWilliams et al., 2005, 2008; McWilliams, 2009), may appear related to PDTP but are actually unrelated. Ben-Ayed (2012b) formulated PDTP as a nonlinear program, and proposed a solution method that links all decision variables to origin hub departure times and solved the problem by explicitly enumerating the possible departure times from the origin hubs. Hamzaoui and Ben-Ayed (2011) proposed a linear formulation for the basic version of the problem that minimizes the time elapsing between pickup and delivery for all shipments. Their work provides more details on the literature review, the problem definition, and the importance of the model. The present paper, which is an extension of the latter one, emphasizes the multiproduct and multiobjective aspects as well as the real-world application and related practical features. The paper is organized in four sections. The next section provides the mathematical formulation of the model and the following section applies the formulation to a case study. The last section concludes the paper with some suggestions for further research.

2. Mathematical formulation PDTP heavily relies on the output of PDNDP, which assigns a fixed route for each origin–destination (OD) pair of customers. PDTP optimizes the pickup and delivery times for all OD pairs by deciding the departure time of each shipment from each node of the corresponding route.  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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Fig. 1. Movements of a parcel from leaving origin to leaving destination.

2.1. Sets, parameters, and variables The movements of the shipments on a two-level parcel distribution network are depicted in Fig. 1. On its way from origin customer (sender) to destination customer (recipient), each shipment goes through a route composed of six nodes: (1) origin customer (OC) i1 , (2) origin station (OS) j1 , (3) origin hub (OH) k1 , (4) destination hub (DH) k2 , (5) destination station (DS) j2 , and (6) destination customer (DC) i2 . Each customer is assigned to a unique station and each station is assigned to a unique hub; therefore, to each OD pair of customers (i1 , i2 ) corresponds a unique route (i1 , j1 , k1 , k2 , j2 , i2 ). The six arcs (movements) on the route, as shown in Fig. 1, are: (1) (i1 , j1 ), (2) ( j1 , k1 ), (3) (k1 , k2 ), (4) (k2 , j2 ), (5) ( j2 , i2 ), and (6) from i2 to a nonfixed location. When modeling parcel distribution problems, the nodes of the network represent whole cities. What we call customer i actually refers to all the customers of city i; what we call station j is the station hosted by city j; and what we call hub k is the hub hosted by city k. Usually, each medium to large city hosts a station while a small city shares its station with at least another city. In Fig. 2, station 10 services only the customers of city 10 while station 3 services those of cities 3 and 25. The route from the customers of city 10 to those of city 25 is (10, 10, 4, 3, 3, 25); it involves the following operations: (i) shipments of OC 10 are consolidated at OS 10, (ii) shipments of OS 10 are consolidated with those of 4, 9, 11, 15, and 22 at OH 4, (iii) shipments destined to cities 3, 19, 20, and 25 are sorted at OH 4 and transported to DH 3, (iv) shipments destined to cities 3 and 25 are sorted at DH 3 and moved to DS 3, and (v) shipments of city 25 are sorted at DS 3 and delivered to DC 25. Shipments can be sent from any city to any city, including from a city i to itself (from customers in i to other customers in i), with the corresponding route being (i, j, k, k, j, i). The input data include the sets of products, nodes, arcs and routes, in addition to arc traveltimes, node processing-times, and customer time-windows. The decision variables, defined for each product and each arc, include the clock time at which the product starts its trip on the arc and the number of calendar days elapsing between the successive departures from the two nodes of the arc. A calendar day elapses whenever the clock time is reset to zero; one calendar day elapses from 23:59 to 0:00, while none elapses from 0:00 to 23:59. Sets are denoted by upper case Latin characters, parameters are denoted by lower case Greek letters, and variables are denoted by lower case Latin letters. Superscripts C, S, H, O, and D are used to designate customer (or city), station, hub, origin, and destination, respectively. Sets P is the set of products. I is the set of customers (cities). J is the set of stations; J ⊆ I (a station is necessarily located in a city). Let s(i) denote the city hosting the station servicing city i: J = {s(i) : i ∈ I}.  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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Fig. 2. COMP’s network design.

K is the set of hubs; K ⊆ I (a hub is located in a city). Let h( j) denote the city hosting the hub servicing the station j that services city i: K = {h( j) : j ∈ J} = {h(s(i)) : i ∈ I}. LA1 A2 is the set of (directed) arcs corresponding to the movements of the shipment from node type A1 to node type A2 . LA1 A2 is one of the following: LCS = {(i1 , j1 ) : i1 ∈ I; j1 = s(i1 ) ∈ J}, LSH = {( j1 , k1 ) : j1 ∈ J;k1 = h( j1 ) ∈ K}, LCC = {(i2 , i2 ) : i2 ∈ I}, LSC = {( j2 , i2 ) : i2 ∈ I; j2 = s(i2 ) ∈ J}, LHS = {(k2 , j2 ) : j2 ∈ J;k2 = h( j2 ) ∈ K}, or LHH = {(k1 , k2 ) : k1 , k2 ∈ K}. The set LCC corresponds to the departure from DC i2 to a nonfixed location. For the sake of harmony of notation, we assume that the courier stays in the same city; the destination of the courier after delivery is irrelevant to the problem anyway. R is the set of 6-tuples corresponding to all OD routes. Each element ri1 i2 ∈ R corresponds to the route from OC i1 to DC i2 referred to as ri1 i2 = (i1 , s(i1 ), h(s(i1 )), h(s(i2 )), s(i2 ), i2 ) or ri1 i2 = (i1 , j1 , k1 , k2 , j2 , i2 ), with (i1 , j1 ) ∈ LCS , ( j1 , k1 ) ∈ LSH , (k1 , k2 ) ∈ LHH , (k2 , j2 ) ∈ LHS , and ( j2 , i2 ) ∈ LSC .

Parameters A A δ pn11 n22 is the travel time of product p ∈ P along the arc (n1 , n2 ) ∈ LA1 A2 . A A γ pn is the processing time of product p ∈ P at the node type A located in city n. γ pn is one of the OC OS OH DH DS DC following: γ pi1 , γ p j , γ pk , γ pk , γ p j , or γ pi2 . 1 2 2 1 [αi , βi ] is the time window at city i, starting at αi ∈ [0, 24) and ending at βi ∈ [0, 24).  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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Decision variables A A x pn11 n22 is the departure clock-time of product p from n1 to n2 , with p ∈ P, (n1 , n2 ) ∈ LA1 A2 . A A A

A A

y pn11 n22 n33 is the integer number of calendar days elapsing between the two successive departures x pn11 n22 A A

and x pn22 n33 , with p ∈ P, (n1 , n2 ) ∈ LA1 A2 , (n2 , n3 ) ∈ LA2 A3 . If specific products are offered in only specific cities, then to each product p corresponds a specific set of cities I p. In such a case, usually a network is designed for each product p, with specific sets of stations (J p), hubs (K p), arcs (LAB p ), and routes (R p ). Also, time window can be expressed as [α pi , β pi ] when it depends on the product. These cases are not considered in this paper to avoid unnecessary complications.

2.2. Constraints Constraints consist of (i) time precedence, which is the core constraint of the model; (ii) vehicle partial sharing, which handles the interaction between the different products; and (iii) variable domains, including time-window and clock-time restrictions. Time precedence Precedence constraints simply ensure that the movement on an arc starts only after the end of the movement on the preceding arc. These constraints allow counting the number of days elapsing between two successive departures on the route for a given shipment. Let the two departures for a A A A A product p be x pn11 n22 ∈ [0, 24) and x pn22 n33 ∈ [0, 24). The time in hours elapsing between the two events A A

A

must be at least the travel time δ pn11 n22 (to arrive from n1 to n2 ) plus the processing time γ pn22 at n2 (in order to be ready to depart from n2 to n3 ). In other words, the second event whose clock time A A A A A A A is x pn22 n33 should occur after x pn11 n22 by at least δ pn11 n22 + γ pn22 . Referring to the day at which the first A A

A A

A

departure occurred as day 0, the sum x pn11 n22 + δ pn11 n22 + γ pn22 can be interpreted as the time elapsing in hours from midnight of day 0 until the shipment is ready to leave node n2 . On the other hand, A A A A A y pn11 n22 n33 being the number of calendar days elapsing from day 0 until the departure time x pn22 n33 , the A A

A A A

time elapsing in hours is x pn22 n33 + 24y pn11 n22 n33 . Since the time elapsing until the shipment departs has to be at least equal to the time elapsing until it is ready to depart, precedence constraints are written A A A A A A A A A A as x pn22 n33 + 24 y pn11 n22 n33 ≥ x pn11 n22 + δ pn11 n22 + γ pn22 , and more specifically: CSH CS CS OS xSH p j k + 24 y pi j k ≥ x pi1 j1 + δ pi1 j1 + γ p j1

p ∈ P; (i1 , j1 ) ∈ LCS ; ( j1 , k1 ) ∈ LSH

(1a)

SHH SH SH OH xHH pk k + 24 y p j k k ≥ x p j k + δ p j k + γ pk

p ∈ P; ( j1 , k1 ) ∈ LSH ; (k1 , k2 ) ∈ LHH

(1b)

HHS HH HH DH xHS pk j + 24 y pk k j ≥ x pk k + δ pk k + γ pk

p ∈ P; (k1 , k2 ) ∈ LHH ; (k2 , j2 ) ∈ LHS

(1c)

1 1

1 2

2 2

1 1 1

1 1 2

1 2 2

1 1

1 2

1 1

1 2

1

2

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HSC HS HS DS xSC p j2 i2 + 24 y pk j i ≥ x pk j + δ pk j + γ p j2 2 2 2

2 2

p ∈ P;(k2 , j2 ) ∈ LHS ; ( j2 , i2 ) ∈ LSC

2 2

SCC SC SC DC xCC pi2 i2 + 24 y p j2 i2 i2 ≥ x p j2 i2 + δ p j2 i2 + γ pi2

p ∈ P; ( j2 , i2 ) ∈ LSC ; (i2 , i2 ) ∈ LCC .

(1d)

(1e)

Vehicle partial sharing The multiproduct case arises when there is more than one product and interaction between different products. The most common form of this interaction is the partial sharing of vehicles (products share vehicles on some arcs but not on other arcs). Two or more products may share the same vehicle on an arc when the load of one of the products is too small to justify the use of a dedicated vehicle. In the absence of other interactions, if some products share all vehicles they can be treated as a single product, and if they share no vehicles the problem can be solved for each product separately (as single product problem). Vehicle sharing is more frequent on station–hub arcs than on hub–hub arcs; but it is most frequent on customer-station arcs. A A Let Qn11n2 2 ⊂ P be the set of products sharing the same vehicle on the arc (n1 , n2 ) ∈ LA1 A2 ; the A A

A A

A A

cardinality |Qn11n2 2 | (number of elements in the set Qn11n2 2 ) is greater than 1 (1 ≤ |Qn11n2 2 | ≤ |P|). Since the products are being transported on the same vehicle, they must have the same departure time on the corresponding arc: CS xCS p1 i1 j1 = x p2 i1 j1

(i1 , j1 ) ∈ LCS ; p1 , p2 ∈ QCS i1 j1

(2a)

SH xSH p j k = xp j k

( j1 , k1 ) ∈ LSH ; p1 , p2 ∈ QSH j k

(2b)

HH xHH p k k = xp k k

(k1 , k2 ) ∈ LHH ; p1 , p2 ∈ QHH k k

(2c)

(k2 , j2 ) ∈ LHS ; p1 , p2 ∈ QHS k j

(2d)

1 1 1

1 1 2

xHS p k

1 2 j2

2 1 1

2 1 2

= xHS p k

2 2 j2

SC xSC p1 j2 i2 = x p2 j2 i2

1 1

1 2

2 2

( j2 , i2 ) ∈ LSC ; p1 , p2 ∈ QSC j2 i2

(2e)

Variable domains The last group of constraints are related to variable domains including time window constraints, which impose that, for each city i, the start of pickup, the end of pickup, the start of delivery, and the end of delivery occur within the interval [αi , βi ]: OC αi1 ≤ xCS pi1 j1 − γ pi1 ≤ βi1

p ∈ P; (i1 , j1 ) ∈ LCS

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(3a)

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αi1 ≤ xCS pi1 j1 ≤ βi1

p ∈ P; (i1 , j1 ) ∈ LCS

(3b)

DC αi2 ≤ xCC pi2 i2 − γ pi2 ≤ βi2

p ∈ P; i2 ∈ I

(3c)

αi2 ≤ xCC pi2 i2 ≤ βi2

p ∈ P; i2 ∈ I.

(3d)

The other variable domain constraints restrict departure times to be clock times and elapsing days to be integers: SH HH HS SC CC xCS pi1 j1 , x p j k , x pk k , x pk j , x p j2 i2 , x pi2 i2 ∈ [0, 24)

(i1 , j1 , k1 , k2 , j2 , i2 ) ∈ R

(3e)

SHH HHS HSC SCC yCSH pi j k , y p j k k , y pk k j , y pk j i , y p j2 i2 i2 ∈ = {0, 1, 2, ...}

(i1 , j1 , k1 , k2 , j2 , i2 ) ∈ R.

(3f)

1 1

1 1 1

1 1 2

1 2

2 2

1 2 2

2 2 2

2.3. Objectives PDTP is a multiobjective problem involving time criteria and sales criteria. The possible multiobjective optimization approaches for the problem are the weighted sum method and the lexicographic method. It is also possible to combine the two methods. Time criteria Usually the satisfaction of customers increases with the decrease of the number of shipping days; “next-day delivery” is more attractive to customers than “two-day delivery.” However, the number of shipping days depends on the cutoff and commitment times. Cutoff time is the latest time to collect a shipment and process it at the station before the vehicle departs from that station to the corresponding hub; all shipments collected after cutoff are processed the following working day. Commitment time, on the other hand, is the time to be promised to customers; the parcel distribution company is committed to make the delivery by this time. Senders who cannot prepare their shipments before the end of business day do not accept early cutoff, and receiving customers who need the shipments for their business activities may require early commitment; moreover, all customers seek the minimum number of shipping days. Customers are most satisfied when their parcels are picked up at the end of business day, shipped overnight, and delivered at the beginning of next business day. The time performance of a parcel distribution timetable is measured by its ability to allow late cutoffs, make early commitments, and reduce shipping days; we refer to these three criteria as “time criteria.” In terms of optimization, time criteria are interpreted as minimizing the number of shipping days, maximizing the pickup  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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time, and minimizing the delivery time. We define the following three objectives for each product p ∈ P and each route ri1 i2 = (i1 , j1 , k1 , k2 , j2 , i2 ) ∈ R: SHH HHS HSC SCC minYpi1 i2 = yCSH pi j k + y p j k k + y pk k j + y pk j i + y p j2 i2 i2

(4a)

O CS maxXpiO1 = xCS pi1 j1 ⇔ min − X pi1 = −x pi1 j1

(4b)

minXpiD2 = xCC pi2 i2 .

(4c)

1 1 1

1 1 2

1 2 2

2 2 2

To explain the conflicting nature of the three objectives above, let us assume that travel time plus facility processing time on an OD route is 25 h, pickup time (processing time at OC) is 3 h, delivery time (processing time at DC) is 3 h, and time window is [9:00, 17:00]. A next-day delivery would be impossible if pickup was required to end after 13:00 (delivery would end after 17:00, i.e., the following day) or delivery was required to end before 12:00 (delivery would start before 8:00. i.e., the preceding day). Therefore, the number of shipping days may be increased by either a late pickup (conflict between first and second objectives) or an early delivery (conflict between first and third objectives). Moreover, when pickup ends at 11:00 delivery ends at 15:00, and when pickup ends at 13:00 delivery ends at 17:00; in this case any gain in cutoff results in an equivalent loss in commitment (conflict between second and third objectives). Sales criteria In addition to time criteria, parcel distribution timetable is also influenced by “sales criteria” reflecting customer and product priorities. Each of the two sets of products and OD customer pairs are partitioned into several homogeneous groups sorted by their importance (two elements of the same group are equally important). We define a category as a combination of a group of products and a group of OD customer pairs. Let Pq ⊆ P be the qth group of products, and IIm ⊆ I × I be the mth group of customer pairs. Assuming nP product groups and nII customer pair groups, we obtain a total of nP × nII categories (nP ≤ |P| and nII ≤ |I × I|). Each of the three objectives above (4a), (4b), and (4c) is applied to each of the nP nII categories, resulting in the following 3nP nII objectives: 1 = min Uqm





p∈Pq (i1 ,i2 )∈IIm

=



p∈Pq



ri

1 i2

∈R:

Ypi1 i2 

SHH HHS HSC SCC yCSH pi j k + y p j k k + y pk k j + y pk j i + y p j2 i2 i2 1 1 1

1 1 2

1 2 2



2 2 2

(5a)

(i1 ,i2 )∈IIm

2 = min Uqm





p∈Pq

i1 ∈I: (i1 ,i2 )∈IIm

−XpiO1 =





p∈Pq

i1 ∈I: (i1 ,i2 )∈IIm , j1 =s(i1 )

−xCS pi1 j1

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(5b)

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p∈Pq

i2 ∈I: (i1 ,i2 )∈IIm

XpiD2 =





p∈Pq

i2 ∈I: (i1 ,i2 )∈IIm

xCC pi2 i2 .

621

(5c)

Weighted sum method The most common approach to multiobjective optimization is the weighted sum method (Keeney and Raiffa, 1976; Steuer, 1989; Ehrgott and Gandibleux, 2002; Marler and Arora, 2004). The application of this method to PDTP requires the estimation of the weight of each of the three time criteria in addition to the weight of each of the nP nII categories related to sales criteria. Although these weights are typically set by the decision maker, they can also be fairly estimated by the operations research analyst. The estimation of time criteria weights can be based on the fact that the quantity 24Ypi1 i2 − XpiO1 + XpiD2 is the time, in hours, elapsing between the end of pickup and end of delivery (call it shipping time); for example, if pickup ends at 17:00 and delivery ends at 11:00 of the following day, the shipping time is 18 h (7 h from 17:00 to midnight plus 11 h from midnight until 11:00), which can be obtained as 24 × 1 − 17 + 11. This suggests that if the decision maker fails to estimate time criteria weights, we can assume that the objective is to minimize the shipping time, which is equivalent to giving the weights 24, 1, and 1 to the objectives (4a), (4b), and (4c), respectively. Accordingly, we get a single objective for every product p and every OD pair (i1 , i2 ) corresponding to the route ri1 i2 = (i1 , j1 , k1 , k2 , j2 , i2 ):   SHH HHS HSC SCC min 24Ypi1 i2 − XpiO1 + XpiD2 = 24 yCSH pi j k + y p j k k + y pk k j + y pk j i + y p j2 i2 i2 1 1 1

−xCS pi1 j1

+

xCC pi2 i2 .

1 1 2

1 2 2

2 2 2

(6)

The next step is the estimation of the sales criteria weights, which provide measurements of the importance of the different categories. These weights allow the model to generate a schedule with the highest time priority being given to the most important products and most important pairs of customers. Once again, when these weights are not provided by the decision maker, they can be estimated by the operations research analyst. The importance of a product can be estimated by its selling price (a product whose shipping price is $10 per kilogram is 10 times more important than another product whose shipping price is $1 per kilogram); and the importance of an OD pair of customers can be estimated by the volume between the two customers (a pair whose volume is 10 kilograms for a given product is 10 times more important than another pair whose volume is only 1 kilogram for the same product). The selling price and volume are simultaneously expressed by the revenue. The importance of a given product transported on a given OD route is reflected by the revenue generated. From a practical point of view, customers expect shipments between large cities to have better time performance than those between small cities, even when the distance between the large cities is longer than that between the small cities; similarly they expect express (expensive) products to have priority on economy products. Let λ pi1 i2 be the relative revenue generated by the   OD pair (i1 , i2 ) for product p with p∈P (i ,i )∈I×I λ pi1 i2 = 1. The use of the relative revenues as 1 2 weights expands the number of categories to |P × I × I| and therefore allows distinguishing between  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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every two products and every two OD pairs. The objective (6) is extended to minU =

  p∈P i1 ,i2 ∈I

=

  λ pi1 i2 24Ypi1 i2 − XpiO1 + XpiD2

 

∈R 1 i2

    SHH HHS HSC SCC CS CC λ pi1 i2 24 yCSH + y + y + y + y + x − x p j 2 i2 i2 pi1 j1 pi2 i2 . pi j k pj k k pk k j pk j i 1 1 1

1 1 2

1 2 2

2 2 2

p∈P ri

(7)

Lexicographic method The other sensible approach to multiobjective optimization in our case is the lexicographic method (Marler and Arora, 2004). PDTP typically has multiple optima as it is usually possible to find a large number of timetables providing the same objective function value. With the lexicographic method, the objective functions are arranged by order of importance. Let Oe denote the eth priority objective function; Oe is more important than Oe+1 . The following optimization problems are solved one at a time: min Oe subject to:(1a)−(1e), (2a)−(2e), (3a)−(3f), and for e >1:Oh = O∗h , h ∈ {1, 2, . . . , e − 1}. (8) O∗h represents the optimum of hth objective function found at the hth iteration. In a pure lexicographic method, the objective Oe is one of the 3nP nII objectives in (5a), (5b), and (5c): l ; l ∈ {1, 2, 3}, q ∈ {1, 2, . . . , nP }, m ∈ {1, 2, . . . , nII }. Oe = Uqm

(9)

However, it is also possible to combine the lexicographic method with the weighted sum method. We can either apply the weighted sum method to the time criteria and the lexicographic method to the sales criteria, or vice versa. In the first case, Oe is one of the following nP nII objectives:   Oe = Uqm = 24Ypi1 i2 − XpiO1 + XpiD2 ; q ∈ {1, 2, . . . , nP }, m ∈ {1, 2, . . . , nII }. (10) p∈Pq (i1 ,i2 )∈IIm

In the second case, Oe is one of the following three objectives:       U1 = λ pi1 i2 Ypi1 i2 ; U 2 = −λ pi1 i2 XpiO1 ; U 3 = λ pi1 i2 XpiD2 . p∈P i1 ,i2 ∈I

p∈P i1 ,i2 ∈I

p∈P i1 ,i2 ∈I

3. Real-world application The formulation developed in the previous section is applied to a real-world case study.  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

(11)

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3.1. Overview of the company and related data The application company, whose real identity is concealed in order to preserve its private information as desired by its managerial body, is referred to as COMP. This one-thousand-employee company is the exclusive licensee of Federal Express Corporation (FedEx) in a country whose gross domestic product is among the top 30 in the world (International Monetary Fund, 2010; World Bank, 2010). FedEx sets standards for COMP without directly interfering in management. Domestic shipments are pure COMP products; the franchising agreement concerns only international products. All COMP operations are carried out within the borders of COMP’s country. Outbound international shipments are processed by COMP until they are loaded on FedEx carriers, and inbound international shipments are processed by COMP once they are unloaded from FedEx carriers. COMP is heavily relying on expediting to cut shipping times to the level that satisfies its customers and gives it a competitive advantage. Expediting consists of skipping some processing operations (e.g. weighting and manifesting) and over-speeding. Over-speeding is not only allowed but is even required by the company that is committed to pay all speeding tickets. However, new government regulations will soon replace the tickets by more severe sanctions ranging from the suspension of the driver’s license to his imprisonment. These regulations will be reinforced by increasing the number of over-speed detectors that will be spread over the entire road network. The company’s top managers realized that it is time to generate a more effective timetable excluding expediting. This research was sponsored by COMP to achieve this goal. The network design of COMP is the one shown in Fig. 2. The number of cities serviced by COMP is 154. The revenue generated by the largest 25 cities exceeds 97.5% of the total revenue of COMP; it was judged that the remaining 129 cities should not be included in the study as their weight is too small to justify the complexity they introduce. Only the 25 selected cities are shown in the figure. The set of cities is denoted by I = {1, . . . , 25}, with the first 24 nodes being stations J = {1, . . . , 24} and the first five being hubs K = {1, . . . , 5}. COMP offers a variety of parcel and cargo services for both domestic and international shipments. These services are being treated as two products, namely priority and economy: P = {1, 2}, with 1 denoting priority and 2 denoting economy. Both are express but the first product is more sensitive to time while the second one is more sensitive to price. They have dedicated vehicles on some arcs and share the same vehicles on other arcs; this partial sharing of vehicles makes them depend on each other. Two types of data (with and without expediting) were provided by the industrial engineers of COMP in cooperation with the operations managers. It was decided to run the model with the current data (with expediting) in order to compare the performance of the generated timetable with that of the existing one. Since practitioners neither have the time nor the theoretical background to understand the model, the easiest way for them to assess it is through its output. Once the model is approved, the standard data (without expediting) are used to generate the timetable to be implemented. 3.2. Objectives as defined and prioritized by the company When asked to define and prioritize their objectives, COMP’s executive directors emphasized the company’s classification of products in two groups (priority and economy) and cities in three  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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groups. The three largest cities are called group A, the second seven cities are called group B, and the remaining cities are called group C; the set I is thus partitioned into three sets IA = {1, 2, 3}, IB = {4, 5, . . . , 10}, and IC = {11, 12, . . . , 25}. The top objective of COMP is to satisfy the customers of priority products in group A cities, its second objective is to satisfy the customers of economy products in A cities, its third objective is to satisfy the customers of priority products in B cities, its fourth objective is to satisfy the customers of economy products in B cities, its fifth objective is to satisfy the customers of priority products in C cities, and its sixth objective is to satisfy the customers of economy products in C cities. Using our terminology (as defined in Section 2.3), we denote by P1 = {1} the higher priority set of products, by P2 = {2} the lower priority set of products, by IIA = {1, 2, 3} × {1, 2, 3} the highest priority set of city pairs, by IIB = {1, 2, . . . , 10} × {1, 2, . . . , 10}\IIA the medium priority set of city pairs, and by IIC = {1, 2, . . . , 25} × {1, 2, . . . , 25}\IIA \IIB the lowest priority set of city pairs; when a pair has two cities belonging to two different city levels it is assigned to the lower city pair level. The number of city pairs is 3 × 3 = 9 at the highest level, 10 × 10 − 9 = 91 at the middle level, and 25 × 25 − 91 − 9 = 525 at the lowest level. Based on the two sets of cities (A, B, and C) and products (1 and 2), we obtain six (i.e. 3×2) categories. The priorities defined by the company can be restated as: (1) P1 , IIA ; (2) P2 , IIA ; (3) P1 , IIB ; (4) P2 , IIB ; (5) P1 , IIC ; and (6) P2 , IIC . All these priorities are based on sales criteria with no mention of time criteria. The absence of time criteria priorities and weights suggests the use of the lexicographic method as in (8) with the objective function Oe being defined as in (10). The model was run six times. The objective of the first run is       SHH HHS HSC SCC CS CC + y + y + y + y + x minO1 = 24 yCSH − x . (12) p j pi pi i i j i pi j k pj k k pk k j pk j i 2 2 2 1 1 2 2 p∈P1

ri

1 i2

∈R:

1 1 1

1 1 2

1 2 2

2 2 2

(i1 ,i2 )∈IIA

In the second run, p ∈ P2 , (i1 , i2 ) ∈ IIA and the constraint O1 = O∗1 is added:       SHH HHS HSC SCC CS CC minO2 = + y + y + y + y + x 24 yCSH − x p j 2 i2 i2 pi1 j1 pi2 i2 pi j k pj k k pk k j pk j i p∈P2

ri

1 i2

∈R:

1 1 1

1 1 2

1 2 2

2 2 2

(13a)

(i1 ,i2 )∈IIA

O1 = O∗1 .

(13b)

O3 is obtained in a similar way with p ∈ P1 , (i1 , i2 ) ∈ IIB and O1 = O∗1 , O2 = O∗2 . In the fourth run, p ∈ P2 , (i1 , i2 ) ∈ IIB and O1 = O∗1 , O2 = O∗2 , O3 = O∗3 . In the fifth run, p ∈ P1 , (i1 , i2 ) ∈ IIC and O1 = O∗1 , O2 = O∗2 , O3 = O∗3 , O4 = O∗4 . In the sixth run, p ∈ P2 , (i1 , i2 ) ∈ IIC and O1 = O∗1 , O2 = O∗2 , O3 = O∗3 , O4 = O∗4 , and O5 = O∗5 . All the runs involve the constraints (1a)–(1e), (2a)– (2e), and (3a)–(3f) defined in the previous section. R The problem is solved using LINGO 11.0 optimization modeling software run on an Intel TM Pentium 4 CPU 2.4 GHz with 1 GB of RAM. The total runtime is 12.23 min, as shown in Table 1. The obtained solution improves the six objectives, with a rate varying from 1.8% to 33.34% and averaging 12%. Surprisingly, this solution was not satisfactory to COMP’s executive directors, who judged that it has several shortcomings including excessive number of shipping days for some OD pairs and unacceptable early cutoff times for some cities.  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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625

Table 1 Results of the six-objective model: comparison with COMP’s values Objective

O1

O2

O3

O4

O5

O6

COMP’s values Model’s solution Runtime (min) Improvement (%)

18.33 18.00 2 1.80

29.75 19.83 2 33.34

27.35 25.18 2 7.96

35.88 31.33 2.43 12.67

33.74 29.76 1.9 11.80

46.21 44.16 1.9 4.44

In trying to steer the conflict between “theoretical” and “practical” optimality, the authors had to reconsider the three phases of the methodology, namely the definition, the formulation, and the solution of the problem. It was decided to concentrate on the problem definition, first because the model has already been extensively reviewed and tested, and second because there were no unreasonable results. Much effort was made in collecting data about the specific requirements of COMP such as the earliest cutoff time for each city and the maximum number of shipping days for each customer pair. Moreover, since the improvement shown in Table 1 did not mean much to the executive directors, it was mandatory to investigate the objectives to ensure that these objectives reflect what the company really wants.

3.3. Improving objectives’ definition and prioritizing In order to help COMP’s managers better define and prioritize their objectives, the authors asked them to select the better timetable (“i” or “ii”) in each of the following situations: (1) Either (i) timetable with less shipping times between the three largest cities and more shipping times between all other cities, or (ii) timetable with moderate shipping times between all cities. (2) Either (i) timetable with less shipping times for priority products and more shipping times for economy products, or (ii) timetable with moderate shipping times for both priority and economy products. (3) Either (i) timetable with early cutoff; late commitment; and one shipping day, or (ii) timetable with late cutoff; early commitment; and two shipping days. (4) Either (i) timetable with early cutoff and early commitment, or (ii) timetable with late cutoff and late commitment. The answers to the first two questions were quite quick, but the last two ones were not as obvious. The time spent in answering the last two questions was so long that it was not possible to ask any further questions; actually, the fourth question remained unanswered. The obtained information can be summarized as follows:

r The answer to the first question was “i.” It means that the three largest cities are more important to COMP than all other cities combined. This is in line with the fact that these three cities are generating more revenue than all other cities. r The answer to the second question, which is “i,” is also influenced by the fact that the revenue generated by priority products is several times higher than that generated by economy products.  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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r The third question was difficult to answer because COMP has been relying on expediting to guarantee late cutoff, early commitment, and next-day delivery between the three major cities. It was not easy for the executive directors to decide which one of the three priorities has to be given up. However, they finally reached a consensus on the higher importance of the number of shipping days. r The failure to provide an answer to the fourth question can be interpreted as indifference between the two choices; late pickup is as important as early delivery. The answers imply that the most important cities and/or products are actually those generating most of the revenue; in other words, the company’s priorities may be revenue driven. However, when ordering the six categories defined by COMP according to their revenues (e.g., the revenue generated by the category P1 , IIA is (i ,i )∈II λ1i i ), the following ranking is obtained: (1) P1 , IIA ; (2) P2 , IIA ; 1 2 A 1 2 (3) P1 , IIC ; (4) P1 , IIB ; (5) P2 , IIC ; and (6) P2 , IIB . The ranking shows that the first product generates more revenue than the second product for all groups of cities; however it shows also that the cities of group C, which are initially conceived to be the least important by COMP’s executive directors, turn out to generate more revenue than the cities of group B. In other words, the prioritizing initially provided by COMP’s executive directors is partly confirmed and partly disconfirmed by the revenue. It was decided to reformulate the objectives based on the revenue (rather than on COMP’s initial prioritizing). The lexicographic method is used as in (8) with the objective function Oe being defined as in (11). Since cutoff and commitment are equally important to the company and are both less important than the number of shipping days, the problem involves the following two objectives: minO1 =

  p∈P i1 ,i2 ∈I

=

λ pi1 i2 Ypi1 i2   SHH HHS HSC SCC λ pi1 i2 yCSH + y + y + y + y p j 2 i2 i2 pi j k pj k k pk k j pk j i

(14a)

      CS λ pi1 i2 XpiD2 − XpiO1 = λ pi1 i2 xCC − x pi2 i2 pi1 j1 .

(14b)

  p∈P ri

1 i2

∈R:

1 1 1

1 1 2

1 2 2

2 2 2

i1 ,i2 ∈I

minO2 =

  p∈P i1 ,i2 ∈I

p∈P i1 ,i2 ∈I, j1 =s(i1 )

Let O∗1 be the optimal value of O1 under the constraints (1a)–(1e), (2a)–(2e), and (3a)–(3f) and COMP’s specific requirements. The second iteration involves O2 under the same constraints plus the requirement O1 = O∗1 . The number of OD cities is 25 × 25 = 625 at both priority levels. Table 2 shows the results of the two-objective model. The generated timetable improves the number of shipping days by 6.36% but deteriorates the cutoff and commitment by 12.59%; overall it improves the shipping time by 5.61%. This improvement is not as important as the one exhibited by the six-objective model. One possible explanation is the effect of the numerous requirements of the company (they were not part of the previous model). Another possible explanation is the better performance of COMP’s current schedule with regard to the objective functions (14a) and (14b). Fortunately, the timetable generated by this model was highly appreciated by COMP’s executive  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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Table 2 Results of the two-objective model: comparison with COMP’s values Objective

Y

XD–XO

24Y+ X D –X O

COMP’s values Model’s solution Runtime (min) Improvement

1.10 1.03 2 6.36%

–2.86 –2.5 2 –12.59%

23.54 22.22 N/A 5.61%

directors. This confirms that the shortcomings of the six-objective model are caused by the definition rather than the formulation or the solution of the problem.

3.4. Implementation The executive directors approved the model based on the data currently used by their existing timetable. The next step was to generate a new timetable based on the new data (longer travel and processing times) and discuss it with lower level managers (including hub and station managers). The discussions with the people on the ground were much more exhaustive and technical. Everyone was asking specific questions about the parts of the timetable related to the vehicle or facility under his/her direct control. There were plentiful of definite inquiries about every single detail in the solution: “Why does this vehicle leave that late?” “Why does this vehicle leave before connecting with the other regions?” “Why is the departure time of this vehicle different from the previous version?” . . . Some of the inquiries were so challenging that they required a thorough analysis of the solution by tracking shipments or making test runs. The discussions provided a unique opportunity to test the correctness of the solution and validate the model. They also led to significant improvements of the input data and to important extensions of the model. On average a meeting is scheduled every week to get the feedback on the previous version, answer the questions, include new requirements, and correct some input data (the solution often revealed inconsistency in the data). A new version is generated few days later and a new meeting is held the following week. The timetable has been traveling back and forth between the researchers and the practitioners for more than 4 months. In the last 2 months the meetings started to converge to a final version that everybody contributed to and took ownership of.

4. Conclusion This paper is concerned with the multiobjective multiproduct PDTP. The problem consists of generating the timetable that best satisfies the objectives of the parcel distribution company. Such objectives combine time criteria (number of shipping days, pickup time, and delivery time) and sales criteria (products and OD pairs). The mathematical formulation includes time-precedence and variable-domain constraints, which apply to single- and multiproduct formulations, in addition to vehicle-sharing constraints, which are specific to the multiproduct case.  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

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The model is applied to a real-world parcel distribution company facing the challenges of new regulations imposing longer travel times. The application was a valuable experience for the authors to learn how to develop a better understanding with practitioners. A successful operations research application requires both a successful model construction (theoretical part) and a successful problem definition (practical part). Without a common ground with the practitioners, it would be impossible for the analyst to define the actual requirements and objectives. There are several possible extensions for this research. Most important one is the combination of PDNDP with PDTP. Instead of having two problems minimizing sequentially cost and time, we can have a single problem minimizing them simultaneously. The choices made by the PDNDP and imposed to the PDTP are certainly cost effective, but they are not guaranteed to be time effective. The combined PDNDP–PDTP is a multiobjective model that is supposed to tradeoff cost and time to come up with the optimal design and schedule for the parcel distribution company. Another important extension of the model is related to the solution method. Our computational experience suggests that the introduction of valid inequalities may be very effective in reducing the computation time of the PDTP solution; this result needs more investigations that can be the object of a further study. Another path of research would be the development of meta-heuristic algorithms, which are proven to be efficient in solving similar problems.

References Ben-Ayed, O., 2010. Hierarchical parcel distribution facility ground network design problem. International Journal of Logistics Economics and Globalisation 2, 3, 250–269. Ben-Ayed, O., 2011. Re-engineering the inter-facility process of a parcel distribution company to improve the level of performance. International Journal of Logistics Research and Applications: A Leading Journal of Supply Chain Management 14, 2, 97–110. Ben-Ayed, O., 2012a. Parcel distribution network design problem. Operational Research International Journal, DOI: 10.1007/s12351-011-0118-2. Ben-Ayed, O., 2012b. Timetabling hub-and-spoke parcel distribution inter-facility network. International Journal of Advanced Operations Management, forthcoming. Crainic, T.G., 2000. Service network design in freight transportation. European Journal of Operational Research 122, 2, 272–288. Crainic, T.G., Laporte, G., 1997. Planning models for freight transportation. European Journal of Operational Research 97, 3, 409–438. Crainic, T.G., Rousseau, J.-M., 1986. Multicommodity, multimode freight transportation: a general modeling and algorithmic framework for the service network design problem. Transportation Research Part B: Methodological 20, 3, 225–242. Ehrgott, M., Gandibleux, X., 2002. Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys. Springer, Berlin. Grunert, T., Sebastian, H.-J., 2000. Planning models for long-haul operations of postal and express shipment companies. European Journal of Operational Research 122, 2, 289–309. ¨ Gunnarsson, H., Ronnqvist, M., Carlsson, D., 2006. A combined terminal location and ship routing problem. Journal of the Operational Research Society 57, 928–938. Hamzaoui, S., Ben-Ayed, O., 2011. Parcel distribution timetabling problem. Operations Management Research 4, 3–4, 138–149. International Monetary Fund., 2010. World Economic Outlook Database. Available at: http://www.imf.org/external/ pubs/ft/weo/2010/01/weodata/download.aspx (accessed July 14, 2010).  C 2012 The Authors. C 2012 International Federation of Operational Research Societies International Transactions in Operational Research 

O. Ben-Ayed and S. Hamzaoui / Intl. Trans. in Op. Res. 19 (2012) 613–629

629

Iyer, A.V., Ratliff, H.D., 1990. Accumulation point location on tree networks for guaranteed time distribution. Management Science 36, 8, 958–969. Kara, B.Y., Tansel, B.C., 2001. The latest arrival hub location problem. Management Science 47, 10, 1408–1420. Keeney, R.L., Raiffa, H., 1976. Decisions with Multiple Objectives: Preference and Value Trade-offs. John Wiley & Sons, New York. Lin, C.-C., Chen, S.-H., 2008. An integral constrained generalized hub-and-spoke network design problem. Transportation Research Part E: Logistics and Transportation Review 44, 6, 986–1003. Marler, R.T, Arora, J.S., 2004. Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization 26, 6, 369–395. McWilliams, D.L., 2009. A dynamic load-balancing scheme for the parcel hub-scheduling problem. Computers & Industrial Engineering 57, 3, 958–962. McWilliams, D.L., Stanfield, P.M., Geiger, C.D., 2005. The parcel hub scheduling problem: a simulation-based solution approach. Computers & Industrial Engineering 49, 3, 393–412. McWilliams, D.L., Stanfield, P.M., Geiger, C.D., 2008. Minimizing the completion time of the transfer operations in a central parcel consolidation terminal with unequal-batch-size inbound trailers. Computers & Industrial Engineering 54, 4, 709–720. Min, H., 1996. Consolidation terminal location allocation and consolidated routing problems. Journal of Business Logistics 17, 2, 235–263. Reuters., 2011a. Stocks, Company Profile, FedEx Corp. Available at: http://www.reuters.com/finance/stocks/company Profile?symbol=FDX.P (accessed July 31, 2011). Reuters., 2011b. Stocks, Company Profile, UPS Corp. Available at: http://www.reuters.com/finance/stocks/company Profile?symbol=UPS.P (accessed July 31, 2011). Steuer, R.E., 1989. Multiple Criteria Optimization: Theory, Computation, and Application. Krieger Pub Co, Malabar, FL. Tan, P.Z., Kara, B.Y., 2007. A hub covering model for cargo delivery systems. Networks 49, 28–39. Wasner, M., Z¨apfel, G., 2004. An integrated multi-depot hub-location vehicle routing model for network planning of parcel service. International Journal of Production Economics 90, 3, 403–419. Wieberneit, N., 2008. Service network design for freight transportation: a review. OR Spectrum 30, 1, 77–112. World Bank., 2010. Gross domestic product 2009. Available at: http://siteresources.worldbank.org/DATASTATISTICS/ Resources/GDP.pdf (accessed July 14, 2010). Yaman, H., 2009. The hierarchical hub median problem with single assignment. Transportation Research Part B: Methodological 43, 6, 643–658. Yaman, H., Kara, B.Y., Tansel, B.C ¸ ., 2007. The latest arrival hub location problem for cargo delivery systems with stopovers. Transportation Research Part B: Methodological 41, 8, 906–919. Zapfel, G., Wasner, M., 2002. Planning and optimization of hub-and-spoke transportation networks of cooperative third-party logistics providers. International Journal of Production Economics 78, 2, 207–220.

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