Accepted Manuscript

Mathematical formulations for a 1-full-truckload pickup-and-deliveryproblem

Michel Gendreau, Jenny Nossack, Erwin Pesch

PII: S0377-2217(14)00875-3DOI: 10.1016/j.ejor.2014.10.053Reference: EOR 12606

To appear in: European Journal of Operational Research

Received date: 11 September 2012Revised date: 21 October 2014Accepted date: 24 October 2014

Please cite this article as: Michel Gendreau, Jenny Nossack, Erwin Pesch, Mathematical formulationsfor a 1-full-truckload pickup-and-delivery problem, European Journal of Operational Research (2014),doi: 10.1016/j.ejor.2014.10.053

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Mathematical Formulations for a 1-Full-Truckload

Pickup-and-Delivery Problem

Michel Gendreaua, Jenny Nossackb, Erwin Peschb

aCIRRELT and MAGI, Ecole Polytechnique de Montreal, C.P. 6079, succ. Centre-ville, Montreal,

Quebec, Canada H3C 3A7

b Department of Management Information Science, Universitat Siegen, Holderlinstraße 3, 57068 Siegen,

Germany

Abstract

We address a generalization of the asymmetric Traveling Salesman Problem where routes

have to be constructed to satisfy customer requests, which either involve the pickup or

delivery of a single commodity. A vehicle is to be routed such that the demand and the

supply of the customers is satisfied under the objective to minimize the total distance

traveled. The commodities which are collected from the pickup customers can be used

to accommodate the demand of the delivery customers. In this paper, we present three

mathematical formulations for this problem class and apply branch-and-cut algorithms to

optimally solve the model formulations. For two of the models we derive Benders cuts

based on the classical and the generalized Benders decomposition. Finally, we analyze

the different mathematical formulations and associated solution approaches on well-known

data sets from the literature.

Keywords: Pickup and Delivery Problem, Asymmetric Traveling Salesman Problem,

Classical Benders Decomposition, Generalized Benders Decomposition

1 Introduction

The problem studied in this paper is a generalization of the asymmetric Traveling Sales-

man Problem (TSP) in which the set of customers is divided into pickup and delivery

customers and where the former supplies and the latter demands one unit of a single com-

modity. A vehicle is to be routed such that the supply and the demand of the customers is

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satisfied while minimizing the total distance traveled. We refer to this routing problem as

one-commodity Full-Truckload Pickup-and-Delivery Problem (1-FTPDP). The term full-

truckload implies unit capacity and unit supply/demand of the vehicle and the customers,

respectively (Parragh et al., 2008b). The 1-FTPDP belongs to the class of many-to-many

Pickup and Delivery Problems (PDP) where each unit of a pickup customer can be used

to accommodate the demand of any delivery customer (Berbeglia et al., 2007). Besides

the exchange of commodities between customers, the depot has the capacity to fulfill the

customers’ supply and demand. For the sake of simplicity, we assume that the depot

has a sufficient number of commodity available and enough space for commodity storage.

Note that this is a general assumption in the literature (refer, e.g., to Hernandez-Perez

Salazar-Gonzalez (2004a) and Martinovic et al. (2008)).

A real-life application of the 1-FTPDP arises, for example, in the pre- and end-haulage

of intermodal container transportation. Intermodal container transportation denotes the

movement of containers by two or more transportation modes (rail, maritime, and road)

in a single transport chain, where the change of modes is performed at bi- and tri-modal

terminals (Macharis Bontekoning, 2004). The route of intermodal transport is namely

subdivided into the pre-, main-, and end-haulage, denoting the route segments from cus-

tomer to terminal, terminal to terminal, and terminal to customer, respectively. The

main-haulage generally implies the longest traveling distance and is typically carried out

by rail or maritime, whereas the pre- and end-haulage are handled by trucks to enable

door-to-door transports. The transportation assignments that arise in the pre- and end-

haulage are the movements of fully-loaded containers from customers to terminals and vice

versa. In addition, empty containers are considered as transportation resources and are

provided by the carrier for freight transportation. The carrier’s objective is to sequence

the fully-loaded container transportations such that the total traveling cost is minimized.

Furthermore, it is part of the optimization to decide where to deliver the empty contain-

ers released at the receiver customers and where to pickup the empty containers for the

shipper customers. This outlined routing problem can be modeled as a 1-FTPDP, where

each receiver customer is regarded as (empty container) pickup customer and each ship-

per customer as (empty container) delivery customer. For further details on the real-life

application, we refer the reader to Zhang et al. (2010) and Nossack Pesch (2013).

The literature on PDPs is quite extensive. Savelsbergh Sol (1995), Berbeglia et al.

(2007), Parragh et al. (2008a), Parragh et al. (2008b) Pillac et al. (2013), and Lahyani et al.

(2015) provide detailed surveys of the recent literature, as well as classification schemes.

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We follow Berbeglia et al. (2007) by differentiating between many-to-many, one-to-many-

to-one, and one-to-one PDPs. The most frequently encountered PDPs are the ones with

a one-to-one structure, where each commodity has a defined pickup and delivery location.

Problems of this type arise, for example, in courier and door-to-door transportation (refer,

e.g., to Cordeau Laporte (2003)). In problems with a one-to-many-to-one relationship,

commodities are initially located at the depots and are delivered to the delivery customers,

whereas the commodities that are picked up at the pickup customers are destined to the

depots. Real-world applications arise, for example, in the delivery of beverages and the

collection of empty bottles (refer, e.g., to Gendreau et al. (1999)). The 1-FTPDP belongs

to the class of PDPs with a many-to-many dependency where the supply of any pickup

customer can be accommodated by any other delivery customer.

The literature on PDPs with a many-to-many relationship is rather limited and mainly

focuses on the single vehicle case. This problem is denoted in the literature as Pickup-and-

Delivery Traveling Salesman Problem (PDTSP). If the PDTSP is restricted explicitly to a

single commodity, it is referred to as 1-PDTSP. Chalasani Motwani (1999) address a special

case of the 1-PDTSP by considering unit supply/demand of the customers and finite vehicle

capacity. The authors call this problem Q-delivery TSP (Q denotes the vehicle capacity).

They propose a 9.5-approximation algorithm for Q ∈ R+ and a 2-approximation algorithm

for Q = 1 and Q =∞. Anily Bramel (1999) present a (7− 3/Q)-approximation algorithm

for the same problem with Q ∈ R+ and refer to it as Capacitated Traveling Salesman

Problem with Pickups and Deliveries. Hernandez-Perez Salazar-Gonzalez (2004a) develop

a branch-and-cut algorithm using Benders decomposition to optimally solve instances of

the 1-PDTSP. The authors consider real-valued supply/demand and finite vehicle capac-

ity. Wang Lim (2006) propose polynomial time algorithms for the same problem with unit

supply/demand on a path and a tree graph topology. Hernandez-Perez Salazar-Gonzalez

(2004b) suggest two heuristics for the 1-PDTSP with real-valued supply/demand and fi-

nite vehicle capacity. One heuristic is based on a nearest neighbor and a 2-opt/3-opt ap-

proach and the other applies the branch-and-cut algorithm presented in Hernandez-Perez

Salazar-Gonzalez (2004a) on restricted feasible sets. Moreover, Martinovic et al. (2008)

solve instances of the 1-PDTSP by an iterative modified simulated annealing algorithm,

Hernandez-Perez et al. (2009) by a hybrid GRASP/VND heuristic, Zhao et al. (2009) by a

genetic algorithm, and Hosny Mumford (2010) by a VNS/SA approach. The PDTSP with

multiple commodities, unit supply/demand, and unit vehicle capacity has been addressed

by Anily Hassin (1992). The authors propose a 2.5-approximation algorithm for this so-

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called swapping problem. Furthermore, Anily et al. (1999) address the swapping problem

on a line and propose an exact, polynomial time algorithm.

The considered 1-FTPDP is NP-hard. To verify its computational complexity, we refer

the reader to Anily Hassin (1992). They prove the NP-hardness of the swapping problem

by showing that even the simplest problem (namely the 1-FTPDP) is NP-hard.

The key contribution of our work is to present various mathematical formulations for

the 1-FTPDP and to analyze their performances in a computational study. The nature of

the 1-FTPDP points to decomposition methods in which the problem is partitioned into

a routing and an assignment problem. We propose two so-called integrated formulations

that are suited for decomposition and which capture the routing and the assignment struc-

ture of the 1-FTPDP. We apply the classical and the generalized Benders decomposition

(Benders, 1962; Geoffrion, 1972) to these integrated formulations and study their compu-

tational performances. Furthermore, we compare the results to a classical asymmetric TSP

formulation.

The remainder of the paper is organized as follows. A detailed description of the various

mathematical formulations are given in Section 2. Branch-and-cut solution algorithms for

the different mathematical models are described in Section 3. In Section 4, we summarize

the results of our computational study which we conducted on several instances to assess

the computational performance of the algorithms. Finally, we conclude our research in

Section 5.

2 Mathematical Formulations for the 1-FTPDP

In the following, we will present the different model formulations for the 1-FTPDP. The

following notation is used throughout the paper. Let 0 denote the depot, CP = {1, . . . , n1}the set of pickup customers, and CD = {n1 + 1, . . . , n2} the set of delivery customers.

Based on the property that the depot is assumed to provide and receive a sufficient amount

of a given commodity, the depot may either be considered as pickup or as delivery customer.

Hence, to ensure the supply/demand of the customers, we add an appropriate number of

depot duplicates to the set of pickup/delivery customers.

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2.1 Asymmetric TSP Formulation

The 1-FTPDP can simply be solved as a classical asymmetric TSP (refer, e.g., to Dantzig

et al. (1954)). The according model formulation is thereby defined on a digraph G′

=

(V′, A

′), where V

′represents the vertex set and A

′the set of directed edges. V

′consists of

the depot 0, the set of pickup customers CP , and the set of delivery customers CD. Directed

edges (i, j) ∈ A′are defined between any pair of vertices i, j ∈ V ′

with i 6= j and symbolize

vehicle movements. The traveling distance between any two locations (i, j) ∈ A′is denoted

by the edge weight c(i, j) ∈ R+. Note that the traveling distance between two locations

may be different, i. e. c(i, j) 6= c(j, i), and edges that correspond to infeasible vehicle

movements have edge weight∞ and are referred to as infeasible edges. For instance, edges

(i, j) ∈ A′among pickup customers, i.e., i, j ∈ CP , i 6= j, and among delivery customers,

i.e., i, j ∈ CD, i 6= j, are infeasible. We incorporate binary decision variables y′ij ∈ {0, 1}

for each directed edge (i, j) ∈ A′to denote whether (y

′ij = 1) or not (y

′ij = 0) edge (i, j)

is traversed by the vehicle. The TSP formulation is further denoted by P TSP and is given

by the following model. Moreover, let y′:= (y

′ij|i, j ∈ V

′, i 6= j).

min∑

(i,j)∈A′c(i, j) · y′

ij (2.1)

s.t.∑

j∈V ′

j 6=i

y′ji = 1 ∀i ∈ V ′

(2.2)

∑

j∈V ′

j 6=i

y′ij = 1 ∀i ∈ V ′

(2.3)

∑

i,j∈Sj 6=i

y′ij ≤ |S| − 1 ∀S ⊂ V ′

(2.4)

y′ij ∈ {0, 1} ∀i, j ∈ V ′

, i 6= j (2.5)

Objective function (2.1) minimizes the total traveling distance. Constraints (2.2) and

(2.3) ensure that each pickup customer and each delivery customer, as well as the depot

is entered and left exactly once. Constraints (2.4) are the classical subtour elimination

constraints that impose route connectivity. Finally, constraints (2.5) define the domains

of the decision variables.

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2.2 Integrated Formulations

Furthermore, the 1-FTPDP can be considered as an integrated problem that simultane-

ously solves a routing and an assignment problem. It is part of the assignment problem

to fulfill the demand and the supply of the customers by determining where to deliver

the commodities that are collected from the pickup customers and where to pick up the

commodities that are required by the delivery customers. The routing problem evaluates

in which order the pickup customers are visited by taking into account the decisions of the

assignment problem.

The integrated models are defined over a mixed graph G = (V,E,A) where V represents

the vertex set, E the set of undirected edges, and A the set of directed edges. V consists of

the depot 0 and the set of pickup CP and delivery customers CD. Directed edges symbolize

vehicle movements and correspond to the routing problem, whereas undirected edges denote

assignments and relate to the assignment problem. To easily differentiate between directed

and undirected edges, a directed edge is denoted by (i, j) and an undirected edge by [i, j]

for vertices i ∈ V and j ∈ V . Directed edges (i, j) ∈ A are solely defined for the depot

and the pickup customers, i.e., i ∈ {0} ∪ CP , j ∈ {0} ∪ CP , i 6= j, and undirected edges

[i, j] ∈ E simply for the pickup and delivery customers, i.e. i ∈ CP , j ∈ CD. The edge

weight incorporates the traveling distance c(i, j) ∈ R+ that is required to get from vertex

i ∈ V to vertex j ∈ V with i 6= j.

In the following, we will explain in more detail how the 1-FTPDP may be partitioned

into a routing and an assignment problem.

Routing problem: Each delivery customer demands one unit of a given commodity by a

pickup customer. Analogously, each pickup customer supplies one unit of a given commod-

ity to a delivery customer. Thus, a vehicle route starts at the depot, alternates between

pickup and delivery customers, and returns to the depot. Because of this alternating

structure, it is possible to solely define the routing problem in terms of the depot and the

pickup customers. The allocation of the delivery customers to the pickup customers is left

to the assignment problem. A route is said to be feasible, if each pickup customer is visited

exactly once and if the standard TSP constraints – assignment and subtour elimination

constraints (refer, e.g., to Cook (2012)) – are satisfied.

Assignment problem: The assignment problem allocates the delivery customers to the

pickup customers by determining where to deliver the commodities that are released by

the pickup customers and where to pick up the commodities for the delivery customers.

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0

Assignment Problem Routing Problem

D CP

CD

1

2

3

4

5

6

7

8

(a) Example Graph

0

Assignment Problem Routing Problem

D CP

CD

1

2

3

4

5

6

7

8

(b) Routing and Assignment So-lution

0

Assignment Problem Routing Problem

D CP

CD

1

2

3

4

5

6

7

8

(c) 1-FTPDP Solution

Figure 1: Example

A solution to the assignment problem is feasible, if each pickup and delivery customer is

assigned exactly once.

We combine the solutions of the routing and of the assignment problem to a solution of

the 1-FTPDP by simply inserting the delivery customers into the solution of the routing

problem right after the assigned pickup customers.

Example: An example with four pickup and three delivery customers is depicted in

Figure 1(a). A vehicle is initially located at depot 0. We add a depot duplicate to the set of

delivery customers to accommodate the supply of the pickup customers. In Figure 1(b), we

present a feasible solution to the routing, as well as to the assignment problem. A solution

to the routing problem is, e.g., given by (0, 1, 3, 2, 4, 0) and a solution to the assignment

problem, e.g., by (1, 6), (2, 8), (3, 5), and (4, 7). The solutions presented in Figure 1(b) have

to be translated into a solution of the 1-FTPDP: The vehicle starts at depot 0 and visits

pickup customer 1. The commodity that is picked up at pickup customer 1 is used to

accommodate the demand of delivery customer 6. Hence, the vehicle continues to delivery

customer 6. Next, pickup customer 3 is visited and the available commodity is picked up

and delivered to delivery customer 5. From customer 5, we continue to pickup customer 2,

to delivery customer (depot) 8, to pickup customer 4, to delivery customer 7, and end at

depot 0. A feasible 1-FTPDP solution is therefore given by (0, 1, 6, 3, 5, 2, 8, 4, 7, 0) and is

presented in Figure 1(c). Observe that some edges that are illustrated in Figure 1(c) are

not part of the edge set A (e.g., directed edge between 7 and 0) and are simply used to

visualize the solution of the 1-FTPDP.

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2.2.1 Integrated Nonlinear Model PNL

We will base both integrated 1-FTPDP formulations on the asymmetric TSP model (Dantzig

et al., 1954). We thus associate a binary decision variable yij ∈ {0, 1} with each directed

edge (i, j) ∈ A to indicate whether (yij = 1) or not (yij = 0) edge (i, j) is traversed

by the vehicle. For the assignment problem, we include for each edge [i, j] ∈ E a bi-

nary decision variable xij ∈ {0, 1} to denote whether (xij = 1) or not (xij = 0) pickup

customer i ∈ CP is assigned to delivery customer j ∈ CD. The problem can be for-

mulated as the following integer nonlinear programming model, denoted by PNL, with

routing variables y := (yij|i ∈ {0} ∪ CP , j ∈ {0} ∪ CP , i 6= j) and assignment variables

x := (xij|i ∈ CP , j ∈ CD).

min∑

j∈CP

c(0, j) · y0j +∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

(c(i, j) + c(j, k)) · yik · xij (2.6)

s.t.∑

j∈{0}∪CP

j 6=i

yji = 1 ∀i ∈ {0} ∪ CP (2.7)

∑

j∈{0}∪CP

j 6=i

yij = 1 ∀i ∈ {0} ∪ CP (2.8)

∑

i,j∈Sj 6=i

yij ≤ |S| − 1 ∀S ⊆ CP (2.9)

∑

j∈CD

xij = 1 ∀i ∈ CP (2.10)

∑

i∈CP

xij = 1 ∀j ∈ CD (2.11)

xij ∈ {0, 1} ∀i ∈ CP , j ∈ CD (2.12)

yij ∈ {0, 1} ∀i, j ∈ {0} ∪ CP , i 6= j (2.13)

Objective function (2.6) minimizes the total traveling distance of the vehicle obtained by

summing over the traveling distance between the depot and the pickup customers (term 1)

and between the depot and the pickup and delivery customers (term 2). Constraints (2.7)

and (2.8) ensure that each pickup customer and the depot are entered and left exactly once,

respectively. Route connectivity is imposed by the classical subtour elimination constraints

(2.9). Constraints (2.10) and (2.11) ensure that each pickup and each delivery customer

is assigned exactly once. Finally, constraints (2.12) and (2.13) define the domains of the

decision variables x and y.

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2.2.2 Integrated Linear Model PL

So far the objective function (2.6) of the model formulation PNL is nonlinear. To linearize

PNL, we introduce variables zijk ∈ {0, 1} which denote whether (zijk = 1) or not (zijk = 0)

pickup customer i ∈ CP is followed by delivery customer j ∈ CD and pickup customer or

depot k ∈ CP ∪ {0}. The linearization of PNL is denoted by PL and is given below with

variables z := (zijk|i ∈ CP , j ∈ CD, k ∈ CP ∪ {0} , i 6= k).

min∑

j∈CP

c(0, j) · y0j +∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

(c(i, j) + c(j, k)) · zijk (2.14)

s.t. Eq. (2.7)− Eq. (2.9) (2.15)∑

j∈CD

zijk = yik ∀i ∈ CP , k ∈ {0} ∪ CP , i 6= k (2.16)

∑

i∈CP

∑

k∈{0}∪CP

k 6=i

zijk = 1 ∀j ∈ CD (2.17)

zijk ∈ {0, 1} ∀i ∈ CP , j ∈ CD, k ∈ {0} ∪ CP , i 6= k (2.18)

yij ∈ {0, 1} ∀i, j ∈ {0} ∪ CP , i 6= j (2.19)

3 Solution Approaches

In the following, we introduce solution approaches that we apply to the different model

formulations. Due to the fact that each of the models incorporates numerous subtour elim-

ination constraints, we base all of our solution approaches on branch-and-cut algorithms.

Branch-and-cut, in short, integrates ideas of cutting plane methods into the branch-and-

bound algorithm by adding valid inequalities (also called cuts) throughout the branch-and-

bound tree. The inequalities are thereby identified by so-called separation procedures. For

a general overview of the branch-and-cut solution methodology, we refer the reader, e.g.,

to Hoffman Padberg (1985).

3.1 Solution Approach for the Asymmetric TSP Formulation

Instead of dealing with all the subtour elimination constraints (2.4) of P TSP simultaneously,

we solve a restricted version of P TSP by considering a subset of the constraints (2.4). The

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resulting solution is then used to set up a separation problem which either proves global

optimality of the solution to the full problem P TSP or identifies a violated constraint and

adds it to the restricted model formulation. This is a well-known procedure for solving the

TSP and is outlined in more detail subsequently.

We start off with an empty set of subtour elimination constraints, solve the restricted

version of P TSP , and obtain an optimal solution y′. We use the following separation

procedure to separate the subtour elimination constraints: We construct a supporting

digraph G = (V , A) where V = V′

and A ={

(i, j) ∈ A′ |y′ij = 1

}and detect the connected

components of G. If G is disconnected, subtour elimination constraints are violated. In

that case, all connected components are detected, the corresponding subtour elimination

constraints are formulated, are added to the restricted version of P TSP , and P TSP is

resolved. If G is connected, an optimal solution to the full model P TSP has been obtained.

Various different cutting planes and separation procedures have been presented in the

literature for the asymmetric TSP (e.g., CAT and COMB inequalities). We, however, re-

strict ourself to the subtour elimination constraints and refer the reader, e.g., to Applegate

et al. (2006), Cook (2012), and Gutin Punnen (2002) for further notes on the TSP.

3.2 Solution Approaches for the Integrated Formulations

If the routing variables y of the model formulations PNL and PL are fixed, the remaining

assignment problems are efficiently solvable. This property points to a Benders decom-

position approach for generating additional Benders cuts for the integrated formulations.

In the following, we apply the classical and the generalized Benders decomposition to the

model formulations PL and PNL, respectively, and obtain equivalent formulations MPNL

and MPL with fewer variables but with a large number of constraints. These formulations

are solved by branch-and-cut algorithms. In the following, we will explain how Benders

decomposition can be applied to the integrated models.

The classical Benders decomposition was originally proposed by Benders (1962) for

linear programming problems where Benders cuts are derived based on linear programming

duality theory. Classical Benders decomposition has been applied successfully to a variety

of problems, including aircraft routing (Desaulniers et al., 1997; Mercier, 2008; Mercier

et al., 2005; Richardson, 1976), locomotive and car assignment (Cordeau et al., 2000,

2001), and vehicle routing (Bektas, 2012; Fisher Jaikumar, 1981; Hernandez-Perez Salazar-

Gonzalez, 2004a; Sexton Bodin, 1985a,b).

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Geoffrion (1972) extended Benders decomposition to a broader class of problems in which

the model formulation is not restricted to be linear and the constraints are generated by

employing nonlinear convex duality theory. The so-called generalized Benders decomposi-

tion has been applied, among others, to a vehicle routing and inventory allocation problem

(Federgruen Zipkin, 1984), a distribution system planning problem (Benchakroun et al.,

1991), and a capacity expansion planning problem (Bloom, 1983). We will describe in the

subsequent sections how the classical and the generalized Benders decomposition can be

applied to PL and PNL, respectively.

3.2.1 Solution Approach for PL

The classical Benders decomposition was proposed for problems of the form

minx,y

cTx + h(y) s.t. Ax +H(y) = 0,x ∈ X ⊆ Rp,y ∈ Y ⊆ Rq (3.1)

where A is a matrix and c a vector of appropriate size, and h(y) and H(y) are convex

functions in y. The basic idea behind Benders decomposition is to partition problem (3.1)

into two problems, a master problem (linear, nonlinear, or discrete in y) and a subproblem

(linear in x).

We obtain the master problem by reformulating model PL into an equivalent model with

fewer variables but with a large number of constraints. Let Y denote the set of feasible

solutions to the routing problem, defined by constraints (2.15) and (2.19). For any vector

y ∈ Y , model formulation PL reduces to the hereafter defined subproblem which we further

refer to as the primal subproblem.

min∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

(c(i, j) + c(j, k)) · zijk (3.2)

s.t.∑

j∈CD

zijk = yik ∀i ∈ CP , k ∈ {0} ∪ CP , i 6= k (3.3)

∑

i∈CP

∑

k∈{0}∪CP

k 6=i

zijk = 1 ∀j ∈ CD (3.4)

zijk ∈ {0, 1} ∀i ∈ CP , j ∈ CD, k ∈ {0} ∪ CP , i 6= k (3.5)

Recall that the primal subproblem is required to be a linear programming problem:

Problem (3.2)-(3.5) can be considered as a classical transportation problem for which the

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constraint matrix, defined by constraints (3.3)-(3.4), is known to be totally unimodular

(refer, e.g., to Schrijver (2003)). Given that the right-hand-sides of constraints (3.3)-(3.4)

are integer, we can solve the primal subproblem efficiently by solving its LP-relaxation.

Moreover, since the primal subproblem is bounded for any value of y ∈ Y , we can de-

termine the objective function value by solving its dual. Let us introduce dual variables

α=(αik|i ∈ CP , k ∈ {0} ∪ CP , i 6= k

)and β=

(βi|i ∈ CD

)associated with the constraints

(3.3) and (3.4), respectively. The dual of (3.2)-(3.5) results in the following dual subprob-

lem.

min∑

i∈CP

∑

k∈{0}∪CP

k 6=i

αik · yik +∑

j∈CD

βj (3.6)

s.t. αik + βj ≤ c(i, j) + c(j, k) ∀i ∈ CP , j ∈ CD, k ∈ {0} ∪ CP , i 6= k (3.7)

αik ∈ R ∀i ∈ CP , k ∈ {0} ∪ CP , i 6= k (3.8)

βj ∈ R ∀j ∈ CD (3.9)

Now, let F denote the dual feasible region defined by (3.7)-(3.9). Observe that F is

independent of the routing variables y. Thus, any conclusion that we draw for F remains

unaffected by the choice of y ∈ Y .

Properties of F :

• Since the traveling distances are assumed to be non-negative, the zero vector lies in

F , i.e., F is a non-empty set.

• Since F is non-empty, we can follow by the weak duality theorem of linear program-

ming that the primal subproblem is either infeasible or feasible and bounded (refer

to, e.g., Bertsimas Tsitsiklis (1997)).

• Since F is non-empty and contains at least one extreme point, the resolution theorem

of linear programming states, that F can be represented as a convex combination of

the extreme points plus a nonnegative linear combination of the extreme rays (refer

to, e.g., Bertsimas Tsitsiklis (1997)).

By applying this resolution theorem to the dual feasible region F , we can determine

the optimal value of the primal and the dual subproblem for any given value of y ∈ Y by

an auxiliary linear programming problem, which is defined in terms of the extreme rays

and extreme points of F . Since the primal subproblem is feasible for any y ∈ Y , the

corresponding dual subproblem is bounded, i.e., the set of extreme rays is empty. Hence,

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the auxiliary problem is solely defined in terms of the extreme points which we further

denote by F P . The auxiliary problem is then given as follows.

min q (3.10)

s.t.∑

i∈CP

∑

k∈{0}∪CP

k 6=i

αik · yik +∑

j∈CD

βj ≤ q ∀(α,β) ∈ FP (3.11)

q ∈ R (3.12)

Observe that problem (3.10)-(3.12) contains only a single decision variable q and an

enormous number of constraints. Constraints (3.11) are known as Benders optimality

cuts, are defined over the set of extreme points, and ensure that q takes the optimal value

of the primal and the dual subproblem, if an optimal solution exists. We can use this

auxiliary problem to formulate an equivalent model of PL by simply replacing the primal

subproblem by problem formulation (3.10)-(3.12). This reformulation is presented next

and is further referred to as master problem/model MPL.

min∑

j∈CP

c(0, j) · y0j + q (3.13)

s.t. Eq. (2.7)− Eq. (2.9) (3.14)∑

i∈CP

∑

k∈{0}∪CP

k 6=i

αik · yik +∑

j∈CD

βj ≤ q ∀(α,β) ∈ FP (3.15)

yij ∈ {0, 1} ∀i, j ∈ {0} ∪ CP , i 6= j (3.16)

πi ∈ R ∀i ∈ CP (3.17)

Instead of dealing with all the Benders and subtour elimination constraints of MPL

simultaneously, we solve in our branch-and-cut method a restricted master problem to op-

timality by considering only a subset of the Benders optimality cuts and subtour elimination

constraints. According to the TSP solution approach, we use the procedure described in

Section 3.1 to separate the subtour elimination constraints. For the Benders optimality

cuts, we initially start off with an empty set of Benders optimality constraints, i.e., F P = ∅.We solve the restricted version of MPL and obtain an optimal solution (y∗, q∗) with value

w. Thereafter we solve the dual subproblem for y∗ and let (α∗,β∗) denote a dual optimal

solution with optimal value u. If q∗ = u, the solution (y∗, q∗) is also an optimal solution

to the full problem MPL. Otherwise, we generate a new Benders optimality constraint by

setting F P := F P ∪ {(α∗,β∗)} and resolve the restricted version of MPL.

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3.2.2 Solution Approach for PNL

Geoffrion (1972) generalized Benders decomposition to a broader class of optimization

problems by employing nonlinear convex duality theory for the generation of Benders

optimality cuts. Benders decomposition was extended by Geoffrion (1972) to problems of

the form

minx,y

f(x,y) + h(y) s.t. G(x,y) = 0,x ∈ X ⊆ Rp,y ∈ Y ⊆ Rq (3.18)

where f(x,y) is a convex function in x, h(y) a convex function in y, and G(x,y) a

vector of scalar functions linear in x. In contrast to the classical Benders decomposition,

f(x,y) and G(x,y) do not require to be convex in x and y jointly.

Following Hoang (1982), we base our notation on that of Geoffrion (1972) in order to

apply the generalized Benders decomposition theory to our problem PNL. Let

f(x,y) :=∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

(c(i, j) + c(j, k)) · yik · xij ,

h(y) :=∑

j∈CP

c(0, j) · y0j ,

G(x,y) :=

(g1(x,y)

g2(x,y)

):=

((∑

j∈CD xij − 1|i ∈ CP )

(∑

i∈CP xij − 1|j ∈ CD)

),

X :={x | 0 ≤ xij ≤ 1, i ∈ CP , j ∈ CD

}, and

Y := { y | Eq. (2.7)-(2.9); yij ∈ {0, 1} , i, j ∈ {0} ∪ CP , i 6= j;

πi ∈ R, i ∈ CP }.

The key idea of the generalized Benders decomposition approach is to partition problem

(3.18) by projecting it onto the y-space. In the words of Geoffrion (1972), the projection

of (3.18) onto y is given by

minyh(y) + v(y) s.t. y ∈ Y ∩W (3.19)

where

v(y) = minxf(x,y) s.t. G(x,y) = 0,x ∈ X (3.20)

and

W = {y|G(x,y) = 0 for fixed x ∈ X} . (3.21)

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The set W contains those values of y for which problem (3.20) is feasible. The opti-

mization problem in (3.20) is further referred to as primal subproblem and is stated below

in detail for our problem setting.

min∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

(c(i, j) + c(j, k)) · yik · xij (3.22)

s.t.∑

j∈CD

xij = 1 ∀i ∈ CP (3.23)

∑

i∈CP

xij = 1 ∀j ∈ CD (3.24)

xij ∈ {0, 1} ∀i ∈ CP , j ∈ CD (3.25)

Observe that problem (3.22)-(3.25) is equivalent to the classical assignment problem

for a fixed value of y ∈ Y . The corresponding constraint matrix, defined by constraints

(3.23)-(3.24), is known to be totally unimodular (refer, e.g., to Burkard et al. (2009))

which legitimates the relaxation of the integrality conditions on x. Moreover, the primal

subproblem is feasible for any value of y ∈ Y and we can conclude that W := Y . Since X

is a nonempty convex set and v(y) is finite for any fixed value of y ∈ Y , we can solve the

primal subproblem by solving its dual. The dual subproblem is written as

v(y) = max(α,β)

[minx∈X

f(x, y)− (α,β)TG(x, y)

], ∀y ∈ Y (3.26)

where α=(αi ∈ R|i ∈ CP

)and β=

(βi ∈ R|i ∈ CD

)are the Lagrange multipliers as-

sociated with the equality constraints g1(x,y) and g2(x,y), respectively. Refer, e.g., to

Bertsekas (1999) for detailed notes on nonlinear duality theory. We can reformulate prob-

lem (3.19) by replacing the primal subproblem (3.20) by its dual (3.26) and obtain

miny∈Y

h(y) + max(α,β)

[minx∈X

f(x,y)− (α,β)TG(x,y)

]. (3.27)

Finally, by introducing the auxiliary variable q to problem (3.27), we can state the

master problem for PNL. The master model formulation, denoted by MNL, is given by the

following optimization problem.

min∑

j∈CP

c(0, j) · y0j + q (3.28)

s.t. Eq. (2.7)− Eq. (2.9) (3.29)

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minx∈X{∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

c(i, j) · yik · xij+

∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

c(j, k) · yik · xij−

∑

i∈CP

αi · (∑

j∈CD

xij − 1)−

∑

j∈CD

βj · (∑

i∈CP

xij − 1) } ≤ q ∀(α,β) (3.30)

yij ∈ {0, 1} ∀i, j ∈ {0} ∪ CP , i 6= j (3.31)

πi ∈ R ∀i ∈ CP (3.32)

q ∈ R (3.33)

In analogy to the master model ML, formulation MNL employs an enormous number of

constraints. Constraints (3.30) are referred to as Benders optimality cuts and are expressed

in terms of optimization problems, for which a closed-form solution is derived next.

A crucial part of the generalized Benders decomposition is the evaluation of L∗(y,α,β).

For a given optimal Lagrange multiplier vector (αt,βt), a closed-form solution of the

Benders optimality cuts L∗(y,αt,βt) can be constructed as follows. First, the expression

of L∗(y,αt,βt) is simplified by restricting the minimization to the terms that are dependent

on x.

L∗(y,αt,βt) :=∑

i∈CP

αti +

∑

j∈CD

βtj+

minx∈X{∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

(c(i, j) + c(j, k)) · yik · xij−

∑

i∈CP

αti ·∑

j∈CD

xij −∑

j∈CD

βtj ·∑

i∈CP

xij }

An optimal solution x∗ of the minimization term is determined by the following obser-

vation:

x∗ij =

0, c(i, j) +∑

k∈{0}∪CP

k 6=i

c(j, k) · yik − αti − βtj ≥ 0

1, c(i, j) +∑

k∈{0}∪CP

k 6=i

c(j, k) · yik − αti − βtj < 0

∀i ∈ CP , j ∈ CD

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Given that constraints (3.29) enforce exactly one yik equal to 1, we can simplify the

expression of L∗(y,αt,βt,γt) by introducing parameters γtijk as follows

γtijk =

0, c(i, j) + c(j, k)− αti − βtj ≥ 0

c(i, j) + c(j, k)− αti − βtj , c(i, j) + c(j, k)− αt

i − βtj < 0

HALLLLLLLLLLLLLLLLLLLoooooooooooo ∀i ∈ CP , j ∈ CD, k ∈ {0} ∪ CP , i 6= k

and rewrite

L∗(y,αt,βt) :=∑

i∈CP

αti +

∑

j∈CD

βtj +∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

yik · γtijk.

According to the previous models, we solve MNL by a branch-and-cut algorithm by con-

sidering only a subset of the Benders optimality cuts and subtour elimination constraints.

We again use the procedure described in Section 3.1 to separate the subtour elimina-

tion constraints. The Benders optimality cuts are separated according to the approach

explained in Section 3.2.1.

3.2.3 Computational Considerations

We tested the straightforward application of the classical and the generalized Benders de-

composition to the model formulations. Due to their moderate computational performance,

we apply some refinements that improve the quality of the Benders cuts.

Here, the master problems, ML andMNL, deal with only a small fraction of the objective

function. We try to move as much as possible of the objective into the master problems

by providing an estimate of the cost c(i, k) incurred when a pickup customer i ∈ CP is

followed by a pickup customer or depot k ∈ CP ∪ {0}. We can easily determine a lower

bound by setting

c(i, k) = minj∈CD

(c(i, j) + c(j, k)) ∀i ∈ CP , k ∈ {0} ∪ CP , i 6= k.

Hence, the objective functions of both master problems, ML and MNL, reformulate to

∑

j∈CP

c(0, j) · y0j +∑

i∈CP

∑

k∈{0}∪CP

k 6=i

c(i, k) · yik + q.

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Obviously, as one moves some costs from the separation problem into the master prob-

lem, the cost definition in the separation problem needs to be adjusted to avoid double-

counting. The objective function of the primal separation problem (3.2)-(3.5) is changed

to

∑

i∈CP

∑

j∈CD

∑

k∈{0}∪CP

k 6=i

(c(i, j) + c(j, k)− c(i, k)) · zijk.

The objective function of the primal separation problem (3.22)-(3.25) is reformulated

analogously. In addition, the Benders optimality cuts have to be adapted in accordance to

Sections 3.2.1 and 3.2.2 and to the redefined objectives.

4 Computational Results

We base our computational study on the benchmark instances of Christofides et al. (1979)

and Solomon (1987) to analyze the algorithmic performance of the different model formu-

lations and the associated branch-and-cut methods. All algorithms have been implemented

in Java under Windows 7 and were run on an Intel Pentium Core 2 Duo, 2.2 GHz PC,

with 4 GB system memory. We use ILOG CPLEX 12.5 Concert technology with standard

settings as branch-and-cut solver for the TSP model P TSP and for the integrated mod-

els MPL and MPNL. Note that subtour elimination constraints and Benders optimality

cuts are only generated if an integer solution has been detected in the course of CPLEX’s

branch-and-cut procedure.

Our data set is based on eight instances derived from the literature, five instances from

Christofides et al. (1979) and three instances from Solomon (1987). The benchmark set of

Christofides et al. (1979) was originally proposed for the Vehicle Routing Problem (VRP)

and consists of 14 instances. Each of these instances contains between 50 to 199 customers

and the customer locations are either generated randomly (instances 1-10) or cluster-based

(instances 11-14). Notice that we exclude instances 4-10 and 13-14 from our computational

study. In our context, they are equivalent to instances 1-3 and 11-12 and either extend

the customer set or impose additional route length constraints which we do not consider

in the 1-FTPDP. Hence, we restrict our study to instances 1, 2, 3, 11, 12 and further refer

to them as C1, C2, C3, C11, C12, respectively. Solomon’s benchmark set was initially

generated for the Vehicle Routing Problem with Time Windows (VRPTW) and consists

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of three instance sets (R, C, and RC) where each set is again subdivided into two subsets

(R=(R1, R2), C=(C1, C2), RC=(RC1, RC2)). Each instance contains 100 customers

and the customer locations are either generated randomly (R), cluster-based (C), or both

randomly and cluster-based (RC). Note that all instances in the instance set R (C, or

RC) use the same customer locations and solely differ in the time windows. Due to these

redundancies, we solely consider a single instance from each instance set R, C, and RC.

We will further refer to these instances as SR, SC, and SRC, respectively.

We adjust the eight instances (C1, C2, C3, C11, C12, SR, SC, SRC) for the 1-FTPDP in

the following way: First, we discard the customer demands and the time windows. There-

after, we randomly decide whether a customer is regarded as pickup or delivery customer.

The traveling distance between any two locations is then determined by the Euclidean dis-

tance plus a service time and is rounded to the nearest integer value. The service time is

thereby randomly drawn from the interval [1, 5] for each pickup-delivery/depot connection

and drawn from the interval [5, 10] for each delivery-pickup/depot connection. For each of

the eight instances, we randomly derive 20 new instances in the same manner and end up

with 8× 20 = 160 instances.

We compare the algorithmic performance of the different model formulations and asso-

ciated branch-and-cut algorithms in terms of the number of generated subtour elimination

constraints and Benders optimality cuts, as well as the required CPU time. The results

are listed below in Table 1. For each instance set, we indicate the underlying data set

(“Data”), the number of pickup (“#PC”) and delivery customers (“#DC”), as well as the

objective function value (“Obj.”) averaged over all respective instances. For each algo-

rithm and each set of instances, we list the average CPU (“CPU”) time (in seconds), the

average number of generated Benders optimality cuts (“#BC”), and the average number of

generated subtour elimination constraints (“#SC”). We indicate by table entry “-”, if a set

of instances has not been solved to optimality in the time frame of one hour. Recall that we

apply the classical and the generalized Benders decomposition approach according to the

description in Section 3.2.1 and 3.2.2 and the algorithmic refinements presented in 3.2.3.

Further note that we obtain the values of the dual variables for the Benders optimality

cuts by solving the primal subproblems.

It can be seen from the data in Table 1 that the classical Benders decomposition out-

performs the generalized Benders approach in terms of the CPU time and the number

of generated Benders optimality cuts and the number of subtour elimination constraints.

Unfortunately, the generalized approach does not solve all instance sets in the defined time

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Table 1: Comp. Results on the TSP and the Integrated Formulations

Data #PC #DC Obj.MPNL MPL PTSP

#BC #SC CPU #BC #SC CPU #SC CPU

C1 5 5 260.85 8.15 3.90 0.10 3.00 2.20 0.05 2.50 0.01C1 10 6 402.35 243.10 45.10 28.68 5.80 4.15 0.22 2.40 0.05C1 10 10 428.30 164.10 40.45 5.08 8.10 4.95 0.24 4.55 0.10C1 15 19 705.05 - - - 22.30 17.10 3.49 5.70 0.29C1 18 18 674.80 - - - 29.35 18.40 3.68 7.10 0.38

C2 5 5 259.80 6.15 3.00 0.08 2.65 2.00 0.05 1.75 0.03C2 10 6 398.00 110.45 32.10 3.83 3.91 3.85 0.17 3.35 0.06C2 10 10 428.55 452.15 61.00 80.51 7.65 5.30 0.23 4.30 0.09C2 15 19 698.75 - - - 29.00 20.25 4.10 5.65 0.25C2 18 18 659.50 - - - 32.25 19.90 4.39 6.95 0.32

C3 5 5 281.25 8.25 3.70 0.09 3.30 2.40 0.06 2.60 0.03C3 10 6 434.50 145.35 35.75 6.03 5.35 4.50 0.18 3.40 0.06C3 10 10 458.90 162.30 41.85 8.17 6.55 4.90 0.20 4.20 0.07C3 15 19 760.10 - - - 25.80 20.45 3.96 6.75 0.33C3 18 18 735.70 - - - 27.10 18.05 3.38 6.00 0.26

C11 5 5 194.50 5.75 2.90 0.07 2.75 1.55 0.02 2.30 0.03C11 10 6 613.95 - - - 5.30 4.60 0.14 3.75 0.06C11 10 10 452.00 1275.60 213.90 175.47 9.25 6.45 0.11 3.45 0.08C11 15 19 965.85 - - - 28.85 20.10 4.94 6.65 0.29C11 18 18 727.80 - - - 25.00 15.80 3.75 6.65 0.25

C12 5 5 131.55 4.30 2.70 0.03 2.15 1.55 0.05 2.00 0.03C12 10 6 343.25 - - - 5.85 5.10 0.22 3.50 0.05C12 10 10 286.40 333.35 69.80 23.24 6.50 4.70 0.19 3.85 0.06C12 15 19 620.00 - - - 24.50 18.35 4.06 6.60 0.31C12 18 18 563.75 - - - 29.30 16.30 4.33 6.65 0.33

SR 5 5 272.95 9.20 3.40 0.09 2.60 2.20 0.04 2.10 0.03SR 10 6 434.40 234.30 51.35 14.37 5.10 3.85 0.16 2.40 0.05SR 10 10 467.65 280.15 58.60 15.04 9.15 5.75 0.26 4.95 0.09SR 20 20 803.95 - - - 42.95 25.60 7.78 8.05 0.41SR 24 24 927.05 - - - 57.75 34.60 23.09 7.65 0.69

SC 5 5 203.20 8.75 3.60 0.13 2.50 1.70 0.05 2.50 0.03SC 10 6 394.35 - - - 5.30 4.45 0.20 3.00 0.06SC 10 10 360.30 - - - 7.70 5.25 0.25 4.05 0.06SC 20 20 752.95 - - - 43.45 24.90 8.75 7.35 0.38SC 24 24 898.50 - - - 75.70 42.00 41.14 9.60 0.73

SRC 5 5 330.75 15.20 5.55 0.14 2.70 2.05 0.04 1.75 0.03SRC 10 6 576.05 - - - 5.10 6.45 0.25 3.30 0.06SRC 10 10 684.90 - - - 8.40 5.90 0.25 3.65 0.06SRC 20 20 1033.35 - - - 40.75 19.45 9.53 7.20 0.37SRC 24 24 1103.50 - - - 62.05 27.75 27.60 7.05 0.68

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frame. It is known from the literature, e.g., refer to Geoffrion Graves (1974) and Mag-

nanti Wong (1981), that the model formulation has a remarkable impact on the quality

of the Benders cuts. The linearization of PNL leads to higher qualified Benders optimal-

ity cuts. Furthermore, the generalized Benders formulations has more difficulties solving

cluster-based instances than randomized instances.

The TSP formulation outperforms both Benders decomposition on all instance sets.

It requires by far the lowest number of subtour elimination constraints and the shortest

CPU times. Compared to both integrated formulations, the TSP formulation is capable of

solving much larger problem instances to optimality. This holds for clustered, as well as

for randomized instances. Summing up, the results show that the TSP formulation shows

the best performance in terms of the number of generated subtour elimination constraints

and the computational times.

5 Conclusion

We address in this research a generalization of the Traveling Salesman Problem, namely

the one-commodity Full-Truckload Pickup-and-Delivery Problem. We present three model

formulations for the 1-FTPDP, a TSP formulation and two integrated formulations. The

integrated formulations are suited for decomposition algorithms and capture the routing

and the assignment structure of the 1-FTPDP. We apply to these model formulations the

classical and the generalized Benders decomposition and solve the resulting formulations

by branch-and-cut algorithms.

We observe in a computational study that the classical Benders decomposition outper-

forms the generalized Benders decomposition on all of the considered instance sets. The

classical Benders decomposition approach computes higher qualified Benders optimality

cuts than the generalized approach. However, the TSP formulation outperforms both

integrated formulations and is capable of solving much bigger instances in less CPU times.

There are two major directions in which to proceed for future research. On the algo-

rithmic side, our implementation of Benders decomposition shows room for improvements.

Accelerating techniques, such as the improvement of Benders cuts by introducing Pareto-

optimal cuts (Magnanti Wong, 1981), may further improve the performance of the algo-

rithms. Moreover, extensions of the 1-FTPDP are further interesting research topics. The

presented model formulations and decomposition approaches can easily be extended to

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multiple vehicles by replacing the reduced TSP by a VRP formulation. Other interesting

variants generalize the 1-FTPDP to multi-commodities or consider customer time-windows

and service times.

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