Core-based cost allocation in the cooperative traveling salesman problem

Accepted Manuscript

Core-Based Cost Allocation in the Cooperative Traveling Salesman Problem A. Kimms, I. Kozeletskyi PII: DOI: Reference:

S0377-2217(15)00719-5 10.1016/j.ejor.2015.08.002 EOR 13153

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

22 June 2014 23 January 2015 5 August 2015

Please cite this article as: A. Kimms, I. Kozeletskyi, Core-Based Cost Allocation in the Cooperative Traveling Salesman Problem, European Journal of Operational Research (2015), doi: 10.1016/j.ejor.2015.08.002

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Highlights • A cooperative TSP model formulation is provided. • Release dates are taken into account. • A cost allocation for the salesmen is defined by means of a cooperative game theory solution concept, i.e. the (least) core.

AC

CE

PT

ED

M

AN US

CR IP T

• Mathematical programming is used to compute a (least) core element.

1

ACCEPTED MANUSCRIPT

Einführung •

Gegenstand: AMPL/CPLEX

CR IP T

– Modellierung und Lösung linearer Optimierungsmodelle Core-Based Cost Allocation in the Cooperative Traveling

•

Ablauf:

Salesman Problem

AN US

– einführende Vorlesung (heute)

A. Kimms and I. Kozeletskyi – Übungen: Aufgaben (5 Serien) auf unserer Homepage

• selbstständige Bearbeitung

Leistungsnachweis:

ED

•

M

• Abgabe per E-Mail January 2015

PT

– durch regelmäßige Abgabe durchdachter Lösungen

CE

2007-10-24 Address of correspondence: Softwarepraktikum Teil 1 Prof. Dr. Alf Kimms and Igor Kozeletskyi Chair of Logistics and Operations Research Mercator School of Management

AC

University of Duisburg–Essen Lotharstr. 65 · 47057 Duisburg · Germany email: alf.kimms@uni–due.de

email: igor.kozeletskyi@uni–due.de URL: http://www.msm.uni–due.de/log/

1

ACCEPTED MANUSCRIPT

Core-Based Cost Allocation in the Cooperative Traveling Salesman Problem Abstract This paper presents a cost allocation scheme for a horizontal cooperation of traveling salesmen that is implemented a priori and provides expected costs for the

CR IP T

coalition members. The cost allocation is determined using the core concept. To compute the value of the characteristic function over the whole planning horizon the TSP with release dates combined with simulation is used. The developed core computation algorithm, based on mathematical programming techniques, provides a core element or, in case of an empty core, a least-core element. To decrease

AN US

the computational effort of core computation a row generation procedure is implemented. The developed computation study tests the computational performance of the solution procedure.

AC

CE

PT

ED

M

Keywords: traveling salesman problem, cooperative game theory, core, cost allocation

1

ACCEPTED MANUSCRIPT

1

Introduction

For logistic service providers horizontal cooperation has a variety of potentials. It offers increase of profitability and competitiveness, capacity optimization, cost reduction and network improvement. The issue of profit distribution among the partners plays the crucial role on a way to the establishment of cooperation. As studies show (see Muir 2009,

CR IP T

2011) there are multiple barriers on the way to practical implementation of horizontal cooperation among logistic service providers, and the lack of profit sharing models belongs to the common ones.

In this paper we develop a cost allocation scheme for a horizontal cooperation of traveling

AN US

salesmen using cooperative game theory. This cost allocation scheme can be implemented before the actual planning horizon (or before the establishment of cooperation agreement) and provides expected costs for coalition members, so they can value their savings and decide if they want to be a part of the partnership. Cooperative game theory offers several

M

well-known allocation concepts: e.g. the core (introduced by Gilles (1959)), the Shapley

ED

value (proposed by Shapley (1953)) and the nucleolus (developed by Schmiedler (1969)). In a previous study (see Kimms and Kozeletskyi (2013)) the cost allocation for the coop-

PT

erative traveling salesman problem was determined using the Shapley value. This paper focuses on the core concept and presents another modeling approach than in Kimms and

CE

Kozeletskyi (2013) where dynamic programming was used. Dynamic programming can be efficiently used for this problem only in combination with approximate dynamic pro-

AC

gramming techniques that give only an approximation of the total costs. This works well for the Shapley value, however for the core, where stability constraints should be proved (see Section 4 for a definition), approximation may lead to the distortion of results. For that reason we developed the mathematical programming model for computation of total costs, that is solved optimally.

This paper has the following structure. In the next section we give a brief literature re2

ACCEPTED MANUSCRIPT view on horizontal cooperation in transportation problems. In Section 3 the optimization model and the resulting exact solution procedure for the cooperative TSP are formulated. Section 4 presents the core computation algorithm for this problem. In Section 5 we show results of our computational study for the performance and the solution quality of the

2

CR IP T

developed algorithm. Section 6 concludes this paper.

Literature Review

The literature on horizontal partnerships in transportation planing focuses mostly on

AN US

categorizing the cooperation forms and empirical studies of possible benefits and barriers. Thus quantitative planing models and especially game-theoretic approaches are barely studied. Cruijssen et al. (2007a) conducted a survey on potential benefits from the horizontal cooperation among logistic service providers in Flanders. An extension of this study was presented in Cruijssen et al. (2010b), where the efficiency of cooperating

M

and non-cooperating road companies is compared. Cruijssen et al. (2010a) presented

ED

a supplier-initiated outsourcing of transportation requests and proposed an allocation approach for resulting savings. Verstrepen et al. (2009) developed a framework for clas-

PT

sification of horizontal cooperation in logistics. Muir (2009, 2011) presented the survey results on identification of drivers, benefits and barriers for horizontal cooperation among

CE

European and North American logistic companies. Cruijssen et al. (2007b) provided a broad literature review on impediments and drivers in horizontal cooperation in transport

AC

and logistics.

An efficient cooperation in transportation planing requires first of all decision support models for planing purposes. Krajewska and Kopfer (2006) developed a general framework for the collaborative transportation planning model and designed a request exchange mechanism based on combinatorial auctions. Wang and Kopfer (2014) proposed a routebased combinatorial auction for the collaborative transportation planning for less-then3

ACCEPTED MANUSCRIPT truckload transportation requests. Wang et al. (2014) studied operational planning of freight forwarders taking into account subcontracting and collaborative planning. Berger and Bierwirth (2010) proposed procedures for reassignment of transportation requests ¨ with limited information transfer. Ozener et al. (2011) studied exchange mechanisms for various scenarios with different information sharing concepts. Nadarajah and Bookbinder (2013) employed guided local search for developing a solution procedure for collaborative

CR IP T

vehicle routing problems with time windows. Adenso-Diaz et al. (2014a) presented a model for connecting deliveries of different transportation companies. In Adenso-Diaz et al. (2014b) a question of predicting possible savings from cooperations is studied.

AN US

Profit-sharing approaches among freight carriers using the Shapley value were discussed in Krajewska et al. (2008). Frisk et al. (2010) presented the Equal Profit Method in the context of of collaborative transport planing, however without proposing any solution procedure that solves the problem of exponential number of constraints in this method.

M

D’Amours and R¨onnqvist (2010) discussed issues on coalition building and sharing resources and benefits in collaborative logistics. Audy et al. (2012) presented a case study

ED

on cost savings among transportation companies from forest industry. Lozano et al. (2013) compared different cooperative game solution concepts for cost allocation on the

PT

example of full-truck-load collaborative planning. Guajardo and R¨onnqvist (2015) stud-

CE

ied coalitional structure and resulting cost allocations in collaborative logistics.

As our study focuses on the cooperation of salesmen and not on the cooperation of cus-

AC

tomers, this makes a crucial difference to the so-called routing games (introduced by Fishburn and Polak (1983) on a TSP example) as they focus on the cooperation of customers in routing problems. This type of cooperative games was also studied by Potters et al. (1992) and extended by G¨othe-Lundgren et al. (1996) and Engevall et al. (2004) to the cooperation of customers in vehicle routing problems.

4

ACCEPTED MANUSCRIPT

3

A Cooperative Traveling Salesman Optimization Problem with Release Dates

In this section we present an optimization model that computes total traveling cost for a given coalition in a cooperative case for the estimated demand outcome over the entire planing horizon. The optimization model is a variation of the multiple traveling sales-

CR IP T

man problem (mTSP) with multiple depots, that takes account of earliest arrival time for every customer (i.e. release dates). We call this problem cooperative traveling salesman problem with release dates (CTSPRD). In addition an optimal solution procedure for this

AN US

problem is introduced.

The cooperation in this approach is defined through the joint planning of routes with the objective of minimizing the total traveling costs of a coalition. It means that in a coalition any salesman can visit any customer from this coalition. Furthermore our model

M

considers cooperative planning that is realized on a short-time basis. As an example, this kind of cooperative planning defined in terms of TSP can be found at companies

ED

that perform a service at the customer’s site (e.g. maintenance, plumber etc.). With such operations capacity restrictions of vehicles (as in VRPs) are not relevant and the

PT

capacity aspect is implied through the working hours (i.e. length of the tour in time units).

CE

Let N = {1, . . . , n} be a set of salesmen with depots {01 , . . . , 0n } and let T = {0, . . . , τ } be our planning horizon divided into periods 0, 1, . . . , τ . With every coalition S ⊆ N, S 6= ∅

AC

we associate the set of customers V (S). We assume that customer sets are disjoint: ∀k, l ∈ N, k 6= l : V ({k}) ∩ V ({l}) = ∅ and for two non-empty coalitions S, T ⊆ N we put that V (S ∪ T ) = V (S) ∪ V (T ). With every customer i from V (S) we associate a time period di ∈ T , when his order is placed. In an a priori consideration di is the result of the simulation procedure. Details on the simulation are given in Section 5. The problem can be represented as a directed graph G = (V (N ), A, C, T ) with the set of arcs A and

5

ACCEPTED MANUSCRIPT the matrix of traveling costs C = (cij ). Furthermore we define for every arc (i, j) ∈ A traveling time tij and the maximal length for every tour tmax .

Figure 1 illustrates the problem description for an example with three salesmen and six customer nodes. The particular demand realization d in this example indicates the earliest period when the customer can be visited. For instance customers v1 and v3 can be planed

CR IP T

in the initial period 0 while other customers appear during the periods 1 to 3. 02 v1

N = {1, 2, 3}

V ({1}) = {v1 , v2 }

V ({2}) = {v3 , v4 }

v6 v4

v5

AN US

V ({3}) = {v5 , v6 }

v3

T = {0, . . . , 3}

d = (0, 2, 0, 1, 3, 2)

v2

01

M

03

Figure 1: Problem Illustration for |N | = 3 and |V (N )| = 6

ED

For the formulation of the optimization model we define two types of decision variables. The binary decision variable xijk :

PT

   1 if the arc (i, j) is used by the salesman k

CE

xijk =

  0 otherwise

AC

identifies the assignment of customers to salesmen and cost minimal routes in the optimal solution. The real-valued decision variable δi indicates the arrival time in node i and is also in combination with a large number M part of subtour elimination constraints. The problem formulation for coalition S ⊆ N, S 6= ∅ is given as follows: minimize F 1 =

X

X

X

k∈N i∈V (S)∪{0k } j∈V (S)∪{0k }

6

cij xijk

(1)

ACCEPTED MANUSCRIPT s.t.

X

xijk = 1

i ∈ V (S),

(2)

xijk = 1

j ∈ V (S),

(3)

k∈S j∈V (S)∪{0k }

X

X

k∈S i∈V (S)∪{0k }

X

X

i ∈ V (S), k, l ∈ S, k 6= l,

(4)

δj + M ≥ δi + tij + M xijk

i ∈ V (S), j ∈ V (S), j 6= i, k ∈ S,

(5)

δi ≥ d i

i ∈ V (S),

(6)

j∈V (S)∪{0k }

j∈V (S)∪{0l }

AN US

xijk +

xjil ≤ 1

CR IP T

X

δi ≤ tmax xijk ∈ {0, 1}

i ∈ V (S),

(7)

i, j ∈ V (S), k ∈ S.

(8)

M

The optimal value of the objective function (1) corresponds to the total traveling cost for the coalition S. Constraints (2) and (3) are standard TSP constraints, which ensure

ED

that every node will be visited exactly once. Constraints (4) prevent that every salesman k cannot switch his affiliation to a different depot during his tour and therefore must

PT

return to his own depot 0k . Without these constraints the following situation can occur. Consider an example with |N | = 2, V (N ) = {1, 2}, d1 = d2 = 0, tmax = 10 and traveling

AC

CE

times are set to 1. The illustration with traveling costs is presented in Figure 2. 1

1

2

02

01 3

1

1

1

Figure 2: Illustration of Constraints (4)

Without constraints (4) the following assignment for variables xijl is feasible: x01 ,1,1 = 1, x1,02 ,2 = 1, x02 ,2,2 = 1, x2,01 ,1 = 1 as it satisfies all remaining constraints. This assignment means that the salesman 1 ends his tour in the depot 02 and the salesman 2 in the depot 7

ACCEPTED MANUSCRIPT 01 . Constraints (4) allows to prevent this kind of solutions. Constraints (5) are subtour elimination constraints, that also set an arrival time in every node. They are formulated as an extension of Miller-Tucker-Zemlin constraints (see Miller et al. 1960). For alternative definitions of subtour elimination constraints we refer to Punnen (2002). The earliest possible arrival time for every node i is stated in constrains (6) and the tour length is bounded through the constraints (7). The maximal length of the tour tmax represents a

CR IP T

capacity restriction in our model. In combination with this restriction tij can be seen not only as a traveling time from i to j but also as a traveling time that includes a service time at the customer j.

AN US

As the problem (1) - (8) is an NP-hard due to the NP-hardness of the classical TSP, we propose an optimal solution procedure that transforms this problem into the TSP with one salesman to reduce the computational effort for solving this problem. The proposed solution procedure is a modification of the transformation procedure proposed by Laporte

M

et al. (1988). This procedure provided good results for the cooperative TSP in a previous

ED

study (see Kimms and Kozeletskyi 2013).

The idea of this procedure is the transformation of graph G into a new graph (we name it

PT

ˇ where every depot 0k is depicted through two nodes 01 and 02 and a dummy node 0 G), k k is inserted. The new set of nodes for a coalition S is Vˇ (S) = 0 ∪ V (S) ∪ (∪k∈S ∪2i=1 {0ik }).

CE

The new set of arcs Aˇ includes the following types of arcs:

AC

• A1 - set of arcs from the dummy node to the first node of each depot 01k . • A2 - set of arcs from the last node of each depot 02k to the dummy node 0. • A3 - set of arcs from the the first node of each depot to the customers. • A4 - set of arcs from customers to the last node of each depot. • A5 - set of arcs from the last node of each depot to the first node of every other depot. 8

ACCEPTED MANUSCRIPT • A6 - set of arcs from each customer to every other customer. • A7 - set of loops for the depot nodes. This transformation requires also redefinition of traveling costs and traveling times. The new traveling costs are:

   ci,0    k     c ij

(i, j) ∈ A3 , i = 01k for some k ∈ S,

(i, j) ∈ A4 , j = 02k for some k ∈ S, (i, j) ∈ A6 .

And the new traveling times are:

(i, j) ∈ A3 , i = 01k for some k ∈ S,

M

ti,0k        −M        tij

(i, j) ∈ A1 ∪ A2 ∪ A7 ,

(i, j) ∈ A4 , j = 02k for some k ∈ S, (i, j) ∈ A5 ,

ED

tˇij =

    0         t0k ,j    

CR IP T

0k ,j

(i, j) ∈ A1 ∪ A2 ∪ A5 ∪ A7 ,

AN US

cˇij =

   0        c

PT

(i, j) ∈ A6 .

The assignment tij = −M for (i, j) ∈ A5 will be explained in the description of the con-

CE

straints of the transformed model. We visualize the transformation of the graph using the example from Figure 1. Figure 3 shows a solution in the transformed graph where all

AC

nodes are visited in one tour. From this solution a single tour of every salesman k can be easily derived, as it corresponds to the sequence of customer nodes between depot nodes 01k and 02k .

9

ACCEPTED MANUSCRIPT

d22

0 v1

d12 v6 v4

v5 v3

d11 d21

d13

d23

CR IP T

v2

ˇ Figure 3: Visualization of a Tour in the Transformed Graph G

AN US

The new binary decision variable yij is 1 if arc (i, j) is selected for the tour and 0 otherwise. In addition we define variable γi that indicates the nodes’ order in a solution and is important to ensure that every customer starts and ends his tour in his own depot. With

M

this additional notation the new optimization problem can be formulated as follows:

minimize F 2 =

X

X

cˇij yij

(9)

yij = 1

i ∈ Vˇ (S),

(10)

yij = 1

j ∈ Vˇ (S),

(11)

δj + M ≥ δi + tij + M yij

i, j ∈ Vˇ (S) \ {0}, i 6= j,

(12)

γj + M ≥ γi + 1 + M yij

i, j ∈ Vˇ (S) \ {0}, i 6= j,

(13)

γ01k ≤ γ02k

k ∈ S,

(14)

y01k ,01k = y02k ,02k

k ∈ S,

(15)

PT

ED

s.t.

i∈Vˇ (S) j∈Vˇ (S)

CE

X

AC

j∈Vˇ

X i∈Vˇ

10

ACCEPTED MANUSCRIPT

δi ≥ d i

i ∈ Vˇ (S) \ {0},

(16)

δ02k ≤ tmax

k ∈ S,

(17)

γi ≥ 0

i ∈ Vˇ (S) \ {0},

(18)

yij ∈ {0, 1}

i, j ∈ Vˇ (S).

(19)

CR IP T

The objective function (9) gives the total traveling cost for coalition S. Constraints (10) (12), (16), (17) have the same meaning as constraints (2), (3), (5) - (7) respectively. This transformation allows a simplification of constraints (7) as now only the depot nodes 02k for all k ∈ S should be considered. The assignment tij = −M for (i, j) ∈ A5 is crucial for the formulation of constraints (16) and (7), as it resets the time counter in variable δi each

AN US

time a tour from a new depot is started. Therefore this allows a correct consideration of the time restriction tmax . Additional constraints (13) - (15) correspond to constraints (4) from the previous formulation and ensure that salesmen can not change their affiliation during the tour and end tours in their own depots. Constraints (13) and (14) state in this

M

context that depots are visited in the correct sequence, and constraints (15) imply that for every k ∈ S both nodes 01k and 02k are simultaneously either selected or not selected

ED

as part of the tour. If they are not selected, then loops are assigned to these nodes and

A Core Computation Algorithm

CE

4

PT

for this purpose the set of arcs A7 has been defined.

Before we start with the core computation procedure we provide a formal definition of the

AC

cooperative traveling salesman game and of the core. In the general case the cooperative cost game Γ is defined as a pair Γ = (N, c), where N is a set of players also called the

grand coalition (in our case the set of salesmen) and c : 2N → R is the characteristic function that associates with every coalition S ⊆ N the value c(S). The value c(S) provides the total cost of coalition S. The cooperative traveling salesman game is defined as a pair (N, c), where the value c(S) equals to the optimal value of the objective function

11

ACCEPTED MANUSCRIPT (1). It was proved in Kimms and Kozeletskyi (2013) that the cooperative TSP game is subadditive and non-monotonic. Subadditivity means that the total cost of any coalition is not larger than the sum of costs of any disjunctive partition, i.e for any S, T ⊆ N such that S ∩ T = ∅ holds c(S ∪ T ) ≤ c(S) + c(T ). The property of non-monotonicity implies that a larger coalition may have lower cost than a smaller one. Proofs for these properties

CR IP T

can also be found in the online supplement of this paper.

A core element is a vector π ∈ RN that completely allocates the total cost c(N ) among all players from the grand coalition N (this property is called efficiency) and assigns to each player i a cost allocation πi that can not be dominated by any other subcoalition

AN US

from N (this property is called stability). The core is as a set of all such vectors π ∈ RN and is formally defined as follows: C(N, c) = {π ∈ RN |

X i∈N

πi = c(N ), ∀S ⊂ N, S 6= ∅ :

X i∈S

πi ≤ c(S).}

(20)

M

The non-monotonicity of the cooperative TSP game allows negative cost allocations to

ED

the players. As Drechsel and Kimms (2010) showed, in case of monotone games one can consider only π ∈ RN + . However the core can be empty, as the stability constraints may

PT

not always be fulfilled. Bondareva (1963) and Shapley (1967) proved the necessary and sufficient condition for the non-emptiness of the core. In case the core is empty we will

CE

refer to the relaxed version of it known as the least-core. For a given  ≥ 0 we consider a

AC

relaxation of the stability constraint and define an -core C (c): C (c) = {π ∈ RN |

X i∈N

πi = c(N ), ∀S ⊂ N, S 6= ∅ :

X i∈S

πi ≤ c(S) + }

(21)

Maschler et al. (1979) define the least-core as an non-empty -core with the smallest : \

C L (N, c) =

≥0:C (c)6=∅

12

C (c).

(22)

ACCEPTED MANUSCRIPT The definition of the core (20) implies that we need to consider all possible subcoalitions of the grand coalitions N i.e. 2|N | − 1 subcoalitions and compute their coalition value defined as the optimal value of the objective function (1) to determine a core element. Drechsel and Kimms (2010) proposed an iterative row generation procedure for the exact computation of a core element in a general case, that considers only a small subset of 2N and therefore significantly reduces the computational effort. In this section we adapt this

CR IP T

procedure for the cooperative TSP to determine the cost allocation.

The core computation algorithm consists of two optimization problems: the master problem that computes a cost allocation π for a given set of coalitions and the subproblem

AN US

which searches for a new coalition that violates the stability condition for the given cost allocation π. These two problems will be solved iteratively until a core element or (in case of an empty core) a least-core element is found.

Master Problem

M

4.1

The master problem is a relaxed version of the constraint satisfaction problem that corre-

ED

sponds to the core definition. Let S be the set of coalitions considered in current iteration (in the first iteration we take S = {{1}, {2}, . . . , {n}, N }) then the master problem is

AC

s.t.

CE

PT

defined as the following linear program:

X

minimize 

(23)

πi = c(N )

(24)

i∈N

X i∈S

πi ≤ c(S) + 

S∈S

≥0

(25) (26)

πi ∈ R

i∈N 13

(27)

ACCEPTED MANUSCRIPT Constraint (24) is an efficiency condition from the core definition and constraint (25) is a relaxed formulation of the stability condition. The optimal value of the objective function indicates whether the core is empty ( > 0) for a given instance of the cooperative TSP or not ( = 0). In case of the empty core the optimal value for  defines the least-core element. The optimal values of πi and  are used as the input data for the subproblem.

Subproblem

CR IP T

4.2

The purpose of this subproblem is to find for π and  from the solution of the master P 0 problem (23) - (27) a coalition S 0 ∈ / S such that i∈S 0 πi > c(S ) + . Therefore it

includes in its formulation the CTSPRD to compute the coalitional value for S 0 . For the

AN US

mathematical formulation of the subproblem the following additional notation is required. The parameter βki indicates if the customer i belongs to the set V ({k}) (βki = 1) or not (βki = 0). The new decision variable zk identifies the set S 0 :

ED

M

zk =

   1 if k ∈ S 0

  0 otherwise.

AC

s.t.

CE

PT

The corresponding mixed-integer formulation of the subproblem is given as follows:

obj = X

X

maximize F 3 = −obj +

X

k∈N

πk zk − 

cij xijk

(28)

(29)

k∈N i∈V (N )∪{0k } j∈V (N )∪{0k }

X

xijk =

k∈N j∈V (N )∪{0k }

X

X

X

X

k∈N i∈V (N )∪{0k }

X

zk βki

i ∈ V (N ),

(30)

zk βkj

j ∈ V (N ),

(31)

k∈N

xijk =

X

k∈N

14

ACCEPTED MANUSCRIPT X

X

(32)

δj + M ≥ δi + tij + M xijk

i ∈ V (N ), j ∈ V (N ), j 6= i, k ∈ N,

(33)

xijk ≤ zk

k ∈ N, i, j ∈ V (N ) ∪ {0k }

(34)

δi ≥ di

i ∈ V (N ),

(35)

δi ≤ tmax

i ∈ V (N ),

(36)

xijk ∈ {0, 1}

i, j ∈ V (N ), k ∈ N,

(37)

zk ∈ {0, 1}

k ∈ N.

(38)

j∈V (N )∪{0k }

j∈V (N )∪{0l }

xjil ≤ 1

CR IP T

i ∈ V (N ), k, l ∈ N, k 6= l,

xijk +

The value of the objective function (28) identifies whether the coalition S 0 should be

AN US

added to the master problem (i.e. F 3 > 0) or the current cost allocation πi determines a (least-) core element. The left hand side of the constraint (29) corresponds to the value of characteristic function c(S 0 ). Constraints (30), (31) and (34) ensure that only depots and customers of salesmen from the set S 0 are considered. Constraints (32), (33), (35),

M

(36) are the same as in formulation (1) - (8) of the CTSPRD.

ED

To the subproblem (28) - (38) we apply the same procedure as in section 3 to transform it into the TSP with one salesman and therefore reduce the computational effort for

PT

solving this problem. The new formulation requires a separation of the set of nodes into

CE

customer nodes V (N ) and depot nodes Vˇ D (N ). Furthermore k(i) denotes for depot i the corresponding salesman k.

AC

maximize F 4 = −obj +

X

k∈N

πk zk − 

(39)

s.t.

obj =

X

X

cij yij

(40)

i∈Vˇ (N ) j∈Vˇ (N )

15

ACCEPTED MANUSCRIPT X

zk βki

i ∈ V (N ),

(41)

zk βkj

j ∈ V (N ),

(42)

yij = zk(i)

i ∈ Vˇ D (N )

(43)

yij = zk(j)

j ∈ Vˇ D (N ),

(44)

yij =

j∈Vˇ (N )

k∈N

X

X

yij =

i∈Vˇ (N )

X

k∈N

j∈Vˇ (N )

X

i∈Vˇ (N )

CR IP T

X

i, j ∈ V (N ) \ {0}, i 6= j,

(45)

γj + M ≥ γi + 1 + M yij

i, j ∈ V (N ) \ {0}, i 6= j,

(46)

γ01k ≤ γ02k

k ∈ N,

(47)

y01k ,01k = y02k ,02k

k ∈ N,

AN US

δj + M ≥ δi + tij + M yij

δi ≥ d i δ02k ≤ tmax

yij ∈ {0, 1}

ED

zk ∈ {0, 1}

M

γi ≥ 0

(48)

i ∈ V (N ) \ {0},

(49)

k ∈ N,

(50)

i ∈ V (N ) \ {0},

(51)

i, j ∈ V (N ),

(52)

k ∈ N.

(53)

PT

Constraints (40) - (44) have the save meaning as constraints (29) - (31), (34) from the initial formulation of the subproblem. Other constraints are identical to the transformed

The Algorithm

AC

4.3

CE

formulation of CTSPRD (9) - (19).

After having presented formulations of all required optimization problems we can summarize the overall procedure for computing the cost allocation as follows: Step 1. Initialization: a. Simulate the release dates for the planning horizon T i.e. values di . b. Define the initial set S = {{1}, {2}, . . . , {n}, N }. 16

ACCEPTED MANUSCRIPT c. Compute coalitional values for all S ∈ S by solving the problem (9) - (19). Step 2. Solve the master problem (23) - (27) for the set S. Step 3. With the values of π and  solve the subproblem in formulation (39) - (53):

b. Define S 0 = {i ∈ N |zi = 1}, c(S 0 ) = obj. c. Update S = S ∪ S 0 . d. Go to Step 2.

CR IP T

a. If F 4 ≤ 0 then the cost allocation is found. Go to Step 4.

AN US

Step 4. If  = 0 then π defines a core element, otherwise is π the least-core element. Step 5. Calculate the percentual value of the cost allocation, i.e. for all i ∈ N : πip =

πi c(N )

(54)

M

The value π gives the cost allocation in absolute values for the simulated demand V (N )

ED

over the planing horizon T , but our goal is to define an allocation rule also for future planing activities that could be applied to the total cost of the grand coalition N . For this

PT

reason we suggest to use the percentual value (54) of the computed (least-) core element as the allocation rule. On the one hand it can be easily applied to the total cost c(N ) in

CE

the future to determine actual cost allocation. On the other hand percentual value gives

AC

a better understanding of cost shares to coalition partners.

If the formation of a cooperation causes certain coordination costs that need to be explicitly considered in the allocation problem, the proposed allocation procedure can be modified using an -core (21), but with some negative-valued , which absolute value depicts coordination costs.

17

ACCEPTED MANUSCRIPT

5

Computational Study

This section presents the results of our computational study where we tested the performance of the developed algorithm from Section 4.3. To determine the values of di for every customer i ∈ V (N ) we assume that they are independent exponentially distributed random variables. That means we associate with every customer i the rate parameter

CR IP T

λi . Using these rate parameters and rounding the obtained random values we generate for every customer i the order time di . For the real-life case the historical data from customers orders can be used to construct the empirical distribution function for every customer and with its help generate possible future arrival times. Furthermore, traveling costs and times for our computational study where generated as uniform distributed ran-

AN US

dom variables.

The key parameters for the proposed algorithm, that influence the computational effort, are the number of players |N | and the number of customer nodes V (N ). We have gener-

M

ated examples with up to 10 salesmen and up to 50 customer nodes. For the evaluation

ED

of our computational study we used following criteria: run-time of the algorithm, number of iterations for core computation (i.e. number of subcoalitions tested), relative savings

PT

(min. and max.) in the coalition compared to the non-cooperative case and emptiness of

CE

the core. The savings are computed for i ∈ N as: si =

c({i}) − πi c({i})

(55)

AC

To keep the computation time within reasonable limits, we limited the solver time to 20000 seconds. The algorithm was implemented using the commercial software AMPL with the solver Gurobi 4.5.1 on a computer with 2.80GHz Duo CPU, 8GB RAM. Table 1 presents the obtained results. In the third column the computation time for the instances is provided. The fourth column (it.) gives a number of additional subcoalitions that were considered for the computation of the cost allocation. The fifth column (c/l-c) shows

18

ACCEPTED MANUSCRIPT whether the obtained cost allocation is in the core (c) or in the least-core (l-c).

5

6

c/l-c

10 15 20 25 30 35 40 45 50 10 15 20 25 30 35 40 45 50 10 15 20 25 30 35 40 45 50 10 15 20 25 30 35 40 45 50

0.3 1.0 5.2 7.7 38.3 64.6 22.4 691.2 381.2 1.0 1.10 9.12 10.4 34.2 15.8 446.3 44.5 3634.9 2.0 8.2 12.2 10.6 22.7 38.5 108.3 378.6 75.6 2.82 13.8 15.2 33.7 58.2 92.8 1398.3 2692.0 1258.9

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

c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c

savings (%) min max 0 62 0 87 0 95 14 81 44 52 25 59 21 58 22 25 0 48 0 120 41 80 31 82 0 72 38 86 38 76 33 62 35 50 35 50 55 100 38 79 47 87 25 83 33 69 0 95 18 74 39 64 38 60 0 100 43 114 31 87 42 79 36 87 25 72 10 71 57 67 20 67

|N |

7

8

V (N )

time

it.

c/l-c

10 15 20 25 30 35 40 45 50 10 15 20 25 30 35 40 45 50 10 15 20 25 30 35 40 45 50 10 15 20 25 30 35 40 45 50

10.3 22.6 45.8 67.9 71.3 61.3 328.2 7872.7 > 20000 10.3 20.2 21.0 38.4 972.5 108.6 1153.2 257.6 > 20000 16.9 20.2 18.7 128.9 66.3 178.4 727.0 5035.6 > 20000 28.0 58.8 80.1 257.2 237.4 > 20000 > 20000 > 20000 > 20000

11 13 17 9 12 9 7 22 17 10 12 11 15 18 25 14 26 10 15 24 9 17 23 24 39 30 43 38 34 -

c c c c c c c c c c c c c c c c c c l-c c c c c c c c c c c -

9

10

savings (%) min max 50 94 43 100 38 100 57 88 31 81 36 87 27 72 12 19 44 117 64 100 29 92 53 91 20 87 25 82 65 78 35 84 57 111 64 100 40 95 69 89 56 94 44 83 53 82 50 79 14 100 60 91 50 109 56 92 38 93 -

CR IP T

it.

AN US

4

time

M

3

V (N )

ED

|N |

PT

Table 1: Computational Results for |N | = 3, . . . , 10

CE

As it can be seen from this table we could handle instances with up to 10 salesmen and with 35 customer nodes within the predefined computation time. Furthermore one in-

AC

stance with an empty core was found. The details on this example can be found in the online supplement to this paper. This means that the cooperative TSP with release dates (CTSPRD) game may have an empty core and is not balanced. The empty core was found for the coalition with |N | = 9 players and 20 customer nodes. The stability condition was violated for |S 0 | = 7 and the violation (value of the ) was 1.7% of the total cost of coalition S 0 . For all other instances a core element was found.

19

ACCEPTED MANUSCRIPT The obtained results also show, that the developed algorithm significantly reduces the number of considered subcoalitions for a (least-) core element, this points out the efficiency of our approach. For instance, for |N | = 9 only 20 coalitions on average (from 511 possible coalitions) were considered. Negative cost allocations in some instances (i.e. savings over 100%) are the implication of non-monotonicity of the cooperative TSP game. The run-

6

CR IP T

time of the algorithm can also be validated as reasonable.

Conclusions

The proposed solution procedure computes a core element for the cooperative traveling

AN US

salesman a priori. A computational study made on randomly generated instances showed the efficiency of the proposed algorithm as well as significant cost savings compared to the non-cooperative case. The found instance with an empty core additionally proves the usage of the least-core as a cooperative TSP game can have an empty core and therefore is not balanced. The least-core can be also implemented for the instances with non-empty

M

core due to fairness issues. For a further discussion on fairness of the core concept we

ED

refer to Drechsel and Kimms (2010). The developed optimization approach can also be applied for a priori calculation of cost allocations in problems with rolling horizon. One

PT

can simulate the appearance of customer orders and incorporate them in the optimization model through the release dates. Additionally the idea of the presented algorithm can

CE

also be extended for further modifications of transportation planing (e.g. vehicle routing

AC

problems).

7

Acknowledgement

This work was done with financial support from the Deutsche Forschungsgemeinschaft grant KI 1272/5-2.

20

ACCEPTED MANUSCRIPT

References ¨ nnqvist, M., (2012), An Empirical Study on Audy, J-F., D’Amours, S., Ro Coalition Formation and Cost/Savings Allocation, International Journal of Production Economies, vol. 136, p. 13 - 27 Adenso-Diaz, B., Lozano, S., Garcia-Carbajal, S., Smith-Miles, K., (2014a),

Transportation Research Part A, vol. 66, p. 268 - 279

CR IP T

Assessing Partnership Savings in Horizontal Cooperation by Planning Linked Deliveries,

Adenso-Diaz, B., Lozano, S., Garcia-Carbajal, S., Smith-Miles, K., (2014b), Analysis of the synergies of merging multi-company transportation needs, Transportmet-

AN US

rica A: Transport Science, vol. 10, p. 533 - 547

Berger, S., Bierwirth, C., (2010), Solutions to the Request Reassignment Problem in Collaborative Carrier Networks, Transportation Research Part E, vol. 46, p. 627 - 638

M

Cruijssen, F., Cools, M., Dullaert, W., (2007a), Horizontal cooperation in logistics: Opportunities and Impediments, Transportation Research Part E, vol. 43, p. 129 -

ED

142

PT

Cruijssen, F., Borm, P., Fleuren, H., Hamers, H. (2010), Supplier-initiated Outsourcing: A Methology to Exploit Synergy in Transportation, European Journal of

CE

Operational Research, vol. 207, p. 763 - 7774 Cruijssen, F., Dullaert, W., Fleuren, H., (2007b), Horizontal Cooperation in

AC

Logistics: A Literature Review, Transportation Journal, vol. 46, p. 22 - 29 Cruijssen, F., Dullaert, W., Joro, T., (2010), Freight Transportation Efficiency

Through Horizontal Cooperation in Flanders, International Journal of Logistics: Research and Application, vol. 13, p. 161 - 178

21

ACCEPTED MANUSCRIPT ¨ nnqvist, M., (2010), Issues in Collaborative Logistics, , in: BjornD’Amours, S., Ro dal, E., et al. (eds.), Energy, Natural Resources and Environmental Economics p. 395 409 Drechsel, J., Kimms, A., (2010), Computing Core Allocations in Cooperative Games with an Application to Cooperative Procurement, International Journal of Production

CR IP T

Economies, vol. 128, p. 310 - 321 ¨ the-Lundgren, M., Va ¨ rband, P., (2004), The Heterogeneous Engevall, S., Go Vehicle Routing Game, Transportation Science, vol. 38, p. 71 - 85

Fishburn, P., C., Pollak, H., O., (1983), Fixed Route Cost Allocation, American

AN US

Mathematical Monthly, vol. 90, p. 366 - 378

¨ the-Lundgren, M., Jo ¨ rnsten, K., Ro ¨ nnqvist, M., (2010),Cost Frisk, M., Go Allocation in Collaborative Forest Transportation, European Journal of Operational Research, vol. 205, p. 448 - 458

M

Gilles, D., B., (1959), Solutions to General Non-Zero-Sum Games, in: Tucker, A.,

ED

W., Luce, R., D., (Ed.), Contributions to the Theory of Games, Volume 4, Princeton University Press, p. 47-85

PT

¨ the-Lundgren, M., Jo ¨ rsten, K., Va ¨ rband, P., (1996), On the Nucleolus of Go

CE

the Basic Vehicle Routing Game, Mathematical Programming, vol. 72, p. 83 - 100 ¨ nnqvist, M., (2015), Operations Research Models for Coalition Guajardo, M., Ro

AC

Structure in Collaborative Logistics, European Journal of Operational Research, vol. 240, p. 147 - 159 Kimms, A., Kozeletskyi, I., (2013), Shapley-Value-Based Cost Allocation in the Cooperative Traveling Salesman Problem under Rolling Horizon Planning, Working Paper, University of Duisburg-Essen Krajewka, M., Kopfer, H., (2006), Collaborating Freight Forwarding Enterprises Request Allocation and Profit Sharing, OR Spectrum, vol. 28, p. 301 - 317 22

ACCEPTED MANUSCRIPT Krajewka, M., Kopfer, H., Laporte, G., Ropke, S., Zaccour, G., (2008), Horizontal Cooperation Among Freight Carries: Request Allocation and Profit Sharing, Journal of the Operations Research Society, vol. 59, p. 1483 - 1491 Laporte, G., Nobert, Y., Taillefer, S., (1988), Solving a Family of Multi-Depot Vehicle Routing and Location-Routing Problems, Transportation Science, vol. 22, p. 161

CR IP T

- 172 Lozano, S., Moreno, P., Adenso-Diaz, B., Algaba, E., (2013), Cooperative Game Theory Approach to Allocating Benefits of Horizontal Cooperation, European Journal of Operational Research, vol. 229, p. 444 - 452

AN US

Maschler, M., Peleg, B., Shapley, L., S., (1979), Geometric Properties of the Kernel, Nucleolus, and Related Solution Concepts, Mathematics of Operations Research, vol. 4, p. 303 - 338

Miller, C., E., Tucker, A., W., Zemlin, R., A., (1960), Integer Programming

ED

Machinery, vol. 7, p. 326 - 329

M

Formulation of Traveling Salesman Problems, Journal of the Association for Computing

Muir, M., (2009), European Supply Chain Horizontal Cooperation, Report, Eye for

PT

Transport

Muir, M., (2011), North American Horizontal Cooperation in the Supply Chain, Re-

CE

port, Eye for Transport

AC

Nadarajah, S., Bookbinder, J., H., (2013), Less-Than-Truckload Carrier Collaboration Problem: Modeling Framework and Solution Approach, Journal of Heuristics, vol. 49, p. 917 - 942 ¨ ¨ Ergun, O., ¨ Savelsbergh, M., (2011), Lane Exchange Mechanism Ozener, O., O., for Truckload Carrier Collaboration, Transportation Science, vol. 45, p. 1 - 17 Potters, J., Curiel, I., J., Tijs, S., H., (1992), Traveling Salesman Games, Mathematical Programming, vol. 53, p. 199-211 23

ACCEPTED MANUSCRIPT Punnen, A., P., (2002), The Traveling Salesman Problem: Applications, Formulations and Variations, in: Gutin, G., Punnen, A., P., (Ed.), (2002), The Traveling Salesman Problem and Its Variations, Kluwer Academic Publishers Schmeidler, D., (1969), The Nucleolus of a Characteristic Function Game, SIAM Journal of Applied Mathematics, vol. 17, p. 1163 - 1170

CR IP T

Shapley, L., S., (1953), A Value for n-Person Games, in: Roth, A., E., (Ed.), (1988), The Shapley Value: Essays in Honor of Lloyd S. Shapley, Cambridge University Press, p. 31 - 40

Verstrepen, S., Cools, M., Cruijssen, F., Dullaert, W., (2009), A Dynamic

AN US

Framewor for Managing Horizontal Cooperation in Logistics, International Journal Logistics Systems and Management, vol. 5, p. 228 - 248

Wang, X., Kopfer, H., (2014), Collaborative Transportation Planning of Less-Than-

M

Truckload Freight, OR Spectrum, vol. 36, p. 357 - 380

Wang, X., Kopfer, H., Gendreau, M., (2014), Operational Transportation Plan-

ED

ning of Freight Forwarding Companies in Horizontal Coalitions, European Journal of

AC

CE

PT

Operational Research, vol. 237, p. 1133 - 1141

24