European Journal of Operational Research 216 (2012) 647–657

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

A game theoretical approach to sharing penalties and rewards in projects q

Arantza Estévez-Fernández ⇑Department of Econometrics and Operations Research and Tinbergen Institute, VU University Amsterdam, De Boelelaan 1105, 1081HV Amsterdam, The Netherlands

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 April 2011Accepted 18 August 2011Available online 25 August 2011

Keywords:Game theoryProject managementDelay and expeditionCost and surplus sharing mechanismBankruptcy and taxation problems

0377-2217/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.ejor.2011.08.015

q The author thanks three referees for their valuament and to René van der Brink, Harold Houba, GeLindner for their comments on a first version of thisPeter Borm for his encouragement and his helpful dis⇑ Tel.: +31 20 5985483; fax: +31 20 5986020.

E-mail address: arantza.estevezfernandez@vu.nl

This paper analyzes situations in which a project consisting of several activities is not realized accordingto plan. If the project is expedited, a reward arises. Analogously, a penalty arises if the project is delayed.This paper considers the case of arbitrary nondecreasing reward and penalty functions on the total expe-dition and delay, respectively. Attention is focused on how to divide the total reward (penalty) among theactivities: the core of a corresponding cooperative project game determines a set of stable allocations ofthe total reward (penalty). In the definition of project games, surplus (cost) sharing mechanisms are usedto take into account the specific characteristics of the reward (penalty) function at hand. It turns out thatproject games are related to bankruptcy and taxation games. This relation allows us to establish nonemp-tiness of the core of project games.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

A project consists of a set of activities, which interconnectionsare known, being completed over a period of time and intendedto achieve a particular aim. The project time is the minimum timeneeded to end all the activities in a project.

Nowadays, many projects require employing and coordinatingdifferent companies to perform specialized jobs, like the creationof high technology devices, the construction of a building, andthe construction of infrastructures (such as highways and rail-roads). Due to its high societal and economical relevance, projectmanagement has become an important and relevant discipline.A good planning of a project is important to reduce the projecttime. Two important methods to schedule and coordinate theactivities in a project are the PERT (Program Evaluation ReviewTechnique) and the CPM (Critical Path Method). During the lastdecades many studies have appeared addressing several issues asthe question of assessing the project success in Pinto and Slevin(1988) and Tubig and Abeti (1990), the role of research in projectsof weapon systems development in Sherwin and Isenson (1967),the problem of determining the amount, location, and timing ofprogress payments in projects from a client’s perspective inDayanand and Padman (2001), behavioral functions in technol-ogy-based innovative projects in Roberts and Fusfeld (1981), the

ll rights reserved.

ble suggestions for improve-rard van der Laan and Inespaper. Special thanks go to

cussion and comments.

classification of projects in types depending on the managementstyle in Shenhar and Dvi (1996), the search of adequate successcriteria for project management in Atkinson (1999) and the studyof project scheduling with resource constrains in Dorndorf et al.(2000) and Möhring et al. (2003). The study of scarce resourcesin project management has originated an extensive literature onthe topic; surveys on this subject are Hartmann and Briskorn(2010) and Weglarz et al. (2011).

An important problem within project management is the delayof a project. The delay in the construction of highways and railwaysconnections brings both economical and social losses since societyhas to wait longer to enjoy their benefits. Delay is one of the mostimportant issues for project managers. Quoting Mr. Rafael Romero,chairman of the Chamber of Constructors of Catalonia, ‘‘The expen-ditures used to prepare a project imply a saving of time andmoney’’ and ‘‘The suitability of a work goes through three funda-mental pillars: security, finishing in the agreed deadline, and thatthe budget is met’’ (see http://www.lukor.com/not-esp/locales/portada/08030207.htm). In many countries, when a public projectis delayed, the government is legally entitled to fine the companyin charge. Also in private projects there is usually a clause in thecontract in which a penalty for the company is agreed upon ifthe project is finished later than planned. In all these cases, if thecompany in charge of the project has subcontracted some special-ized jobs to other firms, and these firms have contributed to the de-lay, it is important to know for which part of the penalty they canbe held responsible for. Opposite to extra costs associated to delayin projects, there may be extra rewards associated to expeditedprojects. As an example, we have the development of newhigh-technology devices. It is well known that development of hightechnology needs the input of different specialized agents. Besides,

Activity Predecessors

A

B

C A,BB

A

C

Fig. 1. Predecessors of the activities and representation of the project in Example2.1.

648 A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657

companies usually compete to be the first to have the newproduction on the market and therefore they promise a bonus totheir employees if the product is finished before the agreed time.The subsequent question is how to share the bonus among theemployees whose efforts resulted in the expedition of the project.The question we address in this paper is how to share the extracosts (or fine) among the agents responsible for the delay of a pro-ject, or the extra rewards in case of expedition.

Albeit delayed project problems and expedited project prob-lems seem to be related, they have a different nature. While anyfirm or agent can delay a project on its own, a coordination of firmsor agents is needed in order to expedite the project. In this paperwe study both the allocation of extra costs and rewards simulta-neously, taking into account the similarities and differences of bothproblems. Cooperative games are a mathematical tool to providean answer to this type of allocation problems. The focus of ourstudy will be on defining a game in an adequate way and analyzingthe corresponding core. The core of a game (Gillies, 1953) providesallocations of the total penalty or the total reward that are stable,i.e. no group of activities can reasonably object to allocations in thecore. The interaction between operations research and cooperativegame theory already goes back to the seventies, for a survey on thetopic see Borm et al. (2001).

The main focus in the literature on project problems has beenon delayed project problems. Branzêi et al. (2002) study delayedproject problems in the framework of taxation problems and pro-pose a specific allocation rule. Bergantiños and Sánchez (2002)analyze two other allocation rules for delayed project problems.A common feature in these papers is however that game theoreti-cal aspects are only indirectly present in analyzing the allocationproblem at hand. Estévez-Fernández et al. (2007) is the first paperto approach the related allocation problem from a direct game the-oretical point of view. Moreover, Estévez-Fernández et al. (2007) isthe first article where both delayed and expedited project prob-lems are analyzed. Still, the paper is restricted to project problemswhere the penalty (reward) function is proportional with respectto the total delay (expedition) of the project. An important aspectof Estévez-Fernández et al. (2007) is that it provides the tools toobtain ‘‘fair’’ allocations of the corresponding penalty or rewardby explicitly considering the structure of the project, i.e. the inter-connections among the different activities.

In this paper, we extend the work in Estévez-Fernández et al.(2007) by both analyzing project problems with arbitrary but non-decreasing penalty and reward functions and by taking into ac-count whether an activity can be started before its plannedstarting time. As in Estévez-Fernández et al. (2007), the analysisis done by defining project games associated to project problemsand considering the corresponding core as the solution set to theunderlying allocation problem. We have decided to focus on the(set-valued solution) core instead of given an explicit one-pointsolution because this allows us to cover a wide variety of projectsituations that can have different needs depending, for instance,on the legal aspects that are applicable, which may vary dependingon the type of project at hand and/or on the country where the pro-ject is realized. Here, two stages are needed to define project gamesfor nondecreasing reward (penalty) functions. In the first stage sur-plus (cost) sharing problems are used. As a first approach to sharethe reward (penalty) of a project, we look at each path in the pro-ject separately. We share the reward (penalty) that the path can beheld responsible for among its activities by making use of a surplus(cost) sharing mechanism. The mechanism is chosen by taking intoaccount the specific characteristics of the reward (penalty) func-tion at hand. In this first stage, the total amount shared amongall the activities may exceed the total reward (penalty) of the pro-ject, hence a second stage is needed to exactly obtain allocations ofthe total reward (penalty). In the second stage, a project game is

defined in the spirit of Estévez-Fernández et al. (2007) using theallocations obtained in the first stage. Although the model hereproposed is inspired in Estévez-Fernández et al. (2007), both gamesneed not coincide when we restrict to proportional reward (pen-alty) functions. In fact, the core of the game defined here alwayscontains the core of the game in Estévez-Fernández et al. (2007).

Our main result is that project games have a nonempty core.Moreover, it turns out that the games associated to the expeditionpart of the project are convex and can be described as the maxi-mum of a number of bankruptcy games (see O’Neill, 1982).

The structure of this paper is as follows. Section 2 recalls somebasic notions from project problems, cost and surplus sharingproblems, cooperative games, and bankruptcy and taxation prob-lems. Section 3 analyzes general project problems. Section 4concludes.

2. Preliminaries

2.1. Projects

A project consists of a set of activities for which the inter-con-nections are known. These activities are completed over a periodof time and intended to achieve a particular aim. Let N denotethe set of activities of a project. Given an activity i 2 N, let Pi denotethe set of predecessors of i, i.e. the set of activities that have to beprocessed before i can start. A project is defined as a 2-tuple(N, {Pi}i2N). Given i 2 N, let Fi be the set of followers of i, i.e. the setof activities that need i to be completed before starting. Formally,Fi = {j 2 Nji 2 Pj}. Moreover, let IPi be the set of immediate predeces-sors of i, i.e. IPi = {j 2 Njj 2 Pi and Pi \ Fj = ;}; note that IPi = ; whenPi = ;.

Given a project (N, {Pi}i2N), a path {i1, . . . , il} is a sequence of activ-ities satisfying the following conditions: (i) IPi1 ¼ ;, (ii) ir 2 IPirþ1 forevery r 2 {1, . . . , l � 1}, and (iii) Fil ¼ ;.

Note that a project can be represented by the collection of all itspaths. Throughout this paper we identify a project (N, {Pi}i2N) withthe set of all its paths {N1, . . . ,Nm}. Moreover, a project can also berepresented by a directed graph where the set of arcs correspondsto the set of activities. In order to avoid multiple arcs, dummyactivities are introduced in the graph (a dummy activity is a ficti-tious activity that does not consume neither time, nor resources).Dummy activities are represented by a dashed arc.

Example 2.1. Fig. 1 gives the set of activities of a project with theircorresponding predecessors and the graphical representation.Here, the set of activities is N = {A,B,C} and the collection of pathsis {N1,N2}, with N1 = {A,C}, and N2 = {B,C}. h

Associated to a project {N1, . . . ,Nm}, there is a duration functionl : N ! Rþ with l(i) denoting the length or duration of activity i 2 N.We assume without loss of generality that the project starts at timezero and that activity i starts as soon as all its predecessors arecompleted. Given a project {N1, . . . ,Nm} and a duration function l,we define the duration of a path Na according to l, D(Na, l), as thesum of the duration of its activities, i.e. DðNa; lÞ ¼

Pi2Na

lðiÞ. The

A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657 649

duration of the project according to l, D(l), is the maximum durationof its paths, i.e. D(l) = max16a6m{D(Na, l)}. The slack of Na accordingto l, slack (Na, l), is the maximum time that the path Na can be de-layed without altering the duration of the project, i.e. slack(Na, l) = D(l) � D(Na, l). We say that a path is critical if it has slackzero. Associated to a project there is also an availability functiona : N ! Rþ with a(i) denoting the amount of time that activity ican start before the planned time if necessary.

Throughout, we use a fixed notation for two specific durationfunctions. We denote by p : N ! Rþ the function representing theplanned or estimated time of the activities and by r : N ! Rþ thefunction giving the real time of the activities after the realizationof the project. We define the delay function d : N ! Rþ as d(i) =(r(i) � p(i))+(:¼max{r(i) � p(i),0}), i.e. d(i) represents the delay ofactivity i. Analogously, we define the expedition functione : N ! Rþ as e(i) = (p(i) � r(i))+, i.e. e(i) represents the expeditionof activity i.

Example 2.2. Consider the project given in Example 2.1. Letp : N ! Rþ be given by p(A) = 15, p(B) = 10, and p(C) = 8. Theplanned duration of the project is D(p) = 23 and the duration andslack of the paths are

DðfA;Cg;pÞ ¼ 23 and slackðfA;Cg;pÞ ¼ 0;DðfB;Cg;pÞ ¼ 18 and slackðfB; Cg;pÞ ¼ 5:

Let a : N ! Rþ be given by a(A) = 0, a(B) = 0, and a(C) = 7, then A andB cannot start before their planned starting time, while activity Ccould start 7 units of time before its planned starting time, i.e. Ccould start at time 8.

Let the real time r : N ! Rþ be given by r(A) = 7, r(B) = 6, andr(C) = 12. The delay and expedition functions are

dðAÞ ¼ ð7�15Þþ ¼ 0; dðBÞ ¼ ð6�10Þþ ¼ 0; dðCÞ ¼ ð12�8Þþ ¼ 4;eðAÞ ¼ ð15�7Þþ ¼ 8; eðBÞð10�6Þþ ¼ 4; eðCÞð8�12Þþ ¼ 0:

Note that since a(C) = 7, C can only start at time 8 and thereforeD(r) = 20. h

2.2. Cost sharing and surplus sharing problems

A cost sharing problem is defined by a tuple (N,q,c) whereN = {1,2, . . . ,n} is the set of agents (or players), q 2 RN

þ is a vectorof nonnegative numbers, with qi representing the demand of agenti 2 N, and c : R! Rþ is a nondecreasing cost function satisfyingc(t) = 0 for t 6 0. A cost sharing mechanism on a class of cost sharingproblems C is a mapping y that assigns to each ðN; q; cÞ 2 C a vectorof cost shares yðN; q; cÞ 2 RN

þ, i.e.P

i2NyiðN; q; cÞ ¼ cP

i2Nqi

� �, satisfy-

ing that if qi = 0, then yi(N,q,c) = 0.Several mechanisms can be found in the literature (see e.g.

Moulin (1987), Moulin and Shenker (1992), and Koster (1999));we recall here one of the most studied cost sharing mechanismsin the literature, the serial cost sharing mechanism, ys. Let (N,q,c)be a cost sharing problem and assume without loss of generalitythat q1 6 q2 6 � � � 6 qn. Let q0 = 0, then the serial cost sharing mech-anism is defined by

ys1ðN; q; cÞ ¼

cðnq1Þn

;

ysi ðN;q;cÞ ¼

cðnq1Þn

þXi

k¼2

cPk�1

j¼0 qjþðn�kþ1Þqk

� �� c

Pk�2j¼0 qjþðn�kþ2Þqk�1

� �n�kþ1

for every i2N nf1g:

Analogously to cost sharing problems, one can think of surplussharing problems. All definitions given above for cost sharing prob-lems can be easily translated to surplus sharing problems.

2.3. Transferable utility (TU) games

A cooperative (TU) game in characteristic function form is an or-dered pair (N,v) where N is a finite set of players and v : 2N ! R

satisfying v(;) = 0. In general, v(S) represents the value of coalitionS, i.e. the joint payoff that can be obtained by this coalition whenits members decide to cooperate.

A cooperative game can reflect costs or rewards. A game reflect-ing costs is denoted by a mapping c, while a game reflecting re-wards is denoted by a mapping v.

The core of a game (N,v) is defined by

CoreðvÞ ¼ x 2 RNXi2N

xi ¼ vðNÞ;Xi2S

xi P vðSÞ for all S 2 2N

�����( )

;

i.e. the core is the set of efficient allocations of v(N) (i.e. v(N) is pre-cisely shared among the players in N) to which no coalition can rea-sonably object. An important subclass of games with nonemptycore is the class of convex games (see Shapley, 1971). A game(N,v) is said to be convex if v(S [ {i}) � v(S) 6 v(T [ {i}) � v(T) forevery i 2 N and every S � T � Nn{i}.

Let (N,c) be a cost game. The corresponding cost savings game(N,v) is defined by

vðSÞ ¼Xi2S

cðfigÞ � cðSÞ

for every S � N.The properties and solution concepts for cooperative cost games

can consequently be derived from the above definitions. In partic-ular, the core of a cost game (N,c) is defined as

CoreðcÞ ¼ x 2 RNXi2N

xi ¼ cðNÞ;Xi2S

xi 6 cðSÞ for all S 2 2N

�����( )

and the equivalent of a convex game for a cost game is a concavegame. A cost game (N,c) is said to be concave if c(S [ {i}) �c(S) P c(T [ {i}) � c(T) for every i 2 N and every S � T � Nn{i}.

2.4. Bankruptcy and taxation problems

A bankruptcy problem is defined by a tuple (N,E,c) whereN = {1, . . . ,n} is the set of agents (or players), E is the estate thatmust be shared among the agents, and c 2 RN is the vector ofclaims of the agents satisfying

Pi2Nci > E. Bankruptcy problems

have being firstly studied from a game theoretical viewpoint inO’Neill (1982). Given a bankruptcy problem (N,E,c), the corre-sponding bankruptcy game, (N,v(N,E,c)), is defined by

v ðN;E;cÞðSÞ ¼ E�Xi2NnS

ci

!þ

for every S � N. In Curiel et al. (1987) it is shown that bankruptcygames are convex.

Taxation problems can be seen as dual of bankruptcy problems.A taxation problem is defined by a tuple (N,E,c) where N = {1, . . . ,n}is the set of agents (or players), E is the tax that must be collectedamong the agents, and c 2 RN is the vector of the abilities to pay ofthe players satisfying

Pi2Nci > E. Given a taxation problem (N,E,c),

the corresponding taxation game, (N,c(N,E,c)), is defined by

cðN;E;cÞðSÞ ¼min E;Xi2S

ci

( )

for every S � N. In Branzêi et al. (2002) it is shown that taxationgames are concave.

650 A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657

3. Project problems

In this section we analyze project problems. For clarity of expo-sition we postpone all proofs to the appendix.

A project problem arises when the planned time of the activitieshas been incorrectly estimated, possibly bringing delay or expedi-tion and associated extra costs or rewards to the project. We studyproject problems in terms of rewards, where a negative reward re-flects costs. A nondecreasing reward function R : R! R is associ-ated to the difference between the planned and real times of theproject, satisfying R(t) 6 0 for t < 0, R(t) = 0 for t = 0, and R(t) P 0for t > 0. A project problem can be described by a 5-tuple({N1, . . . ,Nm},p,r,a,R). For simplicity of notation, we omit the vectorof availabilities a from the description of a project problem when-ever all activities can be started as soon as needed.

When a reward occurs due to the difference between theplanned and real times of a project, one can think of sharing the de-lay or expedition of the project among the activities in a first (lin-ear) stage and allocating the (possibly nonlinear) reward amongthe activities according to the delay or expedition they have beenheld responsible for in a second stage. This approach has been sug-gested in Branzêi et al. (2002) and Branzêi et al. (2010). Moreexplicitly, in Branzêi et al. (2002) it is suggested that ‘‘the delaycost . . . is divided among the agents proportionally to their delayshares. . .’’ while in Branzêi et al. (2010) it is stated that ‘‘. . . thesolution is given in terms of units of time. The related payoffscan then be easily obtained by using a (linear) cost function . . ..In the case when the cost functions are not linear, it is necessaryto apply more sophisticated cost/profit sharing mechanisms’’. Themain problem with this procedure is that the specific relations be-tween the characteristics of the reward function and the structureof the project may be neglected. We show the problems arisingfrom this procedure in the following example in which the alloca-tion of the total difference in time is obtained by considering thecore of the game defined in Estévez-Fernández et al. (2007).

Example 3.1. Consider the project problem ({N1,N2,N3,N4},p,r,R)with N1 = {A}, N2 = {B}, N3 = {C,D}, N4 = {E}; p(A) = 21, p(B) = 19,p(C) = 2, p(D) = 16, p(E) = 17; r(A) = 14, r(B) = 15, r(C) = 7, r(D) = 6,r(E) = 14; and

RðtÞ ¼

50t if t 6 0;50t4 if 0 < t 6 2;100t þ 600 if 2 < t 6 3;t þ 897 if t > 3:

8>>><>>>:Note that d(A) = 0, d(B) = 0, d(C) = 5, d(D) = 0, d(E) = 0, e(A) = 7,e(B) = 4, e(C) = 0, e(D) = 10 and e(E) = 3. The project is representedin Fig. 2.

Here, D(p) = 21 and D(r) = 15, and then the total expedition isD(p) � D(r) = 6 with an associated reward of R(6) = 903. Followingthe suggested two-stage approach, we first share the total expe-dition of 6 among the activities by making use of the associatedlinear project game given in Estévez-Fernández et al. (2007). Weobtain that (2,2,�2,2,2) and (1,2,0,1,2) are two possible shares of

ABC DE

N α D(N α , p) slack(N α , p) D (N α , r )

A 21 0 14

B 19 2 15

CD 18 3 13

E 17 4 14

Fig. 2. Representation of the project given in Example 3.1 and duration of the paths.

the total expedition. First, fix (2,2,�2,2,2) as the allocation of thetotal expedition among the activities. Here, we encounter a firstproblem of an interpretational nature. Activity C is held respon-sible for 2 units of delay, but it is not clear what is the monetaryequivalence of this delay. Is it given by the penalty part or by thebonus part of the reward function? Second, fix (1,2,0,1,2) as theallocation of the total expedition among the activities. Here, weencounter a second problem by losing the relation between thereward function and the structure of the project when translatingexpedition or delay shares to bonus. By dividing the total rewardproportionally to the allocation of expedition shares (1,2,0,1,2),as suggested in Branzêi et al. (2002), an allocation of (150.5,301,0,150.5,301) is obtained. Note that activities A and D are needed toobtain the first two units of expedition, which bring a reward of800. However, the allocation only assigns a total bonus of 301 to Aand D. h

As pointed out in the example above, the inadequacy in firstsharing ‘‘responsibilities’’ among the activities and then the rewardin a second stage is that the possible relation between the rewardfunction and the characteristics of the project are not reflected onthe final share of the total reward. Our approach follows the oppo-site reasoning. We first allocate the reward associated to the delayor expedition of each path (i.e. the total delay or total expeditioncreated by its activities) among its activities. In a second step, weuse cooperative games to share the reward of the project amongall its activities, using the initial allocations as reference points.By doing this, we avoid the interpretational problem of negativeexpedition shares (delays) and we maintain possible relationsbetween the reward function and the structure of the project.The second part of our analysis is inspired on the work carried inEstévez-Fernández et al. (2007).

Associated to a project problem ({N1, . . . ,Nm},p,r,a,R), we definea project game where the set of players is the set of activities andthe value of a coalition is determined by taking into account thesimilarities and differences of both delayed and expedited projects.In determining the value of a coalition we pessimistically assumethat all delayed activities have indeed acted according to realiza-tion and that all expedited activities outside the coalition haveacted according to plan. Then, if the expedition given by the expe-dited activities in the coalition itself is not enough to expedite theduration of the project, the value of the coalition is negative and isdetermined through a cost game. Otherwise, the value of thecoalition is positive and is determined through a reward game. For-mally, given a project problem ({N1, . . . ,Nm},p,r,a,R), a cost sharingmechanism y, and a surplus sharing mechanism z, we denote by(N,uyz) the associated project game, still to be formally defined.Let E denote the set of expedited activities, i.e. E ¼ fi 2 NjeðiÞ > 0g.

3.1. Delay in project problems

If DðpjEnS; rjNnðEnSÞÞP DðpÞ, then the expedition carried by theexpedited activities in S is not enough to expedite the projectand cy(S) reflects the maximum penalty that the coalition can beheld responsible for. If activities in a path are delayed and theycause a delay of the project, we can consider this as a cost sharingproblem where the demands of the activities are their delays andthe cost function is the negative part of the reward function. Forevery a 2 {1, . . . ,m}, consider the cost sharing problem

ðNa; qa;KÞ;

with qai ¼ dðiÞ for every i 2 Na and K(t) = �R(�t) if t > 0 and K(t) = 0

otherwise. Selecting a cost sharing mechanism y by taking intoaccount the type of penalty function at hand, we define ya =y(Na,qa,K). Here, we pessimistically assume that the activities in

A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657 651

Na are not allowed to make use neither of the planned slack, nor ofthe expedition created by other activities in the path and have topay the cost associated to their total delay. In this way, ya

i repre-sents the cost that i is held responsible for the cost associated tothe total delay of the activities in Na. By using these allocations asstarting points, the activities of coalition S that are in path a cannotbe held responsible neither for more than the total cost assigned tothem by the cost sharing mechanism, nor for more than the netdelay of the path as a consequence of the delay of activities in thepath and the expedition of the activities within the coalition. Thevalue of coalition S is then the maximal amount that the coalitioncan be held responsible for with respect to the different pathsinvolved in the coalition. Formally,

cyðSÞ ¼ maxa2PðSÞ

minX

i2Na\S

yai ; K DðNa; ðpjEnS; rjNnðEnSÞÞÞ � DðpÞ

� �( )( );

ð3:1Þ

where PðSÞ represents the set of paths in which S is involved, i.e.PðSÞ ¼ fa 2 f1; . . . ;mgjNa \ S – ;g. Note that cy(N) = K(D(r) � D(p))since min

Pi2Na

yai ; KðDðNa; rÞ � DðpÞÞ

� �¼ KðDðNa; rÞ � DðpÞÞ for

every a 2 {1, . . . ,m}.

Example 3.2. Consider the project problem ({N1,N2,N3},p,r,R) withN1 = {A,B}, N2 = {A,D}, N3 = {C,D}; p(A) = 2, p(B) = 15, p(C) = 13,p(D) = 3; r(A) = 9, r(B) = 11, r(C) = 12, r(D) = 8; and

RðtÞ ¼�t4 � 100 if t < 0;0 if t ¼ 0;t2 þ 200 if t > 0:

8><>:The project is represented in Fig. 3.

Here, D(p) = 17 and D(r) = 20, and then the difference of time isD(p) � D(r) = �3 with an associated reward of R(�3) = �181.Besides, d(A) = 7, d(B) = 0, d(C) = 0 and d(D) = 5; e(A) = 0, e(B) = 4,e(C) = 1 and e(D) = 0; and E ¼ fB;Cg.

For the computation of (N,cy) we use the serial cost sharingmechanism. Associated to each path Na we have the cost sharingproblem (Na,qa,K), with qa

i ¼ dðiÞ and K(t) = 0 if t 6 0 andK(t) = t4 + 100 if t > 0. Then, y1 = y({A,B}, (7,0),K) = (2501,0), y2 =y({A,D}, (7,5),K) = (15786,5050), and y3 = y({C,D}, (0,5),K) =(0,725). Below, the computation of cy({A,B,C}) is explained. Notethat PðfA;B;CgÞ ¼ f1;2;3g. Then,

cyðfA;B;CgÞ ¼ maxa2f1;2;3g

minX

i2Na\S

yai ; KðDðNa;ðpjEnS; rjNnðEnSÞÞÞ�DðpÞÞ

( )( )¼maxfminf2501þ0;Kð20�17Þg;

minf15786;Kð17�17Þg;minf0;Kð20�17Þgg¼maxfminf2501;181g;minf15786;0g;minf0;181gg¼maxf181;0;0g ¼ 181:

All coalitional values are given in Table 1.It can be checked that the core of the game is

CoreðcyÞ ¼ convfð0;0;�175;356Þ; ð0;0;0;181Þ;ð2320;�2320;0;181Þ;ð2501;�2320;0;0Þ; ð2145;�2145;�175;356Þ;ð181;0;�175;175Þ;ð181;0;0;0Þg:

A B

C D

N α D(N α , p) slack(N α , p) D(N α , r )

AB 17 0 20

AD 5 12 17

CD 16 1 20

Fig. 3. Representation of the project given in Example 3.2 and duration of the paths.

Note that this game is not concave by taking i = B, S = {A}, andT = {A,D} since cy({A,B}) � cy({A}) = �2320 j �2145 = cy({A,B,D}) �cy({A,D}). h

Theorem 3.1. Let ({N1, . . . ,Nm}, p, r,a,R) be a project problem and let ybe a cost sharing mechanism. Then, (N,cy) has a nonempty core.

3.2. Expedition in project problems

If DðpjEnS; rjNnðEnSÞÞ < DðpÞ, then the expedition created by theexpedited activities in S is enough to expedite the project andvz(S) reflects the amount of reward from the expedition that thecoalition may claim. Before starting with the description of theexpedited part of project games, we give the following examplethat illustrates the ideas behind the model.

Example 3.3. Consider the project problem ({N1,N2,N3,N4},p,r,R)in Example 3.1. In this project problem, D(p) = 21 and D(r) = 15,therefore there is a total expedition of D(p) � D(r) = 6 and a rewardof R(6) = 903.

Note that the critical path N1 = {A} is always indispensable toexpedite the project. Moreover, due to the delay of activity C,activity D is also indispensable to expedite the project. Wesubsequently explain how this expedition is achieved. First,suppose that the delayed activities (activity C) act according torealization while the expedited activities act according to plan.Second, assume that both the expedited activities in N3, i.e. activityD, and the activities in N1 act according to realization whileactivities in N2 and N4 act according to plan, then the project isexpedited in 2 with an associated reward of R(2) = 800, and N2

becomes critical. Note that N1 and N3 are exclusively responsiblefor a reward of 800, and that N2 becomes indispensable to continueexpediting. Third, suppose that also activities in N2 act according torealization, then the project has an extra expedition of 2 with anassociated marginal reward of R(4) � R(2) = 901 � 800 = 101 forwhich N1, N2, and N3 are responsible. Last, suppose that allactivities act according to realization, then the project has an extraexpedition of 2 with an associated marginal reward of R(6) � R(4) =903 � 901 = 2 for which N1, N2, N3 and N4 are responsible. Thedescription of the sequence in which realization times are consid-ered for the activities is given in Table 2.a and the contribution ofthe paths to the reward obtained by the expedition of the project issummarized in Table 2.b.

Note that the sum of both the first and third rows in Table 2.bgives the total reward. This type of decomposition by means oflevels of expedition plays an important role in the definition of theexpedited part of project games. h

In order to define the value corresponding to the expedited partof a project game, we first optimistically allocate the part of theexpedition that each path could contribute to among its activities.In a second step, we define the value of the game using the initialallocations as reference points.

Before defining vz, we need to introduce some notation. Wedenote by rslack(Na,p,r) the amount of remaining slack of a pathwith respect to the planned duration if only its delayed activitiesact according to realization, i.e. rslackðNa; p; rÞ ¼ slackðNa; pÞ�P

i2NadðiÞ. Note that rslack(Na,p,r) can be negative, meaning

that the delayed activities have consumed all the initial slack andwould produce a delay on the project, as a whole, of�rslack(Na,p,r) if the expedited activities had acted according toplan. We denote by J1 the set (of indexes) of paths with remainingslack less than or equal to zero:

J1 ¼ fa 2 f1; . . . ;mgjrslackðNa; p; rÞ 6 0g:

Table 1Coalitional values in Example 3.2.

S {A} {B} {C} {D} {A,B} {A,C} {A,D} {B,C} {B,D} {C,D} {A,B,C} {A,B,D} {A,C,D} {B,C,D} N

cy(S) 2501 0 0 356 181 2501 2501 0 356 181 181 356 2501 181 181

Table 2Description of the decomposition of the total rewards by levels of expedition in Example 3.3.

a: Sequence in which realization times are considered for the activities

p(A) = 21 p(A) = 21 r(A) = 14 r(A) = 14 r(A) = 14p(B) = 19 p(B) = 19 p(B) = 19 r(B) = 15 r(B) = 15p(C) = 2, p(D) = 16 r(C) = 7, p(D) = 16 r(C) = 7, r(D) = 6 r(C) = 7, r(D) = 6 r(C) = 7, r(D) = 6p(E) = 17 p(E) = 17 p(E) = 17 p(E) = 17 p(E) = 17

N1 21 21 14 14 14N2 19 19 19 15 15N3 18 23 13 13 13N4 17 17 17 17 14

b: Decomposition of the total reward

Phase 1 Phase 2 Phase 3

N1 800 101 2N2 0 101 2N3 800 101 2N4 0 0 2

2 The idea behind the model is the following. By being pessimistic, we consider thatall delayed activit ies are planned according to realization and ifDðpÞ � DðpjE ; rjNnE Þ < 0, i.e. the delayed activities could bring delay to the project ontheir own, then the reward function for the expedited activities changes toe

652 A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657

Recursively, we define for k P 2

Jk ¼ a 2 f1; . . . ;mg n[k�1

l¼1

Jl

( �����rslackðNa;p; rÞ 6 rslackðNb;p; rÞ

for all b 2 f1; . . . ;mg n[k�1

l¼1

Jl

);

i.e. Jk contains all paths that would have smallest remaining slack ifthe paths in J1, . . . , Jk�1 were not present. Set rslack(J1) :¼ 0 and letrslack(Jk) denote the remaining slack of the paths in Jk for k P 2,i.e. rslack(Jk) = rslack(Na,p,r) for each a 2 Jk, k P 2. Let g be such thatrslack(Jg) < D(p) � D(r) 6 rslack(Jg+1) if D(p) � D(r) > 0 and g = 0otherwise. For k = 1, . . . ,g, we define Fk as the marginal contributionof the paths in J1, . . . , Jk to the total reward associated to the expedi-tion. Formally,

Fk ¼RðrslackðJkþ1ÞÞ � RðrslackðJkÞÞ if 1 6 k < g;

RðDðpÞ � DðrÞÞ � RðrslackðJgÞÞ if k ¼ g:

(

Note thatPg

k¼1Fk ¼ RðDðpÞ � DðrÞÞ since R(rslack(J1)) = 0.For every a 2 {1, . . . ,m}, consider the surplus sharing problem

ðNa; �qa;RaÞ with1 �qai ¼minfeðiÞ;minj2Na\Fi

faðjÞgg for every i 2 Na(i.e. since activity i needs to coordinate with its followers to expeditethe project, i cannot expedite the path e(i) if its followers do not haveenough availability to start as soon as allowed by i’s expedition) andRa(t) = R(t + (rslack(Na,p, r))+) � R((rslack(Na,p, r))+) if t P 0 andRa(t) = 0 otherwise.

Selecting a surplus sharing mechanism z (taking into accountthe type of reward function at hand), we denote by za the alloca-tion proposed by our mechanism to the surplus sharing problemðNa; �qa;RaÞ, i.e. za ¼ zðNa; �qa;RaÞ. Here, za

i is the maximum amountthat i can claim according to the surplus sharing mechanism if itspath is awarded with the total expedition that it can bring to theproject. Then, we optimistically define the vector of maximal re-wards, fz, by f z

i ¼maxa:Na3i zai

� �for every i 2 N, i.e. f z

i is the maxi-mum reward that activity i can claim from the expedition of theproject when the surplus sharing mechanism z is considered.

Next, we define vz(S) representing the sum over all k 2 {1, . . . ,g}of those specific parts of the contribution to the total reward Fk for

1 Note that if all followers of i in path a have enough availability to start as soon asneeded, then �qa

i ¼ eðiÞ.

which the activities outside the coalition that are in paths ofSk

l¼1Jl

cannot be held responsible for anymore at that phase. Formally,2

vzðSÞ ¼Xg

k¼1

Fk �wkzðSÞ

� �; ð3:2Þ

where wkzðSÞ represents the part of the contribution to the total

reward Fk that players in S maximally would have to concede toplayers in the paths corresponding to

Skl¼1Jl outside S, taking into

account earlier concessions from the previous phases. Formally,

wkzðSÞ ¼min

Xi2ðSk

l¼1NJlÞnS

f zi �

Xk�1

l¼1

wlzðSÞ; F

k

8><>:9>=>; ð3:3Þ

for all k 2 {1, . . . ,g}, where NJl¼S

a2JlNa. Note that wk

z is non-negative. Moreover, vz(N) equals the total expedition of the projectsince wk

zðNÞ ¼ 0 for any k 2 {1, . . . ,g}.

Example 3.4. Consider the project problem ({N1,N2,N3,N4},p,r,R)in Example 3.1. In this project problem D(p) = 21 and D(r) = 15,therefore there is a total expedition of D(p) � D(r) = 6 and a rewardof R(6) = 903. Besides,

eðAÞ ¼ 7; eðBÞ ¼ 4; eðCÞ ¼ 0; eðDÞ ¼ 10 and eðEÞ ¼ 3;

rslackðN1; p; rÞ ¼ 0� 0 ¼ 0; rslackðN2;p; rÞ ¼ 2� 0 ¼ 2;rslackðN3; p; rÞ ¼ 3� 5 ¼ �2; and rslackðN4;p; rÞ ¼ 4� 0 ¼ 4;

J1 ¼ f1;3g; J2 ¼ f2g; and J3 ¼ f4g;F1 ¼ RðrslackðJ2ÞÞ � RðrslackðJ1ÞÞ ¼ Rð2Þ � Rð0Þ ¼ 800;

F2 ¼ RðrslackðJ3ÞÞ � RðrslackðJ2ÞÞ ¼ Rð4Þ � Rð2Þ ¼ 101; and

F3 ¼ RðDðpÞ � DðrÞÞ � RðrslackðJ3ÞÞ ¼ Rð6Þ � Rð4Þ ¼ 2:

RðtÞ ¼ ðRðt þ DðpÞ � DðpjE ; rjNnE ÞÞÞþ . In this way, the expedited activities need toallocate eRðDðpjE ; rjNnE Þ � DðrÞÞ among themselves. To solve this allocation problem, theideas behind bankruptcy games are followed taking into account the levels ofexpedition.

Table 3Coalitional values in Example 3.4.

S {A} {B} {C} {D} {E} {A,B} {A,C} {A,D} {A,E} {B,C} {B,D} {B,E} {C,D} {C,E} {D,E}

vz(S) 0 0 0 0 0 0 0 800 0 0 0 0 0 0 0

S {A,B,C} {A,B,D} {A,B,E} {A,C,D} {A,C,E} {A,D,E} {B,C,D} {B,C,E} {B,D,E} {C,D,E}

vz(S) 0 901 0 800 0 800 0 0 0 0

S {A,B,C,D} {A,B,C,E} {A,B,D,E} {A,C,D,E} {B,C,D,E} N

vz(S) 901 0 903 800 0 903

A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657 653

For the computation of (N,vz) we use the serial surplus sharingmechanism. Associated to each path Na we have the surplus sharingproblem:

ðN1; �q1;R1Þ : N1 ¼ fAg; �q1 ¼ ð7Þ;

R1ðtÞ ¼RðtÞ if t 6 0;0 if t > 0;

and z1 ¼ ð904Þ;

ðN2; �q2;R2Þ : N2 ¼ fBg; �q2 ¼ ð4Þ;

R2ðtÞ ¼Rðt þ 2Þ � Rð2Þ if t P 0;0 if t > 0;

and z2 ¼ ð103Þ;

ðN3; �q3;R3Þ : N3 ¼ fC;Dg; �q3 ¼ ð0;10Þ;

R3ðtÞ ¼RðtÞ if t 6 0;0 if t > 0;

and z3 ¼ ð0;907Þ;

ðN4; �q4;R4Þ : N4 ¼ fEg; �q4 ¼ ð3Þ;

R4ðtÞ ¼Rðt þ 4Þ � Rð4Þ if t P 0;0 if t > 0;

and z4 ¼ ð3Þ;

which gives f z = (904,103,0,907,3). Below, the computation ofvz({A,B,D}) is explained.

w1z ðfA;B;DgÞ ¼min f z

C ;F1

n o¼minf0;800g ¼ 0;

w2z ðfA;B;DgÞ ¼min f z

C �w1z ðfA;B;DgÞ;F

2n o

¼minf0� 0;101g ¼ 0;

w3z ðfA;B;DgÞ ¼min f z

C þ f zE �w1

z ðfA;B;DgÞ�w2z ðfA;B;DgÞ;F

3n o

¼minf0þ3� 0� 0;2g ¼ 2

and therefore

vzðfA;B;DgÞ ¼ F1 �w1z ðfA;B;DgÞ

� �þ F2 �w2

z ðfA;B;DgÞ� �

þ ðF3 �w3z ðfA;B;DgÞÞ

¼ ð800� 0Þ þ ð101� 0Þ þ ð2� 2Þ ¼ 901:

All coalitional values are given in Table 3.It can be checked that the core of the game is

CoreðvzÞ ¼ convfð0;0;0;901;2Þ; ð0;0;0;903;0Þ; ð0;101;0;800;2Þ;ð0;103;0;800; 0Þ; ð901;0;0;0;2Þ; ð903;0;0;0;0Þ;ð800;101;0;0;2Þ; ð800;103;0;0; 0Þg: �

Given a project problem ({N1, . . . ,Nm},p,r,a,R) and a surplussharing mechanism z, the corresponding game (N,vz) can bedescribed as the maximum of as many bankruptcy games as levelsof expedition in the project, where the bankruptcy problem

associated to level of expedition k 2 {1, . . . ,g} is (N,Ek,cz,k) with

Ek = R(rslack(Jk+1)) if k < g and Eg = R(D(p) � D(r)), and cz;ki ¼ f z

i if

i 2 [kl¼1NJl

and cz;ki ¼ 0 otherwise. Besides, it turns out that (N,vz)

is convex.

Theorem 3.2. Let ({N1, . . . ,Nm}, p, r,a,R) be a project problem and let zbe a surplus sharing mechanism. Then,

vzðSÞ ¼ maxk2f1;...;gg

fv ðN;Ek ;cz;kÞðSÞg

for every S � N.

Convexity of (N,vz) can be shown following the same lines as inthe proof of Theorem 4.2 in Estévez-Fernández et al. (2007). Wehave decided to provide this new proof since it is more under-standable and less technical than that in Estévez-Fernández et al.(2007). The simplicity of this new proof is rooted in both the rela-tion between the expedited part of a project game and a number ofrelated bankruptcy games, and the special structure of the claimsof the related bankruptcy games.

Theorem 3.3. Let ({N1, . . . ,Nm}, p, r,a,R) be a project problem and let zbe a surplus sharing mechanism. Then, (N,vz) is convex.

3.3. Project games

Finally, we define the associated project game (N,uyz) by

uyzðSÞ ¼�cyðSÞ if DðpjEnS; rjNnðEnSÞÞP DðpÞ;

vzðSÞ if DðpjEnS; rjNnðEnSÞÞ < DðpÞ

(ð3:4Þ

for every S � N. Note that if the project has been delayed, thenuyz = �cy since DðpjEnS; rjNnðEnSÞÞP DðrÞP DðpÞ and therefore projectgames need not be convex (see Example 3.2).

As an illustrative example of the computation of project games,we compute the project game and its corresponding core for theproject problem in Examples 3.1 and 3.4.

Example 3.5. Consider the project problem ({N1,N2,N3,N4},p,r,R)in Examples 3.1 and 3.4 which is represented in Fig. 2. For thecomputation of (N,uyz), we first compute (N,cy) and (N,vz). In orderto compute (N,cy), note that only activity C, which is in path N3, hasbeen delayed and hence y1 = (0), y2 = (0), y3 = (�R(�2),0) = (100,0)and y4 = (0). Recall that we have used the serial surplus sharingmechanism for the computation of (N,vz) (see Example 3.4). Allcoalitional values are given in Table 4.

It can be checked that the core of the game is

CoreðuyzÞ ¼ convfð0; 0;�100;1001;2Þ; ð0; 0;�100;1003;0Þ;ð0; 0;0;901;2Þ; ð0;0; 0;903;0Þð0;101;�100;900;2Þ;ð0;103;�100;900; 0Þ; ð0;101; 0;800;2Þ; ð0;103; 0;800;0Þð901; 0;�100;100;2Þ; ð903;0;�100;100; 0Þ;ð901; 0;0;0;2Þ; ð903;0; 0;0;0Þð800;101;�100;100;2Þ;ð800;103;�100;100; 0Þ; ð800;101;0; 0;2Þ;ð800;103;0; 0; 0Þg: �

The following example illustrates how availabilities of theactivities are taken into account when solving a project problem.

Example 3.6. Consider the project problem ({N1,N2},p,r,a,R) withN1 = {A,C} and N2 = {B,C}; p(A) = 15, p(B) = 10, and p(C) = 8; r(A) = 7,r(B) = 6, and r(C) = 12; a(A) = 0, a(B) = 0, and a(C) = 7; and

Table 4Computation of the project game in Example 3.5.

S {A} {B} {C} {D} {E} {A,B} {A,C} {A,D} {A,E} {B,C} {B,D} {B,E} {C,D} {C,E} {D,E}

DðpjEnS ; rjNnðEnSÞÞ 23 23 23 21 23 23 23 19 23 23 21 23 21 23 21�cy(S) 0 0 �100 0 0 0 �100 0 0 �100 0 0 0 �100 0vz(S) 0 0 0 0 0 0 0 800 0 0 0 0 0 0 0uyz(S) 0 0 �100 0 0 0 �100 800 0 �100 0 0 0 �100 0

S {A,B,C} {A,B,D} {A,B,E} {A,C,D} {A,C,E} {A,D,E} {B,C,D} {B,C,E} {B,D,E} {C,D,E}

DðpjEnS ; rjNnðEnSÞÞ 23 17 23 19 23 19 21 23 21 21�cy(S) �100 0 0 0 �100 0 0 �100 0 0vz(S) 0 901 0 800 0 800 0 0 0 0uyz(S) �100 901 0 800 �100 800 0 �100 0 0

S {A,B,C,D} {A,B,C,E} {A,B,D,E} {A,C,D,E} {B,C,D,E} N

DðpjEnS ; rjNnðEnSÞÞ 17 23 15 19 21 15�cy(S) 0 �100 0 0 0 0vz(S) 901 0 903 800 0 903uyz(S) 901 �100 903 800 0 903

Table 5Computation of the project game in Example 3.6.

S {A} {B} {C} {A,B} {A,C} {B,C} {A,B,C}

DðpjEnS ; rjNnðEnSÞÞ 22 27 27 20 22 27 20�cy(S) 0 0 �164 0 0 �164 0vz(S) 201 0 0 227 201 0 227uyz(S) 201 0 �164 227 201 �164 227

Table 6Coalitional values in Example 4.1.

S {A} {B} {C} {A,B} {A,C} {B,C} N

�uðSÞ �1 �3 0 �4 0 �2 �3uyz(S) �1 �3 0 �4 �1 �3 �3

654 A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657

RðtÞ ¼t3 � 100 if t < 0;0 if t ¼ 0;t3 þ 200 if t > 0;

8><>:represented in Fig. 1.

Note that since activity C can only start 7 units of time beforethe planned starting time (i.e. at time 8), the expedition of A is notfully exploited. Hence, D(Na,r) = 20 since C cannot start before time8. In this project problem, D(p) = 23 and D(r) = 20, therefore there isa total expedition of D(p) � D(r) = 3 and a reward of R(3) = 227.Besides,

dðAÞ ¼ 0; dðBÞ ¼ 0; and dðCÞ ¼ 4;

eðAÞ ¼ 8; eðBÞ ¼ 4; and eðCÞ ¼ 0;

rslackðN1;p; rÞ ¼ 0� 4 ¼ �4 and rslackðN1;p; rÞ ¼ 5� 4 ¼ 1;

J1 ¼ f1g and J2 ¼ f2g; and

F1 ¼ Rð1Þ � Rð0Þ ¼ 201 and F2 ¼ Rð3Þ � Rð1Þ ¼ 26:

For the computation of (N,uyz), we first compute (N,cy) and (N,vz).Since only activity C has been delayed and C needs the expeditionof A to compensate its own delay, we have that, for any cost sharingmechanism y, cy(S) = 164 if S = {C} or S = {B,C} and cy(S) = 0 otherwise.For the computation of (N,vz) we use the serial surplus sharing mech-anism. Associated to each path Na we have the surplus problem:

ðN1; �q1;R1Þ : N1 ¼ fA;Cg; �q1ðAÞ ¼minfeðAÞ; aðCÞg ¼minf8;7g ¼ 7

and �q1ðCÞ ¼ 0;

R1ðtÞ ¼ RðtÞ for t > 0 and R1ðtÞ ¼ 0 otherwise; and z1 ¼ ð543;0Þ:

ðN2; �q2;R2Þ : N2 ¼ fB;Cg; �q2ðBÞ ¼minfeðBÞ; aðCÞg ¼minf4;7g ¼ 4

and �q2ðCÞ ¼ 0;

R2ðtÞ ¼ Rðt þ 1Þ � Rð1Þ for t > 0 and R2ðtÞ ¼ 0 otherwise; and

z2 ¼ ð124;0Þ:

All coalitional values are given in Table 5.

It can be checked that the core of the game is

CoreðuyzÞ ¼ convfð391;0;�164Þ; ð365;26;�164Þ; ð201;26;0Þ;ð227;0;0Þg: �

Theorem 3.4. Project games have a nonempty core.

In many project problems the general manager of the projectdoes not have legal authority to oblige delayed activities to com-pensate expedited activities for their contribution to decrease thetotal delay of the project. In this situation, a set-valued solutionshould satisfy: if the project is neither delayed, nor expedited, thenthere should be a solution in which nobody is neither punished,nor rewarded, i.e. if D(r) = D(p), then the zero vector should be apossible solution; if the project is delayed, then there should be asolution in which the delayed activities pay exactly the total costassociated to the total delay, i.e. expedited activities are not com-pensated; if the project is expedited, then there should be a solu-tion in which expedited activities get exactly the total rewardassociated to the total expedition, i.e. the delayed activities donot have to compensate expedited activities.

Let D be the set of delayed activities, i.e. D ¼ fi 2 NjpðiÞ < rðiÞg,and recall that E is the set of expedited activities, i.e. E ¼fi 2 NjpðiÞ > rðiÞg.

Theorem 3.5. Let ({N1, . . . ,Nm} , p, r,a,R) be a project problem, let yand z be a cost and surplus sharing mechanism, respectively, and let(N,uyz) be the associated project game.

1. If D(p) < D(r), then there exist x 2 Core(uyz) such that xi = 0 forevery i 2 N n D.

2. If D(p) = D(r), then 0 2 Core(uyz).3. If D(p) > D(r), then there exist x 2 Core(uyz) such that xi = 0 for

every i 2 N n E.

4. Final remarks

We have generalized the work in Estévez-Fernández et al.(2007) by both considering non decreasing reward functions and

A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657 655

by taking into account whether an activity can be started before itsplanned starting time. It can be easily seen that in project problemswhere the reward function is proportional to the expedition of theproject, the core of the game proposed in Estévez-Fernández et al.(2007) is always contained in the core of our project game. Besides,the core of the game in Estévez-Fernández et al. (2007) does notneed to satisfy the properties in Theorem 3.5 as illustrated in thefollowing example.

Example 4.1. Consider the project problem ({N1},p,r,R) withN1 = {A,B,C}; p(A) = 7, p(B) = 8, and p(C) = 10; r(A) = 8, r(B) = 11,and r(C)=9; and R(t) = t. In this problem, D(p) � D(r) = �3; d(A) = 1,d(B) = 3, and d(C) = 0; e(A) = 0, e(B) = 0, and e(C) = 1. The values ofthe game in Estévez-Fernández et al. (2007), ðN; �uÞ, and our projectgame are given in Table 6. For the computation of (N,uyz) we haveused y(N,q,K) = q and zðN; �q;RÞ ¼ �q.

It can be checked that Coreð�uÞ ¼ fð�1;�3;1Þg while Core(uyz) =conv{(�1,�3,1), (�1,�2,0), (0,�3,0)}. h

We have used cooperative games to find solutions to projectproblems. The associated project game, in our opinion, providesan adequate thought experiment to evaluate coalitional influenceand the core of this game provides a suitable answer to the alloca-tion problem at hand.

Contrary to our focus on finding suitable allocations satisfyingsome basic properties, Castro et al. (2007) concentrate on findinga game related to project problems satisfying some ‘‘desirable’’properties. They put forward the properties of separability, non-manipulability by splitting, and independent slack, and propose acooperative game to share the total delay or expedition of a projectsatisfying these three properties. One can question the desirabilityof these three properties when we concentrate on the allocation ofthe rewards (or penalties) created by a project that has not per-formed as planned. For instance, the property of separability saysthat if a project can be decomposed in two different (sub) projects(i.e. if there is a node used by all paths in the project), then theassociated game can be decomposed as the sum of the two gamesassociated to the two corresponding (sub) projects. Note that thetotal reward of the project does not need to equal the sum of therewards of the (sub) projects since we allow for non additivereward functions too. Hence, in our setting, separability does notneed to be satisfied by project problems, let alone by the associatedgames.

In our opinion, it is not the properties of the game as a wholethat are relevant, but rather the properties of the derived solutions(except of course from adequately modeling the coalitionalpossibilities).

As mentioned above, Castro et al. (2007) define a cooperativegame to share the total delay or expedition of a project. If one usesthe core of this game to share the total reward in a project problemwhere the reward function is proportional to the total expeditionor delay of the project, one encounters unwanted features in theallocations proposed by the core. It turns out that for projects inwhich the corresponding graph is a line, the game in Castro et al.(2007) is additive (i.e. the value of a coalition equals the sum ofthe individual values of its members). Therefore, the core of theirgame does not need to satisfy any of the properties proposed inSection 3.

Appendix A

Proof of Theorem 3.1. Let a 2 f1; . . . ;mg be such that DðrÞ ¼DðNa; rÞ, i.e. Na is responsible for the total delay of the project.Consider the taxation problem ðN; Ea; caÞ given by Ea ¼ KðDðNa; rÞ�DðpÞÞ, ca

i ¼ yai if i 2 Na and ca

i ¼ 0 if i 2 N n Na. Then,

cyðNÞ ¼ KðDðNa; rÞ � DðpÞÞ ¼minXi2Na

yai ; K DðNa; rÞ � DðpÞð Þ

8<:9=;

¼ minXi2N

cai ; Ea

( )¼ cðN;Ea ;caÞðNÞ;

where the first equality follows because DðrÞ ¼ DðNa; rÞ, the second

one is a direct consequence of KðDðNa; rÞ � DðpÞÞ 6 KP

i2NadðiÞ

� �¼P

i2Naya

i since R is nondecreasing and hence K is also nondecreasing,and the third one is a direct consequence of the definition of Ea andca. Moreover, for every S � N we have

cyðSÞ ¼ maxa2PðSÞ

minX

i2Na\S

yai ; KðDðNa; ðpjEnS; rjNnðEnSÞÞÞ � DðpÞÞ

( )( )

P minX

i2Na\S

yai ; KðDðNa; ðpjEnS; rjNnðEnSÞÞÞ � DðpÞÞ

8<:9=;

P minX

i2Na\S

yai ; KðDðNa; rÞ � DðpÞÞ

8<:9=; ¼min

Xi2S

cai ; Ea

( )

¼ cðN;Ea ;caÞðSÞ;

where the second inequality follows because R is nondecreasingand therefore K is also nondecreasing, and the second equality isa direct consequence of the definitions of Ea and ca.

Since ðNa; cðN;Ea ;caÞÞ is concave, we know that there is an x 2CoreðcðN;Ea ;caÞÞ and then

Pi2Nxi ¼ cðN;Ea ;caÞðNÞ ¼ cyðNÞ and

Pi2Sxi 6

cðN;Ea ;caÞðSÞ 6 cyðSÞ for every S � N, and therefore x 2 Core (cy). h

Proof of Theorem 3.2. Since z is fixed, we denote cz,k by ck. Weproceed by induction on g. Let g = 1 and S � N, then

vzðSÞ ¼ F1 �w1z ðSÞ ¼ F1 �min

Xi2NnS

f zi ; F

1

( )

¼max F1 �Xi2NnS

f zi ;0

( )¼ v ðN;E1 ;c1ÞðSÞ:

Next, assume that the result is satisfied for 1, . . . ,g � 1. Let S � N,then

vzðSÞ ¼Xg

k¼1

Fk �Xg�1

k¼1

wkzðSÞ �min

Xi2Sg

k¼1NJk

� �nS

f zi �

Xg�1

k¼1

wkzðSÞ; F

g

8><>:9>=>;

¼maxXg

k¼1

Fk �X

i2ðSg

k¼1NJkÞnS

f zi ;Xg�1

k¼1

Fk �Xg�1

k¼1

wkzðSÞ

8><>:9>=>;

¼maxXg

k¼1

Fk �X

i2Sg

k¼1NJk

� �nS

f zi ; max

k2f1;...;g�1gfv ðN;Ek ;ckÞðSÞg

8><>:9>=>;

¼maxXg

k¼1

Fk �X

i2ðSg

k¼1NJkÞnS

f zi

0B@1CAþ

; maxk2f1;...;g�1g

fv ðN;Ek ;ckÞðSÞg

8><>:9>=>;

¼max Eg �Xi2NnS

cgi

!þ

; maxk2f1;...;g�1g

fv ðN;Ek ;ckÞðSÞg

8<:9=;

¼max v ðN;Eg ;cg ÞðSÞ; maxk2f1;...;g�1g

fv ðN;Ek ;ckÞðSÞg

¼ maxk2f1;...;gg

fv ðN;Ek ;ckÞðSÞg;

where the third equality follows by induction, the fourth one is aconsequence of maxk2f1;...;g�1gfv ðN;Ek ;ckÞðSÞgP 0, and the fifth one

656 A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657

follows becausePg

k¼1Fk ¼ RðDðpÞ � DðrÞÞ ¼ Eg and because, by defi-nition of cg, we have cg

i ¼ f zi if i 2

Sgk¼1NJk

and cgi ¼ 0 otherwise. h

Before showing the convexity of (N,vz), we need to introducesome notation. Note that, by definition of cz,k, we have thatcz;k0

i 6 cz;ki for every i 2 N and every k0 6 k. For U � N, let kðU;vzÞ

denote the smallest index satisfying vzðUÞ ¼ v ðN;EkðU;vz Þ ;cz;kðU;vz ÞÞðUÞ,i.e.

kðU; vzÞ ¼minfk 2 f1; . . . ; ggjvzðUÞ ¼ v ðN;Ek ;cz;kÞðUÞg:

Proof of Theorem 3.3. Let ({N1, . . . ,Nm},p,r,R) be an expeditedproject problem, let z be a cost sharing mechanism, and let (N,vz)be the expedited part of the project game. Since z is fixed, wedenote cz,k by ck. Let i 2 N and S � T � Nn{i}, we have to show thatvz(S [ {i}) � vz(S) 6 vz(T [ {i}) � vz(T).

Note that (N,vz) satisfies monotonicity, i.e. vz(S) 6 vz(T) for everyS � T � N, since bankruptcy games are monotonic together withTheorem 3.2. Therefore, if vz(S [ {i}) � vz(S) = 0 or vz(S) = vz(T), thenthe condition is satisfied by monotonicity of (N,vz). We can thenassume without loss of generality that vz(S [ {i}) > vz(S) andvz(T) > vz(S) by monotonicity of (N,vz). This assumption impliesthat vz(S [ {i}) > 0 and vz(T) > 0 since vz is nonnegative. We distin-guish between two cases.

Case 1: kðS [ fig;vzÞ 6 kðT;vzÞ. Then,

vzðS [ figÞ � vzðSÞ ¼ EkðS[fig;vzÞ �Xj2NnS

ckðS[fig;vzÞj þ ckðS[fig;vzÞ

i

� EkðS;vzÞ �Xj2NnS

ckðS;vzÞj

!þ

6 EkðS[fig;vzÞ �Xj2NnS

ckðS[fig;vzÞj þ ckðS[fig;vzÞ

i

� EkðS[fig;vzÞ �Xj2NnS

ckðS[fig;vzÞj

!þ

6 ckðS[fig;vzÞi 6 ckðT;vzÞ

i ¼ EkðT;vzÞ �X

j2NnTckðT;vzÞ

j

þ ckðT;vzÞi � EkðT;vzÞ þ

Xj2NnT

ckðT;vzÞj

6 EkðT[fig;vzÞ �X

j2NnTckðT[fig;vzÞ

j þ ckðT[fig;vzÞi

� EkðT;vzÞ �X

j2NnTckðT;vzÞ

j

!

¼ vzðT [ figÞ � vzðTÞ;

where the first equality is a direct consequence of thedefinition of k applied to S [ {i} and S and becausevz(S [ {i}) > vz(S)(P0) by assumption; the first inequality followsby definition of k applied to coalition S; the second inequality is adirect consequence of (x)+ = max{0,x}; the third inequality fol-lows because kðS [ fig;vzÞ 6 kðT; vzÞ and by definition of ck wehave ckðS[fig;vzÞ 6 ckðT;vzÞ; the last inequality follows by definitionof k applied to coalition T [ {i} and vz(T [ {i}) P vz(T) > 0; andthe last equality is a direct consequence of the definition of kand vz(T [ {i}) P vz(T) > vz(S)(P0) by assumption.

Case 2: kðS [ fig;vzÞ > kðT;vzÞ. In this case we show that theequivalent condition vz(S [ {i}) + vz(T) 6 vz(T [ {i}) + vz(S)is satisfied.

vzðS[ figÞþ vzðTÞ ¼ EkðS[fig;vzÞ �X

j2NnðS[figÞckðS[fig;vzÞ

j

þ EkðT;vzÞ �X

j2NnTckðT;vzÞ

j

¼ EkðS[fig;vzÞ �X

j2NnðS[figÞckðS[fig;vzÞ

j

þ EkðT;vzÞ �Xj2NnS

ckðT;vzÞj þ

Xj2TnS

ckðT;vzÞj

6 EkðS[fig;vzÞ �X

j2NnðS[figÞckðS[fig;vzÞ

j

þ EkðT;vzÞ �Xj2NnS

ckðT;vzÞj

!þ

þXj2TnS

ckðT;vzÞj

6 EkðS[fig;vzÞ �X

j2NnðS[figÞckðS[fig;vzÞ

j

þ EkðS;vzÞ �Xj2NnS

ckðS;vzÞj

!þ

þXj2TnS

ckðT;vzÞj

6 EkðS[fig;vzÞ �X

j2NnðS[figÞckðS[fig;vzÞ

j

þ EkðS;vzÞ �Xj2NnS

ckðS;vzÞj

!þ

þXj2TnS

ckðS[fig;vzÞj

¼ EkðS[fig;vzÞ �X

j2NnðT[figÞckðS[fig;vzÞ

j

þ EkðS;vzÞ �Xj2NnS

ckðS;vzÞj

!þ

6 EkðT[fig;vzÞ �X

j2NnðT[figÞckðT[fig;vzÞ

j

þ EkðS;vzÞ �Xj2NnS

ckðS;vzÞj

!þ

¼ vzðT [ figÞþ vzðSÞ;

where the first equality is a direct consequence of the definition of kapplied to S [ {i} and T together with the assumptionvz(S [ {i}) > vz(S)(P0) and vz(T) > vz(S)(P0); the second inequalityfollows by definition of k applied to S; the third inequality is a directconsequence of kðT; vzÞ < kðS [ fig;vzÞ and by definition of ck wehave ckðT;vzÞ 6 ckðS[fig;vzÞ; the last inequality follows by definition ofk applied to coalition T [ {i} together with vz(T [ {i}) P vz(T) > 0; fi-nally, the last equality follows by definition of k together withvz(T [ {i}) P vz(T)(>0) by monotonicity of vz. h

Proof of Theorem 3.4. Let ({N1, . . . ,Nm},p,r,a,R) be a projectproblem, let y and z be a cost and a surplus sharing mechanism,respectively, and let (N,uyz) be the associated project game. IfD(p) 6 D(r), then DðpÞ 6 DðrÞ 6 DðpjEnS; rjNnðEnSÞÞ for every S � N anduyz(S) = �cy(S) for every S � N. Then, (N,uyz) has a nonempty coreby Theorem 3.1.

If D(p) > D(r), then uyz(N) = vz(N) and uyz(S) 6 vz(S) for everyS � N. By Theorem 3.3, we know that (N,vz) is convex and thereforeCore (vz) – ;. Let x 2 Core (vz), then

Pi2Nxi ¼ vzðNÞ ¼ uyzðNÞ andP

i2Sxi P vzðSÞP uyzðSÞ for every S � N, and therefore x 2 Core(uyz). h

Proof of Theorem 3.5

1. If D(p) < D(r), then DðpÞ < DðrÞ 6 DðpjEnS; rjNnðEnSÞÞ for every S � Nand uyz(S) = �cy(S) for every S � N. Let a 2 f1; . . . ;mg be suchthat DðrÞ ¼ DðNa; rÞ, i.e. Na is responsible for the total delay of

A. Estévez-Fernández / European Journal of Operational Research 216 (2012) 647–657 657

the project. Consider the taxation problem ðN; Ea; caÞ given byEa ¼ Kð

Pi2Na

dðiÞ �P

i2NaeðiÞ � slackðNa; pÞÞ, ca

i ¼ yai if i 2 Na

and cai ¼ 0 if i 2 N n Na. Note that ca

i ¼ 0 for every i 2 Na n D.By the proof of Theorem 3.4, we know that CoreðcðN;Ea ;caÞÞ �CoreðcyÞ. Moreover, it is well known that any x 2

CoreðcðN;Ea ;caÞÞ satisfies 0 6 x 6 ca, and therefore xi = 0 for everyi 2 N n D.

2. If D(p) = D(r), then DðpÞ ¼ DðrÞ 6 DðpjEnS; rjNnðEnSÞÞ for every S � N,and therefore uyz(S) = �cy(S) 6 0 for every S � N. Moreover,uyz(N) = 0 and then 0 2 Core(uyz).

3. If D(p) > D(r), then uyz(N) = vz(N) and uyz(S) 6 vz(S) for everyS � N. Then, Core (vz) � Core (uyz). Since f z

i ¼ 0 for everyi 2 N n E, we have vz(S [ {i}) = vz(S) for every i 2 N n E and there-fore xi = 0 for every x 2 Core (vz) and i 2 N n E. h

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