Processing games with restricted capacities

European Journal of Operational Research 202 (2010) 773–780 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor Stochastics and Statistics Processing games with restricted capacities Hans Reijnierse a,*, Peter Borm a, Marieke Quant a, Marc Meertens b a b CentER and Department of Econometrics and OR, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands Reaal Insurances, Utrecht, The Netherlands a r t i c l e i n f o Article history: Received 10 June 2008 Accepted 8 June 2009 Available online 24 June 2009 Keywords: Scheduling Individual capacity Cooperation Core allocation a b s t r a c t This paper analyzes processing problems and related cooperative games. In a processing problem there is a finite set of jobs, each requiring a specific amount of effort to be completed, whose costs depend linearly on their completion times. The main feature of the model is a capacity restriction, i.e., there is a maximum amount of effort per time unit available for handling jobs. There are no other restrictions whatsoever on the processing schedule. Assigning to each job a player and letting each player have an individual capacity for handling jobs, each coalition of cooperating players in fact faces a processing problem with the coalitional capacity being the sum of the individual capacities of the members. The corresponding processing game summarizes the minimal joint costs for every coalition. It turns out that processing games are totally balanced. The proof of this statement is constructive and provides a core element in polynomial time. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction Consider the situation in which a number of jobs have to be completed, each requiring its own amount of effort, and in which there is a capacity constraint to process jobs. The terminology has been chosen very general in order to let several interpretations fit. Jobs can involve maintenance problems, the manufacturing of products, computational tasks, or investments under periodic budget raises. Capacity constraints can be induced by limited availability of labor and/or engine power, by periodic supplies of raw material, by maximum computational speed of a computer facility, or by budget. In these examples, effort represents performance of men and/or machinery, or volumes of raw material, computing power or money. In all cases, capacity means the maximum available effort per time unit. It is assumed that for each time unit that a job is incomplete, a fixed cost has to be paid. The objective is to find a processing schedule, taking the capacity constraint into account, to perform all jobs such that the total costs are minimized. There are no restrictions on the schedule with respect to, for instance, pre-emption, semi-activeness or serial vs. parallel planning. We have baptized this type of problem a processing problem with restricted capacity or processing problem for short. It turns out that in order to minimize costs in a processing problem with restricted capacity, the jobs have to be performed one by one. So, until all jobs have been completed, all capacity should be used on one job at a time. Thus, it suffices to find an optimal order on the jobs. From this observation it follows that from an operations research point of view, processing problems with restricted capacity and the wellknown sequencing problems with one machine and aggregated (weighted) completion times are equivalent. Applying Smith’s rule (Smith, 1956), i.e., process the jobs in the order of decreasing urgencies, provides in both problems an optimal order on the jobs. Here, the urgency of a job is defined to be equal to the costs that it generates per time unit divided by its processing demand. However, the problems diverge when analyzed in a cooperative game theory framework. Problems are extended to situations in which each job belongs to a (different) player and each player has an individual capacity to handle jobs. Besides minimizing total costs, costs have to be allocated to each player individually. In order to find fair allocations, a cooperative game is constructed. The approach to associate a cooperative game to an operation research problem is quite common in the literature (see Borm et al. (2001) for an overview). Sequencing problems with one machine have been analyzed from a game-theoretical point of view in several ways, starting from the basic paper by Curiel et al. (1989) (see Curiel et al. (2002) for an overview). In this paper we associate games to processing situations. These games are called processing games (with restricted capacities) and differ from sequencing games. This diversion is due to two main differences between a processing and a sequencing situation. The first difference is that in a processing situation the players have individual and generally different capacities to handle jobs, while in a sequencing situation with one machine there are no individual capacities: in fact, the machine * Corresponding author. E-mail address: J.H.Reijnierse@uvt.nl (H. Reijnierse). 0377-2217/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2009.06.014 774 H. Reijnierse et al. / European Journal of Operational Research 202 (2010) 773–780 processes all jobs with a constant capacity. The second difference is that in a processing situation with restricted capacities there is no fixed initial order in which the jobs stand in line in front of a machine. So, in a processing situation there are no initial restrictions nor rights on the order in which players may process their jobs. Maniquet (2003) studies a related model: queueing problems in which there is no fixed initial order on the jobs but, in contrast to our model, a fixed capacity to process jobs and equal processing times for all jobs. Although, in this setting, there is no clear-cut definition of an associated coalitional game, two specific dual games are considered and their Shapley value is characterized. Actually, our model can be also applied to the setting of Maniquet (2003) by interpreting individual capacities to be individual rights. Here, the rights are not given by an order, but by, e.g., the proportion of shares of the machine that the owner of a job possesses (see Section 3 for an example). In order to obtain the exact setting of Maniquet (2003), one has to further restrict to equal processing times and equal initial rights. One advantage of our more general model is that it allows for a clear-cut definition for the costs of a cooperating coalition. If a coalition is formed, cost savings can be made by helping each other by means of using a player’s capacity to precipitate the job of another coalition member. To put it differently, the members of the coalition have at their disposal the sum of their individual capacities in order to complete all jobs of the coalition. This situation can be modeled as a processing problem and as a result one can easily determine an optimal schedule and its costs. However, the problem of minimizing the total costs is supplemented with the problem of dividing these costs among the players involved. The latter is of a typical game-theoretical nature and in order to solve it, we analyze the complete processing game with respect to core elements. Here, a processing game is a cooperative cost-game, in which the costs of a coalition equal the costs of an optimal schedule of its corresponding processing problem. The main result of the paper states that every processing game with restricted capacities is totally balanced, i.e., every subgame of a processing game has a non-empty core. To prove this statement, we construct from a given processing situation an exchange economy with land. In this Debreu-type of exchange economy (Debreu, 1959) each player initially owns a part of a perfectly divisible two-dimensional commodity, referred to as land. One dimension is time and the other one is effort per time unit. In the context of processing situations, one can interpret this commodity as an agenda. In order to complete their jobs, players must make reservations in the agenda, i.e., a player must book a block of time and capacity, which is sufficiently large to complete his job. A price is introduced such that the market clears, i.e., no part of the agenda is booked by more than one player. Clearing the market will, as usual, lead to a price equilibrium. From this price equilibrium, we construct an allocation contained in the core of the processing game. Since a subgame of a processing game is again a processing game, we obtain totally balancedness. Advantages of the constructed core element are that it can be computed in polynomial time and that it has a natural interpretation (Section 3). Furthermore, we show that the allocation is independent on which optimal schedule is chosen (in case of coinciding urgencies). As a consequence, the allocation depends continuously on the processing times, capacities and costs coefficients. The paper is organized as follows. In Section 2, we introduce the formal model of a processing problem with restricted capacity. Section 3 studies total balancedness of processing games. Section 4 provides a proof for the main result, following the line described above. 2. Processing problems with restricted capacity A processing problem P with restricted capacity can be described by a tuple hJ; p ¼ ðpj Þj2J ; a ¼ ðaj Þj2J ; bi: Here, J is a finite set of jobs that need to be completed. The vector p in RJþ contains the processing demands or efforts of the jobs, furthermore a in RJþ is the vector of cost coefficients and b is a strictly positive real denoting the maximum available effort per time unit, or shortly capacity. The costs for job j to be incomplete for a period of time t equals aj  t. A feasible schedule can be described by a map F : J  Rþ ! Rþ . The value Fðj; tÞ for job j in J at time t in Rþ , can be interpreted as the cumulative amount of effort which has been used for job j up to time t. In order to be feasible, F has the following properties: (i) Fðj; tÞ is weakly increasing in t for all j 2 J. (ii) Fðj; 0Þ ¼ 0 for all j 2 J. P (iii) j2J ½Fðj; tÞ  Fðj; sÞ 6 b  ðt  sÞ for every s; t 2 Rþ with s 6 t. Property (iii) states that for each segment ½s; t of time, the total effort spent on all jobs together is restricted linearly in the length of the segment by the capacity constraint. We denote F as the family of all feasible schedules. Given a feasible schedule F in F, the completion time T j ðFÞ of job j in J is defined by T j ðFÞ :¼ infft 2 Rþ jFðj; tÞ P pj g: We allow T j ðFÞ to be infinity. The objective is to find a feasible schedule such that the sum of costs over all jobs is minimized. This minimum is expressed by cðPÞ :¼ min F2F X aj  T j ðFÞ: j2J Observe that the minimum exists and therefore the value cðPÞ is well-defined for every processing problem P. In order to minimize total joint costs in an arbitrary processing problem, one must process the jobs one by one. Indeed, if, given schedule F, at some moment in time two jobs i and j are partly finished, and if job i is the first of the two to be completed, one can reschedule as follows. First use all capacity initially used for either i or j to job i until job i has been completed. After that moment, use all capacity that has been used for i or j to work on job j. Formally, the new schedule F 0 is given by H. Reijnierse et al. / European Journal of Operational Research 202 (2010) 773–780 775 F 0 ði; tÞ ¼ minfFði; tÞ þ Fðj; tÞ; pi g; F 0 ðj; tÞ ¼ Fði; tÞ þ Fðj; tÞ  F 0 ði; tÞ; F 0 ðk; tÞ ¼ Fðk; tÞ for all k 2 N n fi; jg: It is not difficult to verify that F 0 is another feasible schedule in which job i is finished earlier than in schedule F, and every other job has the same completion time. This observation shows that a processing problem boils down to a sequencing problem with one machine. The (equivalent) processing time of a job j can be found by dividing the processing demand pj by the capacity constraint b. Therefore, an optimal schedule can be found by applying the well-known Smith’s rule, i.e., process the jobs in the order of decreasing urgencies, in which the urgency of job j in J is given a by pjj . a Proposition 2.1. cf. Smith (1956) Let P be a processing problem such that J ¼ f1; . . . ; jJjg and the jobs are numbered such that ap1 P    P pjJj . 1 jJj Then it is optimal to process the jobs in the order 1; 2; . . . ; jJj and cðPÞ ¼ jJj 1X ai  ½p1 þ    þ pi : b i¼1 The following example illustrates the profitability of using the total capacity for one job at a time. Example 2.1. Suppose, a farmer has to harvest three acres with different types of crop, say type 1, 2 and 3. The tasks require 20, 30 and 10 days of work for one man, respectively. His workforce consists of himself and five employees. He has contracts with distributors to deliver the types of crop, but he is already over time. Every extra day of delay results in penalties of size 24, 30 and 6, respectively. The farmer wants to harvest the acres in such a way that the total sum of penalties will be minimal. This problem can be modeled as the processing problem P :¼ hJ; p; a; bi, in which J :¼ f1; 2; 3g; p :¼ ð20; 30; 10Þ; a :¼ ð24; 30; 6Þ and b :¼ 6. One approach to complete the jobs, is dividing the capacity b over the jobs proportionally to their processing demands. Then after 10 days all jobs are finished simultaneously. This approach corresponds with the schedule F defined as follows: Fð1; tÞ :¼ 2t; Fði; tÞ :¼ pi Fð2; tÞ :¼ 3t and Fð3; tÞ :¼ t for t 2 ½0; 10; for t > 10: It yields a total cost of 600. Another approach is to finish the jobs one after another. If the jobs are done in the order (1, 2, 3), the corresponding schedule F 0 will be F 0 ð1; tÞ :¼ F 0 ð2; tÞ :¼ F 0 ð3; tÞ :¼  6t if 6t 2 ½0; 20; 20 if 6t > 20; 8 if 6t 2 ½0; 20; > <0 6t  20 if 6t 2 ½20; 50; > : 30 if 6t > 50; 8 if 6t 2 ½0; 50; > <0 > : 6t  50 if 6t 2 ½50; 60; 10 if 6t > 60: The schedule F 0 induces completion times TðF 0 Þ ¼ 20 50 ; 6 ; 10 6
, which yield a total cost of 390. 3. Processing games with restricted capacities In this section we introduce processing situations with restricted capacities and associated processing games. Examples are provided and the main result of the paper is stated: processing games are totally balanced. An explicit core element is provided. In a processing situation hN; J; p ¼ ðpj Þj2J ; a ¼ ðaj Þj2J ; ðbi Þi2N i there is a finite set of players N, in which each player i in N is endowed with a strictly1 positive capacity bi , in order to perform jobs. Each job j in J has a processing demand pj in Rþ and cost coefficient aj in Rþ . As long as job j is incomplete, it generates a cost of size aj per time unit. Each player has to complete one specific job in J. Since each player is obliged to a different job, there is a one-to-one correspondence between players and jobs. So, no confusion occurs when the processing demand and the cost coefficient of the job of player i are denoted by pi and ai , respectively. This one-to-one correspondence simplifies notations and proofs, but is not essential for our results. The model can be extended to situations in which players are obliged to several jobs or jobs are in the interest of several players. These generalizations have been studied in Quant et al. (2008). Let S # N be a coalition of players who decide to cooperate. This coalition has to its disposal the individual capacities of all of its memP bers, i.e., coalition S can maximally generate an amount of effort of size bðSÞ :¼ i2S bi per time unit. The aim of coalition S is to complete all jobs of its members, such that aggregate costs are minimized. This is modeled by the processing problem PðSÞ :¼ hJðSÞ; ðpi Þi2S ; ðai Þi2S ; bðSÞi; in which JðSÞ denotes the set of jobs corresponding to players in S. Proposition 2.1 provides a method to calculate an optimal schedule such that the aggregate costs of coalition S are minimized. However, constructing such a schedule is only part of the problem. That is, in addition 1 It is possible to allow for nonnegative personal capacities as long as the total capacity is required to be positive. Coalitions without any capacity will have worth infinity in the associated TU-game. The proposed allocation remains to be well defined and a core element. 776 H. Reijnierse et al. / European Journal of Operational Research 202 (2010) 773–780 to minimizing total costs, the problem remains how to allocate these costs among the players in S in a fair way. To study this problem, we analyze a processing game hN; cP i in which cP : 2N ! Rþ is the mapping defined by cP ðSÞ :¼ cðPðSÞÞ for all S # N: The processing game hN; cP i is a so-called cost-game. A cost-game is a transferable utility game, or TU-game, but instead of rewards it depicts costs of coalitions. Because of the different interpretation, the definitions of standard properties and solution concepts for TU-games have to be adjusted. For the sake of completeness we provide the concepts that will be considered. The core of a cost-game hN; ci is defined by CðcÞ :¼ ( x 2 RN X i2N xi ¼ cðNÞ and X ) xi 6 cðSÞ for all S  N : i2S A game with a non-empty core is called balanced. A TU-game is said to be totally balanced whenever every subgame2 has a non-empty core. A cost-game hN; ci is said to be concave if cðSÞ þ cðTÞ P cðS [ TÞ þ cðS \ TÞ for all S; T # N. As is well-known, concavity of a cost-game implies (totally) balancedness. The main goal of this paper is to prove that every processing game has a non-empty core. We revisit Example 2.1 to show a processing game hN; cP i. It points out in particular that a processing game is in general not concave and that there can be players to whom are assigned a negative cost (i.e., a reward) in any core allocation. As a result, solutions based on a proportional type of cost allocation with respect to processing demands and/or capacities will in general not generate core allocations. Example 2.1 (continued). This time the three acres are owned by different farmers. Farmers 1 and 2 have small farms and no employees. Farmer 3 has three employees. In the processing situation comporting with the story, the player set N consists of the players 1, 2 and 3 of which the processing demands are given by p :¼ ð20; 30; 10Þ, the cost coefficients are given by a :¼ ð24; 30; 6Þ and the individual capacities of the players are b :¼ ð1; 1; 4Þ. Observe that the players are numbered in such a way that ap1 P ap2 P ap3 . According to Proposition 2.1, the corresponding 1 2 3 processing game hN; cP i is given by 1 ½24  20 þ 30  ð20 þ 30Þ þ 6  ð20 þ 30 þ 10Þ ¼ 390; 6 cP ðf1; 2gÞ ¼ 990; cP ðf1; 3gÞ ¼ 132; cP ðf2; 3gÞ ¼ 228; cP ðNÞ ¼ cP ðf1gÞ ¼ 480; cP ðf2gÞ ¼ 900 and cP ðf3gÞ ¼ 15: Observe that cP ðNÞ þ cP ðf2gÞ ¼ 1290 > 1218 ¼ cP ðf1; 2gÞ þ cP ðf2; 3gÞ, so the game is not concave. Furthermore, if x 2 CðcP Þ, x1 þ x3 6 132; x2 þ x3 6 228; x1 þ x2 þ x3 ¼ 390: Hence, 390 þ x3 ¼ x1 þ x2 þ 2  x3 6 360. As a result, this yields x3 < 0. Note that player 3 is rewarded for his participation in every core allocation because of his relatively large capacity. It is left to the reader to verify that the allocation (195, 310, 115) is contained in the core CðcP Þ. The story can be modified in order to express that the model can also be applied for single queueing problems with no initial order, but with initial rights instead. Suppose that the three farmers have together bought an agricultural machine. Farmers 1 and 2 have each invested one sixth of the price of the machine and farmer 3 has paid the remaining two thirds. It is natural to assume that each coalition of farmers is entitled to occupy the machine proportionally to the aggregated invested money of its members. The core allocation of the example above has been found by applying the proof of the following theorem, which is the main result of this paper. Theorem 3.1. A processing game is totally balanced. A proof can be found in Section 4, in which the allocation x in RN , given by xi :¼ i ai X bðNÞ pk þ k¼1 " # " # n n n X X pi 1 b X pk 1 ak  i a‘  ai þ  ak þ   bðNÞ 2 bðNÞ k¼1 bðNÞ 2 k¼iþ1 ‘¼kþ1 for all i 2 N, will be proven to be a core allocation of the processing game hN; cP i, provided that N :¼ f1; . . . ; ng and ap11 P    P apnn . Let us give an interpretation of this core allocation. Since the urgencies are ordered in the way described above it is optimal for the grand coalition N to first use the total capacity bðNÞ on the job of player 1, then on the job of player 2 and so on. According to this schedule, player i Pi 1 has to wait for a period of time with length bðNÞ k¼1 pk until his job has been completed. As a result his individual direct costs will be ai  i 1 X p: bðNÞ k¼1 k ð1Þ If each player i would pay this amount, the costs are divided in an efficient way. It would not be very fair though. A player whose job is placed at the end of the line should be compensated. Furthermore, players who have a relatively large capacity bi should be rewarded. For this reason, besides the actual costs (1), a tax is introduced on the jobs. The tax proceeds then will be used to subsidize the players with large capacities. More particularly, the sum of the tax deposits is redivided proportionally to the capacities of the players. We now explain the reasoning 2 For each coalition S, the subgame hS; cjS i of hN; ci is defined by cjS ðTÞ ¼ cðTÞ for all T # S. H. Reijnierse et al. / European Journal of Operational Research 202 (2010) 773–780 777 behind the explicit format of the tax deposits. At each moment of time t, a cost-rate is introduced. The player whose job is in process must pay this rate. The cost-rate at time t equals X ak  ½the proportion of job jk that has not been finished yet at time t: k2N pi pi During a period of time with length bðNÞ , all players are working on the job of player i. Player i must pay ak  bðNÞ for each player k whose job is still waiting to be processed. This is exactly the loss of player k because of the fact that the job of player i is processed before his own job. pi , since the mean proportion of his own job that has not been finished yet during its processing time Additionally, player i has to pay 12  ai  bðNÞ 1 equals 2. The sum of these amounts equals the tax deposit si of player i: si :¼ " # n X pi 1 ak :  ai þ  bðNÞ 2 k¼iþ1 ð2Þ Finally, the total amount of collected tax money is returned to the players, proportional to their individual capacities. This yields a subsidy for player i of n bi X sk : bðNÞ k¼1 ð3Þ Subtracting expression (3) from the sum of the expressions (1) and (2), yields the amount player i has to pay according to the core allocation x: xi :¼ i ai X bðNÞ p k þ si  k¼1 n bi X sk ; bðNÞ k¼1 ð4Þ Let us return once more to the processing situation arising from Example 2.1. Example 2.1 (continued). Let hN; J; p; a; bi be the processing situation with N :¼ f1; 2; 3g; J :¼ fj1 ; j2 ; j3 g; p :¼ ð20; 30; 10Þ; a :¼ ð24; 30; 6Þ and b :¼ ð1; 1; 4Þ. The players are numbered in such a way that the optimal order is (1, 2, 3). We already stressed out that the allocation (195, 310, 115) is a core allocation of the corresponding processing game. This allocation arises as follows. The first part consists of the individual direct costs of (expression (1)) and equals 24  1  20 ¼ 80; 6 30  1 1  50 ¼ 250 and 6   60 ¼ 60: 6 6 The tax that the players have to pay is (expression (2)) s1 ¼   20 1   24 þ 30 þ 6 ¼ 160; 6 2 s2 ¼   30 1   30 þ 6 ¼ 105 and 6 2 s3 ¼   10 1   6 ¼ 5: 6 2 According to expression (3) the players are subsidized 1  ½s1 þ s2 þ s3  ¼ 45; 6 1  ½s1 þ s2 þ s3  ¼ 45 and 6 4  ½s1 þ s2 þ s3  ¼ 180; 6 respectively. The core allocation x becomes ð80 þ 160  45; 250 þ 105  45; 60 þ 5  180Þ ¼ ð195; 310; 115Þ: Observe that the direct costs as well as the tax deposits are based on the given optimal order of decreasing urgencies. At first sight, the core allocation x depends therefore on the optimal order chosen. The following proposition shows that this is not the case. Proposition 3.2. hN; J; p; a; bi be a processing situation. The core allocation x as given by (4) does not depend on the choice of which optimal order is used to process the jobs. Proof. Two optimal orders can be obtained from each other by a series of switches of two adjacent jobs with equal urgencies. It is sufficient to show that x does not change at each of these switches. Assume that one optimal order is ð1; . . . ; nÞ and that players i and i þ 1 have aiþ1 . We have to show that x and  x coincide, with x and  x denoting the allocations which correspond to the orders equal urgencies: apii ¼ piþ1 ð1; . . . ; nÞ and ð1; . . . ; i  1; i þ 1; i; i þ 2; . . . ; nÞ, where i and i þ 1 have been switched, respectively. The vectors of taxes corresponding to , respectively. these orders are denoted by s and s We first show that the total amount of taxes paid in both orders is the same. Note that for players k unequal to i and i þ 1, the taxes sk k coincide. It is shown below that the sum of the taxes paid by i and i þ 1 does not change either: and s si þ siþ1 ! ! n n X X pi 1 pi piþ1 1 ¼ ai þ a‘ þ aiþ2 þ a‘  aiþ1 þ bðNÞ 2 bðNÞ bðNÞ 2 ‘¼iþ2 ‘¼iþ2 ! ! n n X X p 1 p p 1 ¼ i ai þ a‘ þ iþ1  ai þ iþ1 aiþ1 þ a‘ ¼ si þ siþ1 : bðNÞ 2 bðNÞ bðNÞ 2 ‘¼iþ2 ‘¼iþ2 n X p 1 ¼ i ai þ a‘ bðNÞ 2 ‘¼iþ1 ! n X p 1 þ iþ1 aiþ1 þ a‘ bðNÞ 2 ‘¼iþ2 ! The third equality follows from the fact that i and i þ 1 have equal urgencies. Since the total sum of amount of taxes is equal in both orders, it is immediately clear that xk ¼ xk for all k unequal to i and i þ 1. We now prove that xi ¼ xi . 778 H. Reijnierse et al. / European Journal of Operational Research 202 (2010) 773–780 i ai X n i n X b X a X p p 1 xi ¼ pk þ si  i sk ¼ i pk þ i  aiþ1 þ i ai þ ak bðNÞ k¼1 bðNÞ k¼1 bðNÞ k¼1 bðNÞ bðNÞ 2 k¼iþ2 ¼ i ai X bðNÞ pk þ k¼1 n piþ1 b X i  i sk ¼ xi :  ai þ s bðNÞ bðNÞ k¼1 !  n bi X sk bðNÞ k¼1 The third equality uses the fact that i and i þ 1 have the same urgency. In the same way it can be proved that xiþ1 and xiþ1 coincide. h Because of this result, the allocation x is uniquely determined by a; b and p. It is clear that x is continuous in points with just one optimal order. Proposition 3.2 shows that it is also continuous in points with more than one optimal order. Corollary 3.3. The core allocation x is continuous in a; b and p. We conclude this section with an example that compares the solution of Maniquet (2003) and ours. Example 3.1. Consider a sequencing situation of three jobs, J ¼ ð1; 2; 3Þ, without an initial order. Assume that all processing
times are of unit length, as Maniquet (2003) requires, and take a ¼ ð12; 6; 0Þ. To express equal initial rights we set b ¼ 13 ; 13 ; 13 . Maniquet (2003) proposes the allocation (15, 9, 0), where we propose (19, 10, 5). The striking difference is that we reward players with low costs to take position at the end of the line, where Maniquet (2003) only compensates them. What is to be preferred is left to the reader. 4. Proof of Theorem 3.1 Let us first give an outline of the proof. Given a processing situation, we construct an exchange economy and find a price equilibrium. This equilibrium is situated in the core of the economy. It induces a core allocation of the original processing game. A similar technique has been used in Klijn et al. (2000) to construct core elements of permutation games. An (initially) empty agenda is given. It will be a two-dimensional commodity. Of course time is one dimension, the other one is effort per time unit. In principle, there is no time restriction. The amount of effort per time unit is bounded by the capacity bðNÞ of the grand coalition. At each moment of time, one can buy any (measurable) part of the capacity available. Because of the two dimensions, it is customary to speak of land rather than of an agenda. So, we consider a Debreu-type of exchange economy (Debreu, 1959) in which each player initially owns a part of a perfectly divisible good, land. This type of economies have been studied in several papers (see for instance Legut et al., 1994). In order to complete their jobs, players must make reservations in the agenda. Only if a player books a block of time and effort per time unit sufficiently large to process his job, it will be completed. A price is chosen such that the market clears, i.e., no part of the agenda is booked by more than one player. This gives rise to a feasible schedule F 2 F. Clearing the market will, as usual, lead to a price equilibrium, which is situated in the core of the exchange economy. It is converted to a core element of the processing game. This will end the proof.3 Let hN; J; p; a; bi be a processing situation. Throughout this section we assume, without loss of generality, that N ¼ f1; . . . ; ng and that a1 P    P apnn . p1 Let EðPÞ :¼ hN; ðL; B; kÞ; ðAi ; V i Þi2N i be an exchange economy with land, in which:  A commodity, modeled by a measure space ðL; B; kÞ has to be reallocated among the group of players N. Here, L :¼ ½0; bðNÞ  Rþ denotes a piece of land, B is the Borel-r-algebra of L and k : B ! Rþ denotes the Lebesgue-measure on L. P  P S  Each player i in N has endowment Ai :¼ ½bi   Rþ in which ½bi  denotes the interval k<i bi ; k6i bi . Observe that i2N Ai ¼ L and kðAi \ Ak Þ ¼ 0 whenever i – k.  Each player i has reservation value V i ðBÞ for all sets B in B defined by V i ðBÞ :¼ ai  T i ðBÞ; n o R t R bðNÞ in which T i ðBÞ :¼ inf t 2 Rþ 0 0 1B ðx; sÞ dx ds P pi . Here, 1B ðÞ is the indicator-function of set B. T i ðBÞ denotes the moment of time at which the job of player i will be finished in the case that part B of the land (agenda) is used to work on his job. In case subset B is not sufficient, T i ðBÞ equals infinity.  Player i has a quasi-linear utility function U i : B  R ! R that denotes his valuation for bundles of land and money. It is defined by U i ðB; yÞ :¼ V i ðBÞ þ y for all B 2 B and y 2 R: S S Let S # N be a coalition. An S-redistribution is a set fBi gi2S of k-measurable subsets of L, with i2S Bi # i2S Ai and kðBi \ Bj Þ ¼ 0 whenever P S i – j. Let fBi gi2S be an S-redistribution and z 2 R such that i2S zi ¼ 0. Then the set fðBi ; zi Þgi2S is called an S-reallocation. If fðBi ; zi Þgi2N is an N-reallocation then an S-reallocation fðC i ; yi Þgi2S is called an improvement upon fðBi ; zi Þgi2N if V i ðC i Þ þ yi > V i ðBi Þ þ zi for all i in S. An N-reallocation fðBi ; zi Þgi2N is a core allocation if no coalition S has an improvement upon fðBi ; zi Þgi2N . An N-reallocation fðBi ; zi Þgi2N is called a price equilibrium if there exists a price density function p : L ! R such that (budget constraints) (i) Pp ðBi Þ þ zi ¼ Pp ðAi Þ for all i 2 N, (ii) If V i ðCÞ þ y > V i ðBi Þ þ zi for certain C # L; y 2 R and i 2 N, then Pp ðCÞ þ y > Pp ðAi Þ, in which Pp ðBÞ :¼ 3 R 1 R bðNÞ 0 0 (maximality conditions) 1B ðx; tÞ  pðx; tÞ dx dt for all B # L. The core allocation is stated right after Theorem 3.1. So far, we have not found a direct proof (by plugging the allocation into the core inequalities). 779 H. Reijnierse et al. / European Journal of Operational Research 202 (2010) 773–780 Given an exchange economy with land EðPÞ we define a TU-game hN; v EðPÞ i with the value for coalition S # N as follows: ( X v EðPÞ ðSÞ :¼ sup ) V i ðC i Þ fC i gi2S is an S-redistribution ; i2S i.e., the maximum social welfare in the sub-economy in which only the actions of the players in coalition S are considered. The TU-game hN; v EðPÞ i is in fact the TU-game hN; cP i, as the following lemma demonstrates. Lemma 4.1. EðPÞðSÞ ¼ cP ðSÞ for all S # N. Proof. Take S # N, say S :¼ fið1Þ; . . . ; iðsÞg with ið1Þ <    < iðsÞ. Define, for all j with 1 6 j 6 s: Bj :¼ [ ½bi   i2S 1  ½piðjÞ ; bðSÞ in which ½piðjÞ  denotes the interval hP ‘<j pið‘Þ ; P ‘6j pið‘Þ i with length piðjÞ . Clearly, fBj g16j6s is an S-redistribution. Furthermore, it is easy to verify that s X v EðPÞ ðSÞ ¼ V iðjÞ ðBj Þ: j¼1 Hence, s X v EðPÞ ðSÞ ¼ V iðjÞ ðBj Þ ¼  j¼1 s X aiðjÞ  TðBj Þ ¼  j¼1 s X aiðjÞ ½pið1Þ þ    þ piðjÞ  ¼ cP ðSÞ: bðSÞ j¼1  The following lemma provides a relation between the existence of a price equilibrium in the exchange economy EðPÞ and the non-emptiness of the core of the TU-game hN; cP i. In fact, using a standard argument, one can show that if fðBi ; zi Þgi2N is a price equilibrium, then the corresponding vector ðV i ðBi Þ þ zi Þi2N is contained in the core CðcP Þ. The proof is left to the reader. Lemma 4.2. If the exchange economy EðPÞ has a price equilibrium, then the TU-game hN; cP i has a non-empty core. So, the existence of a price equilibrium in EðPÞ implies balancedness of the TU-game hN; cP i and thus also balancedness of the costgame hN; cP i. Therefore, the proof of Theorem 3.1 boils down to the following proposition. Proposition 4.3. The exchange economy with land EðPÞ has a price equilibrium. Proof. Denote for all i in N the interval lðtÞ :¼ Observe that 8 ai < pi 1 bðNÞ if t 2 ½pi =bðNÞ; P if t P pj : :0 hP j<i pj ; P j6i pj i with length pi bðNÞ as ½pi =bðNÞ and define j2N l : Rþ ! Rþ is weakly decreasing. Let fBi gi2N be the N-redistribution defined by Bi :¼ ½0; bðNÞ  ½pi =bðNÞ: Furthermore, we define the price density function pðx; tÞ :¼ Z p : L ! R by 1 lðsÞ ds: t We prove that the N-redistribution fBi gi2N can be extended to a price equilibrium supported by this price density function. To do so, we first need to calculate the prices of Ai and Bi , respectively. Let us first calculate for every i 2 N the price of Bi . Take i 2 N, then Pp ðBi Þ ¼ Z 1 0 bðNÞ Z 0 1Bi ðx; tÞ  pðx; tÞ dx dt ¼ Z ½pi  bðNÞ Z bðNÞ pðx; tÞ dx dt ¼ 0 Z ½pi  bðNÞ bðNÞ Z 0 Z t 1 lðsÞ ds dx dt ¼ bðNÞ Z ½pi  bðNÞ Z 1 lðsÞ ds dt: t Note that Z ½pi  bðNÞ Z 1 lðsÞ ds dt ¼ t Z ½pi  bðNÞ Z t 1 bðNÞ X pk lðsÞ ds þ k6i 1 ai 1 X p t ¼    2 pi bðNÞ k6i k Z n X k¼iþ1 !2 ½pk  bðNÞ þ ½p  i t2bðNÞ ! lðsÞ ds dt ¼ Hence, from Eq. (5) it follows that for all i 2 N, Z ½pi  bðNÞ Z t 1 lðsÞ ds dt ¼ ½pi  bðNÞ ! Z n X 1 X ak  pk  t dt þ ½p  dt i bðNÞ k6i pi bðNÞ bðNÞ k¼iþ1 ai n pi X ak 1 1 ai pi   ðpi Þ2 þ  ðaiþ1 þ    þ an Þ ¼  bðNÞ k¼iþ1 bðNÞ 2 bðNÞ2 pi bðNÞ2   pi 1  a þ a þ    þ a ¼  : n i iþ1 bðNÞ2 2 Pp ðBi Þ ¼ bðNÞ Z   pi 1  ai þ aiþ1 þ    þ an ¼ si :  bðNÞ 2 ð5Þ 780 H. Reijnierse et al. / European Journal of Operational Research 202 (2010) 773–780 So, for every i 2 N the price of Bi equals the tax deposit si . Similarly, it can be derived that for every i 2 N the price of Ai equals the subsidy for player i. Indeed, Pp ðAi Þ ¼ Z 0 ¼ bi 1 Z bðNÞ 1Ai ðx; tÞ  pðx; tÞ dx dt ¼ 0 X k2N pk bðNÞ Z 0 1 Z pðx; tÞ dx dt ¼ bi Z 1 0 ½bi  Z 1 lðsÞ ds dt ¼ bi t   X 1 b sk :  ak þ akþ1 þ    þ an ¼ i  2 bðNÞ k2N  2 XZ k2N ½pk  bðNÞ Z 1 lðsÞ ds dt t The fifth equality can be derived exactly along the lines of Eq. (5). Define for all i 2 N, zi :¼ Pp ðAi Þ  Pp ðBi Þ ¼ bi X sk  si :  bðNÞ k2N Then fðBi ; zi Þgi2N is an N-reallocation which clearly satisfies the budget constraints with respect to the price density p : L ! R. Now we prove that the maximality conditions also hold. To obtain a contradiction, suppose there exists C # L; y 2 R and i 2 N such that and Pp ðCÞ þ y 6 P p ðAi Þ: V i ðCÞ þ y > V i ðBi Þ þ zi Because P p ðAi Þ ¼ Pp ðBi Þ þ zi , these two inequalities yield V i ðCÞ  Pp ðCÞ > V i ðBi Þ  Pp ðBi Þ: ð6Þ Since the price density function pðx; tÞ does not depend on x and is decreasing in t we can assume without loss of generality that pi  for a certain number t. Define the function f : Rþ ! R by C :¼ C t ¼ ½0; bðNÞ  ½t; t þ bðNÞ   Z tþ pi Z 1 bðNÞ pi f ðtÞ :¼ V i ðC t Þ  Pp ðC t Þ ¼ ai  t þ lðfÞ df ds:  bðNÞ bðNÞ t s Observe that f is differentiable on Rþ and 0 f ðtÞ ¼ ai  bðNÞ  "Z 1 p i tþbðNÞ lðfÞ df  Z t 1 # lðfÞ df ¼ ai þ bðNÞ  Hence, f 0 is also differentiable on Rþ and f 0 ðtÞ ¼ bðNÞ  p Z i tþbðNÞ lðfÞ df: t  h  i pi  lðtÞ 6 0. This inequality follows from the fact l t þ bðNÞ l is weakly decreasing. So, f is a concave function. Therefore its maximal value is taken in t whenever f 0 ðtÞ ¼ 0. Hence, f takes its maximal value in the point P 1 t :¼ bðNÞ j<i pj . Therefore, 1 X f ðtÞ 6 f p bðNÞ j<i j ! ¼ V i ðBi Þ  P p ðBi Þ for all t 2 Rþ : This contradicts Eq. (6) and as a result it follows that the N-reallocation fðBi ; zi Þgi2N is a price equilibrium supported by the price density function p. h The proof of Theorem 3.1 is now straightforward. Proof of Theorem 3.1. Because the exchange economy with land EðPÞ has a price equilibrium, the TU-game hN; cP i has according to Lemmas 4.1 and 4.2 a non-empty core. This means there exists a vector x in RN such that xðNÞ ¼ cP ðNÞ and xðSÞ P cP ðSÞ for all S # N. Equivalently, there exists a vector x in RN such that xðNÞ ¼ cP ðNÞ and xðSÞ 6 cP ðSÞ for all S # N. Hence, the cost-game hN; cP i is balanced. The reason for the cost-game hN; cP i to be totally balanced, is the fact that the processing game restricted to a coalition S  N is again a processing game and thus balanced. According to Lemma 4.2, the vector ðV i ðBi Þ þ zi Þi2N , in which Bi and zi are defined for all i in N as in the proof of Proposition 4.3, is a core allocation in the cost-game hN; cP i. Elaborating this expression provides the core allocation stated below Theorem 3.1. h References Borm, P.E.M., Hamers, H., Hendrickx, R.L.P., 2001. Operations research games: A survey. TOP 9, 139–216. Curiel, I., Hamers, H., Klijn, F., 2002. Sequencing games: A survey. In: Borm, P., Peters, H. (Eds.), Chapters in Game Theory in Honour of Stef Tijs. Kluwer Academic Publishers, The Netherlands, pp. 27–50. Curiel, I., Pederzoli, G., Tijs, S.H., 1989. Sequencing games. European Journal of Operations Research 40, 344–351. Debreu, G., 1959. Theory of Value. John Wiley and Son Inc., New York. Klijn, F., Tijs, S.H., Hamers, H., 2000. A simple proof of the total balancedness of permutation games. Economics Letters 69, 323–326. Legut, J., Potters, J.A.M., Tijs, S.H., 1994. 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