On the Balancedness of Multiple Machine Sequencing Games Herbert Hamers
Flip Klijn
Jeroen Suijs
Department of Econometrics and CentER, Tilburg University, P.O.Box 90153, 5000 LE Tilburg, The Netherlands
November 1, 1999 Abstract: This paper takes a game theoretical approach to sequencing situations with m parallel and identical machines. We show that in a cooperative environment cooperative m-sequencing games, which involve n players, give rise to m-machine games, which involve m players. Here, n corresponds to the number of jobs in an m-sequencing situation, and m corresponds to the number of machines in the same m-sequencing situation. We prove that an m-sequencing game is balanced if and only if the corresponding m-machine game is balanced. Furthermore, it is shown that m-sequencing games are balanced if m 2 f1; 2g. Finally, if m 3, balancedness is established for two special classes of m-sequencing games. Furthermore, we consider a special class of m-sequencing situations in a non cooperative setting and show that a transfer payments scheme exists that is both incentive compatible and budget balanced. Journal of Economic Literature Classification Number: C71 Keywords: (non) cooperative games, sequencing situations
1
Introduction
Scheduling or sequencing problems find their origin in the processing and manufacturing industries, where several jobs have to go through a number of machines before production is completed. In this regard one can think for example of the production, assembly, and testing of different consumer electronic products. Besides the problems in the manufacturing industries, scheduling problems also arise in business, computing, and the service industries, like the allocation problem of airport landing slots to planes or the scheduling of different programs on computers.
1
The example we like to consider is that of a multidivisional firm, whose divisions make use of a joint repair and maintenance facility. When a division requires service from this facility, this division is down and, furthermore, she incurs costs for the duration of the downtime. Since the firm wants to reduce these costs to a minimum, she needs to determine the optimal order in which the repair and maintenance facility serves the divisions that require service. A subsequent problem the firm may face, is how to allocate the minimal total costs from downtime to the divisions. The need for such an allocation can be, for example, for accounting reasons in order to determine the overall performance of each division. One obvious way to do this is just allocating to each division the costs arising from their actual downtime. This allocation, however, may not be considered ‘fair’ by all divisions. For let us assume that there exists some initial schedule for the divisions that require service, say, the schedule based on the first come first served principle, then the optimal schedule can differ from the initial schedule. In that case, there are divisions that initially would receive service at an early point in time, but whose service is delayed in the optimal schedule due to other, high priority divisions that require service first. Since high priority divisions increase the downtime of low priority divisions, one may argue that it is fair that the high priority divisions are responsible for part of the increases in downtime and the costs arising from it of the low priority divisions. The question that remains is what allocations of the minimal total costs of downtime may be considered to be fair. A similar argument holds for the problem of scheduling airport landing slots to planes. High priority planes delay the flight time of low priority planes, which increases the costs of the low priority planes. So, it might be considered fair to transfer part of these additional costs to the high priority planes. These transfer payments could be incorporated in the landing fees for airplanes. The question that again remains is, what transfer payments may be considered to be fair. The allocation of costs is one of the main issues that are addressed in cooperative game theory. Cooperative game theory deals with situations involving several persons who can benefit from cooperation. In the context of our scheduling example, this means that divisions can decrease the total costs of downtime by cooperating with each other, that is, rearrange their positions in the serving order. Establishing a relation between scheduling problems and cooperative games then enables us to apply cooperative game theory to obtain fair cost allocations. In this regard, the core of a cooperative game comes to mind. Roughly speaking, a core allocation divides the costs in such a way that for each group of divisions their total costs of downtime in the optimal schedule plus the additional costs allocated to them does not exceed the minimal total costs of downtime that this group can obtain by exchanging their places in the initial schedule. Core allocations, however, need not always exist. If a core allocation exists for a cooperative game, then this game is called balanced. In this paper, we show that for a particular class of scheduling problems the corresponding cooperative game is balanced. For the examples discussed above this means that a fair allocation of the costs exists.
Cooperative games arising from scheduling or sequencing problems are called sequencing games. Sequencing games were introduced in Curiel, Pederzoli, and Tijs (1989). They considered the class of one-machine sequencing situations in which no restrictions like due dates and 2
ready times are imposed on the jobs and the weighted completion time criterion was chosen as the cost criterion. It was shown for the corresponding sequencing games that they are convex and, thus, that the games are balanced. Hamers, Borm, and Tijs (1995) extended the class of onemachine sequencing situations considered by Curiel et al. (1989) by imposing ready times on the jobs. In this case the corresponding sequencing games are balanced, but are not necessarily convex. For a special subclass of sequencing games, however, convexity could be established. Similar results are also obtained in Borm and Hamers (1998) in which due dates are imposed on the jobs. Instead of imposing restrictions on the jobs, van den Nouweland, Krabbenborg, and Potters (1992) extended the number of machines. They considered m-machine sequencing situations with respect to flow shops and a dominant machine. Convexity was established for the special class in which the first machine is dominant. In general the corresponding sequencing games need not be balanced. This paper considers sequencing situations with m parallel and identical machines in which no restrictions on the jobs are imposed. Again, the weighted completion time criterion is used. Furthermore, each agent has one job that has to be processed on precisely one machine. These sequencing situations, which will be referred to as m-machine sequencing situations, give rise to the class of m-sequencing games. A formal description of the model and the corresponding games can be found in Section 2. In Section 3 we present our results with respect to the non-emptiness of the core of msequencing games. For 1-machine sequencing situations the corresponding class of sequencing games coincides with the class of sequencing games introduced by Curiel et al.(1989). Our first main result is that an m-sequencing game is balanced if and only if the corresponding m-machine game is balanced. Then it readily follows that sequencing games arising from 2-machine sequencing situations are balanced. Finally, our second main result states that for two special subclasses of m-machine sequencing situations, with m 3, the related games are balanced. For the proof of this result we turn to the class of permutation games, which are totally balanced (cf. Tijs et al. (1984)). Finally, in Section 4, we will discuss a non cooperative model related to m-machine sequencing situations. The approach will be that there is no common knowledge about the cost of the divisions incurred by the duration of downtime. Each division has private information about her cost of downtime. Furthermore, she has to, though not necessarily, reveal this information to the firm. We will present a transfer payment scheme that is incentive compatible and budget balanced, i.e. it will force the divisions to reveal their true cost with respect to downtime to the firm.
2
The model
This section describes the sequencing situations with m parallel and identical machines, which will be referred to as m-machine sequencing situations, and the corresponding class of msequencing games. In an m-machine sequencing situation each agent has one job that has to be processed on precisely one machine. Each job can be processed on any machine. The finite set of machines 3
is denoted by M = f1; :::; mg and the finite set of agents is denoted by N = f1; :::; ng. We assume that each machine starts processing at time 0 and that the processing time of each job is independent of the machine the job is processed on. The processing time of the job of agent i is denoted by pi 0. We assume that every agent has a linear monetary cost function ci : [0; 1) ! IR defined by ci(t) = i t where i > 0 is a (positive) cost coefficient. By a one to one map b : N ! f1; :::; mg f1; :::; ng we can describe on which machine and in which position on that machine the job of an agent will be processed. Specifically, b(i) = (r; j ) means that agent i is assigned to machine r and that (the job of) agent i is in position j on machine r. Such a map b will be called a (processing) schedule. In the following an m-machine sequencing situation will be described by (M; N; b0 ; p; ), where M = f1; :::; mg is the set of machines, N = f1; :::; ng the set of agents, b0 the initial N schedule, p 2 IRN + the processing times, and = ( i )i2N 2 IR++ the cost coefficients. The starting time t(b; i) of the job of agent i if processed in a semi-active way according to a schedule b equals X t(b; i) = pj ; j 2N :b(j ) b(i)
where b(j ) b(i) if and only if the job of the agents j and i are on the same machine (i.e. b(j )1 = b(i)1) and j precedes i (i.e. b(j )2 < b(i)2). Consequently, the completion time C (b; i) of the job of agent i with respect to b is equal to t(b; i) + pi : The total costs cb (S ) of a coalitions S N with respect to the schedule b is given by
cb(S ) =
X i2S
i(C (b; i)):
We will restrict attention to m-machine sequencing situations (M; N; b0 ; p; ) that satisfy the following condition: the starting time of a job that is in the last position on a machine with respect to b0 is smaller than or equal to the completion time of each job that is in the last position with respect to b0 on the other machines. Formally, let ik be the last agent on machine k with respect to b0, then for any k 2 M we demand that
t(b0; ik ) C (b0; is) for all s 2 M:
(1)
This condition states that each job that is in the last position of a machine cannot make any profit by joining the end of a queue of any other machine. These schedules can arise in the following way. Let the agents enter one by one the machines before the processing starts. If an agent enters he will choose the queue of a machine that gives him the shortest waiting time. The (maximal) cost savings of a coalition S depend on the set of admissible rearrangements of this coalition. We call a schedule b : N ! f1; :::; mg f1; :::; ng admissible for S with respect to b0 if it satisfies the following two conditions: (i) Two agents i; j 2 S which are on the same machine can only switch if all agents in between i and j on that machine are also members of S ; (ii) Two agents i; j 2 S which are on different machines can only switch places if the tail of i and the tail of j are contained in S . The tail of an agent i is the set of agents that follow agent i on his machine, i.e. the set of agents k 2 N with b(i) b(k ): 4
The set of admissible schedules for a coalition S is denoted by BS . An admissible schedule for coalition N will be called a schedule. By defining the worth of a coalition as the maximum cost savings a coalition can achieve by means of admissible schedules we obtain a cooperative game called an m-sequencing game. Formally, for an m-machine sequencing situation (M; N; b0 ; p; ) the corresponding m-sequencing game (N; v ) is defined by
X v(S ) = max f i[C (b0; i) , C (b; i)]g b2B S i2S
for all coalitions S 2 2N nf;g and v (;) = 0. Here 2N denotes the collection of all subsets of N . Clearly, the m-sequencing game (N; v ) is superadditive, that is, two disjoint coalitions do not suffer from forming one large coalition. Cooperative game theory focuses on fair division rules for the value of v (N ) of the grand coalition. A core allocation x = (xi )i2N 2 IRN divides the value v (N ) among the players in such a way that no coalition has an incentive to split off, i.e.,
x(N ) = v(N ) and x(S ) v(S ) for all S 2 2N ; P where x(S ) = i2S xi for all S 2 2N . The core C (N; v ) consists of all core allocations.
A
game is called balanced if its core is non-empty.
3
On the balancedness of m-sequencing games
In this section we present our results with respect to the balancedness of m-sequencing games. For 1-machine sequencing situations the corresponding class of sequencing games coincides with the class of sequencing games introduced by Curiel et al. (1989), implying that 1sequencing games are balanced. Given an m-machine sequencing situation we define a new cooperative game with m players. Let (M; N; b0 ; p; ) be an m-machine sequencing situation and let (N; v ) be the corresponding m-sequencing game. The set of players whose jobs are on machine k 2 M according to the initial schedule b0 will be denoted by Nk (b0). Then an m-machine game (M; w) is defined by
w(K ) := v( S Nk (b0)) , k 2K
X
k2K
v(Nk (b0));
for every coalition K M of machines. The worth w(K ) of a coalition of machines K M is the extra cost savings the machines in K can make when they decide to cooperate with each other. The next theorem says that the core of an m-sequencing game is non-empty whenever the core of the m-machine game is non-empty, and vice versa. Theorem 3.1 Let (M; N; b0 ; p; ) be an m-sequencing situation. Let (N; v ) be the corresponding m-sequencing game and let (M; w) be the corresponding m- machine game. Then (N; v ) is balanced if and only if (M; w) is balanced. 5
The proof of this theorem and that of all forthcoming theorems, except the proof of Theorem 3.2, has been relegated to the appendix. By definition m-machine games are superadditive. Hence, 2-machine games are balanced and thus also 2-sequencing games are balanced by Theorem 3.1. Theorem 3.2 Let (M; N; b0 ; p; ) be such that jM j = 2. Then the corresponding 2-sequencing game (N; v ) is balanced. Theorem 3.1 implies that in order to check whether an 3-sequencing game with n players is balanced or not, it is sufficient to compute w(f1; 2g); w(f1; 3g); w(f2; 3g), and w(f1; 2; 3g) (w(fk g) = 0 for all k 2 M ), and then check whether this 3-machine game is balanced or not. The following example illustrates this. Example 3.1 Let M = f1; 2; 3g, N = f1; :::; 20g, = (1; : : : ; 1), and processing times and the initial schedule b0 as in Figure 1. Let (N; v ) be the corresponding 3-sequencing game and (M; w) be the corresponding 3-machine game. Some calculations give w(f1; 2g) = 3; w(f1; 3g) = 7; w(f2; 3g) = 0, and w(f1; 2; 3g) = 7. Clearly, (7; 0; 0) 2 C (M; w). Hence, the game (M; w) is balanced. By Theorem 3.1 (N; v ) is balanced. Note that checking the balancedness of the game (N; v ) directly would imply the greater effort of taking into account the 220 coalitions of jobs. M1
1
2
1
M2
6
3
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Figure 1: The schedule b0
20
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21
Now consider m-sequencing situations in which all cost coefficients are equal to one. The next theorem says that the corresponding m-sequencing games are balanced. Theorem 3.3 Let (N; v ) be the m-sequencing game that arises from an m-machine sequencing situation (M; N; b0 ; p; ) in which i = 1 for all i 2 N . Then (N; v ) is balanced. In Theorem 3.3 we assumed that all cost coefficients are equal to one. This implies that the class of m-sequencing games generated by the unweighted completion time criterion is a subclass of the class of balanced games. Clearly, the balancedness result also holds true in the case that all cost coefficients are equal to some positive constant c > 0. Furthermore, a slight adaptation of the proof of Theorem 3.3 gives a similar result for m-sequencing situations with identical processing times instead of identical cost coefficients. 6
Theorem 3.4 Let (N; v ) be the m-sequencing game that arises from an m-machine sequencing situation (M; N; b0 ; p; ) in which pi = 1 for all i 2 N . Then (N; v ) is balanced. The following example shows that if condition (1) is not satisfied, then the corresponding
m-sequencing game need not be balanced. Example 3.2 Let M = f1; 2; 3g, N = f1; : : : ; 5g, p = (2; 2; 1; 2; 2), and = (1; 1; 1; 1; 1). The initial schedule b0 is given in Figure 2. Let (N; v ) be the corresponding 3-sequencing P game. Suppose x 2 C (N; v ) is a core allocation. Then 1 = v (N ) = x(N ) i2N v (i) = 0 + 1 + 0 + 0 + 1 = 2. This contradiction shows that the core is empty. Hence the game (N; v) is not balanced.
M1
2
1
4
2
M2
3 1
M3
5
4 2
4
Figure 2: The schedule b0
For m-machine sequencing situations (m 3) with the weighted completion time criterion, the balancedness of the corresponding m-sequencing games is an open problem. If we follow the approach in this paper we need an optimal order for a coalition S (K ). The problem of finding such an optimal order, however, is difficult in the sense that it is NP-hard.
4
A non cooperative approach
In this section we study m-machine sequencing situations in which all processing times are equal to one and the cost coefficients are private information to the players. These situations give rise to a non cooperative game. It will be shown that for these models there exists an incentive compatible and budget balanced transfer payments scheme. Consider again a multidivisional firm that wants to reduce the costs that divisions incur due to downtime. In case the cost coefficient i of downtime are common knowledge, we have learned that the firm can determine an optimal schedule such that the total cost of downtime are 7
minimized. These costs coefficients , however, need not be common knowledge. In case each i is private information for division i, and thus not known to the firm, then there is an asymmetric information problem. In order to minimize the total cost of downtime, the firm needs to collect information about the cost coefficient i of each division. When collecting this information though, the firm and the division have opposing goals. The firm wants to minimize the total cost of downtime, whereas each division wants to minimize her own cost of downtime. As a ^i. To result, divisions have an incentive to mislead the firm about their true cost coefficient circumvent this incentive problem, the firm has to introduce a scheme of tax/subsidy payments that induces the truthful behavior of the divisions. This means that given such a scheme of transfer payments, each division has an incentive to reveal her true cost coefficient ^i to the firm. In other words, misleading the firm does not pay. Next, let us put this problem in a mathematical framework. Let (N; A; ) be a non cooperative game, where N is the set of divisions (players), A = (A1; :::; An) the strategy spaces and K = (K1; :::; Kn) the payoff functions. A strategy for division i is revealing her cost coefficient i. So, Ai = f i j i > 0g. Note that i may differ from division i’s true cost coefficient ^i. Given the revealed cost coefficients = ( 1 ; :::; n), the firm determines an optimal schedule b corresponding to the m-sequencing situation (M; N; b; ; p). The cost for division i then equal the cost of downtime in the schedule b based on her true cost coefficient ^i , that is
X
i( ) = ,
j :b (j ) b (i)
^i
X
^i : j :b (j )
= ,
2
2
The second equality is a result of the assumption that all processing times are equal to one. Note that these m-machine sequencing situations in which the costs are private information can also be described as a public decision making problem (see e.g. Green and Lafont (1979)). In the above described non cooperative game there is no incentive problem if jN j = k and m k for some k 2 f2; 3; : : :g, since each division is served immediately. In all other cases, each division has an incentive to reveal a cost coefficient as high as possible. Since the strategy space is unbounded, it is easy to verify that there do not exist Nash Equilibria. To prevent divisions from revealing any other cost coefficient than the true ones, we introduce a transfer payments scheme t : A ! IRN , which assigns to each vector = ( 1 ; :::; n) a vector of transfer payments. The interpretation is that division i has to pay the amount ti ( ) if ti ( ) < 0 and receives the amount ti( ) if ti ( ) > 0. A transfer payment is called incentive compatible if revealing the true cost coefficient is a dominant strategy for each player. Formally, this means ^ 2 A we have that that for all i 2 N , all 2 A and true cost coefficients
i(^ i; ,i) + ti(^ i; ,1) i( ) + ti( ); (2) where ,i = ( 1 ; :::; i,1; i+1 ; :::; n). Although incentive compatibility informs the firm correctly of the cost coefficient i of each division, it is not sufficient to minimize costs. For this to be the case, the transfer payments also need to be budged balanced. A transfer scheme t is called budget balanced if X ti( ) = 0 for all 2 A: (3) i2N
8
It would be desirable for the firm to design a transfer payment scheme that is both incentive compatible and budget balanced. In many situations, (see e.g. Green and Laffont (1997)) these properties are not simultaneously satisfied. Suijs (1996), however, showed that there exists an incentive compatible and budget balanced transfer scheme for a class of one machine sequencing situations. Here, we extend this transfer payment scheme for m machine sequencing situations. Let N = f1; :::; ng, n 3. Define for each i 2 N and each 2 A the transfer payments scheme X 1 X 1 i( ) = , j j 2m 2m j :b 2 (j )b 2 (i) (4) , 2m(n1 ,2) ( j , k ); j 2N nfig k2N nfig:b 2 (k)>b 2 (j ) where b denotes an optimal order for the m-machine sequencing situation (M; N; b; ; p) in which pi = 1 for all i 2 N . Now, we are able to formulate the next theorem which will be proven in the appendix.
Theorem 4.1 Let (N; A; ) be a non cooperative game arising from m-machine sequencing situations in which all processing times are equal to one. If j N j 3, then the transfer payments scheme defined in (4) satisfies incentive compatibility and budget balancedness. A similar result as stated in Theorem 4.1 can be obtained in case the processing time is variable and the cost coefficient is constant. In this case, however, it is less realistic to consider the processing time as private information, since the real service time will be revealed after maintenance. The general case leads to an open problem, since the payoff function in the non cooperative game depends on an optimal order that is related to the revealed cost coefficient. Again the NP-hardness of the problem prevents us from defining a payoff function and transfer payments scheme for the general case.
Appendix Proof of Theorem 3.1. First, we prove the ‘only if’ part. Let x 2 C (N; v ). For k
2 M we define
yk := x(Nk (b0)) , v(Nk (b0)): We show that y
2 C (N; w). Let K M . Then,
X x(Nk (b0)) , v(Nk (b0)) k 2K k 2K X S 0 v( Nk (b )) , v(Nk(b0))
y(K ) =
X
k 2K
k2K
= w(K ): S For K = M we have an equality, since x(
0 )) = x(N ) = v (N ) = v (S 0 k2M Nk (b )).
k2M Nk (b 9
Second, we prove the ‘if’ part. Let y 2 C (M; w). For convenience we introduce some notation. Let nk (b0 ) be the number of jobs on machine k with respect to b0, i.e. nk (b0 ) = j Nk (b0) j, and for any machine k 2 M let k : Nk (b0) ! f1; : : :; nk (b0)g be the initial order on machine k , i.e. k (i) < k (j ) if and only if b0(i) b0(j ) for all i; j 2 Nk (b0 ). For a job i 2 Nk (b0) and any k 2 M we define
xi := v(fj : k (j ) k (i)g) , v(fj : k (j ) < k (i)g);
(
and
nk (b0); x^i := xxi + y ifif k ((ii)) 6= i k k = nk (b0). We prove in four steps that x^ 2 C (N; v ). The first step shows that x^ is efficient. This follows from X X X x^i = x^i i2N k2M i2Nk (b ) X = [v(Nk(b0)) + yk ] k 2M X X = yk + v(Nk (b0)) k 2M k2M X = w(M ) + v(Nk (b0)) k2M X X = v(N ) , v(Nk (b0)) + v(Nk(b0)) 0
= v(N ):
k2M
k2M
For the second step of the proof, we need some definitions. A coalition S Nk (b0 ) is connected with respect to k if for all i; j 2 S and p 2 Nk (b0 ) such that k (i) < k (p) < k (j ) it holds that p 2 S . A connected coalition S U Nk (b0) is a component of U if S [ fig is not connected for every i 2 U nS . The components of U form a partition of U , denoted by U=k . Take T N . Define Tk = T \ Nk (b0) for k 2 M , and let T~k be the component with respect to k of Tk that contains the last player on machine k . Formally,
T~k := fS 2 Tk =k : k,1(nk (b0)) 2 S g: Note that T~k is the empty set if k,1 (nk (b0 )) 62 Tk . Next, let T~k be non-empty and i1; i2; : : :; i~tk 2 T~k be the elements of T~k such that k (i1) < k (i2) < < k (i~tk ). Then X X xi = v(fj : k (j ) k (i)g) , v(fj : k (j ) < k (i)g) i2T~k
=
let
i2T~k ~tk X l=1
v(fj : k (j ) k (il)g) , v(fj : k (j ) < k (il)g)
= v(fj : k (j ) k (i~tk )g) , v(fj : k (j ) < k (i1)g) = v(Nk (b0)) , v(Nk (b0)nT~k); 10
(5)
where the third equality follows from
v(fj : k (j ) k (il)g) = v(fj : k (j ) < k (il+1)g) for 1 l < t~k : In the third step, let S 2 Tk =k be such that S 6= T~k . Since the subgame (Nk (b0 ); vjNk (b ) ) 0
is a 1-machine sequencing game it follows that this game is convex (cf. Curiel et al. (1989)). Hence, the marginal vector (xi )i2Nk (b0 ) 2 C (Nk (b0 ); vjNk (b0 ) ) (cf. Shapley (1971)). This implies that X X (6) x^i = xi vjNk(b0)(S ) = v(S ): i2S i2S
P x^ v(T ). i2T i 0 1 X X @X xi + ( xi) + yk A
Finally, in the fourth part we show that
X i2T
x^i =
X
X
k2M S 2 Tk~=k i2S S 6= Tk
X
X
k2M S 2 Tk~=k
XS 6 TkX =
v(S ) +
+ xi ~ ~ k 6=; i2Tk Xk2M : TX = v(S ) + k2M S 2 Tk~=k
XS 6
=
Tk
k2M : T~k 6=;
X
k2M : T~k 6=;
X k2M : T~k 6=;
i2T~k
yk
yk
+ [v(Nk (b0)) , v(Nk (b0)nT~k )] ~k 6=; Xk2M : TX X v(S ) + v( S Nk (b0)) , v(Nk (b0)) k2M S 2 Tk~=k S 6= Tk
k2M : T~k 6=;
X + [v(Nk (b0)) , v(Nk (b0)nT~k )] ~k 6=; Xk2M : TX = v(S ) + v( S Nk (b0)) ,
k2M S 2 Tk~=k
X
S 6= Tk
X
k2M S 2 Tk~=k
= v(T );
S 6= Tk
k2M :T~k6=;
v(S ) + v( S
k2M :T~k6=;
T~k )
k2M :T~k 6=;
X k2M : T~k 6=;
v(Nk (b0)nT~k )
where the first inequality follows from (6), the second inequality follows from y 2 C (M; w), and the third inequality follows from the superadditivity of v . The second equality follows from (5). This completes the proof of Theorem 3.1. 2 Proof of Theorem 3.3. First, we show that we can restrict attention to m-sequencing games that arise from m-machine sequencing situations in which each machine initially has to process an equal number of jobs. 11
Second, we prove that the corresponding m-machine games correspond to permutation games, as introduced by Tijs et al. (1984). Third, we show that m-machine games are balanced. From Theorem 3.1 we can then conclude that m-sequencing games are balanced. Let (M; N; b0 ; p; ) be an m-machine sequencing situation in which i = 1. An optimal schedule ^b(N ) of coalition N is established (see e.g. Conway, Maxwell, and Miller (1967)) by first ordering the jobs of the players in N in a non-decreasing order, i.e., pi1 pi2 ::: pin where fi1; i2 ; :::; ing = N . Second, assign the jobs, after numbering the machines, in rotation to the machines:
Job of player i1 i2 ::: im j im+1 im+2 ::: i2m j ::: j in,r ::: in Machine 1 2 ::: m j 1 2 ::: m j ::: j 1 ::: r ::: m Hence, for an optimal schedule that is obtained by the above described procedure, we can conclude that each machine in f1; :::; rg has an equal number of jobs and that each machine in fr +1; :::; mg has an equal number of jobs. Moreover, the number of jobs on the first r machines is one higher than the jobs on the last n , r machines. We can, however, construct a m-machine sequencing situation such that there exists an optimal schedule of its grand coalition, induced by ^b(N ), in which each machine serves the same number of jobs. This m-machine sequencing situation is obtained by adding dummy jobs with processing time zero and cost coefficient one to the original m-machine sequencing situation. To see this, let l = maxk2M nk (b0) be the length of the longest queue waiting for a machine w.r.t. b0 in (M; N; b0 ; p; ). Then for each machine k we put l , nk (b0) jobs in front of the existing queue, so that a total of l jobs is waiting for service by machine k . Now we have a new m-machine sequencing situation (M; N; b0; p; ) with N the set of jobs, that is, N together with ml , n dummy jobs, b0 the new initial serving order, p the new vector of cost coefficients. Note that for i 2 N it holds that
b0(i) = b0(i) + l , nk (b0) p i = pi i = i (= 1) and for i 2 N nN it holds that
pi = 0 i = 1 and
fb0(i) j i 2 N nN g = f(k; 1); (k; 2); :::; (k; l , nk (b0)) j k 2 M g:
The next lemma gives a relation between the m-sequencing games of the above described mmachine sequencing situations. The proof is omitted since it follows straightforwardly from the described procedure to find an optimal order and the fact that all new (dummy) jobs in the constructed m-machine sequencing situation have processing time zero. Lemma 1 Let
(N; v) be the m-sequencing game corresponding to (M; N; b0; p; ) in which 12
i = 1 for all i 2 N . Let (N; v) be the m-sequencing game corresponding to (M; N; b0; p; ). Then
v(S ) = v(S ) = v(S [ T ) for all S N; T N nN:
From Lemma 1 immediately follows Corollary 1. Corollary 1 Let (N; v ) be the m-sequencing game corresponding to (M; N; b0 ; p; ) in which i = 1 for all i 2 N . Let (N; v) be the m-sequencing game corresponding to (M; N; b0; p; ). Then C (N; v ) 6= ; if and only if C (N; v) 6= ;. So, for the proof of the balancedness of m-sequencing games we may restrict attention to m-machine sequencing situation (M; N; b0 ; p; ) where exactly l jobs are scheduled on each machine in the initial order b0 . Henceforth we therefore only consider m-machine sequencing situations where initially each machine has to process an equal number of jobs and in which all cost coefficient are equal to one. Before we continue, we first define the class of permutation games. Let A = [aij ]ni=1;jn=1 be a square matrix of size n. Then a permutation game (N; r) is defined by
X r(S ) = max [aii , ai(i)] for all S N; 2 S i2S
where S is the set of permutations of coalition S . Note that permutation games are totally balanced (cf. Tijs et al.(1984)). To introduce a square matrix that defines the permutation game that arises from an m-machine sequencing situation (M; N; b0 ; p; ), we need to take into account the following observations. Since exactly l jobs are scheduled on each machine in the initial order b0 as well as in an optimal S order bS (K ) for the jobs S (K ) = k2K Nk (b0), we can reduce the set BS (K ) of admissible orders to BS(K) = fb 2 BS(K) : nk (b) = l for all k 2 M g:
Thus B S (K ) is the set of all admissible orders that schedule exactly l jobs on each machine. Then given an order b 2 B S (K ) the total (waiting) costs for jobs S (K ) equals
cb(S (K )) =
l X i XX k2K i=1 j =1
pb, (k;j) = 1
X Xl k2K j =1
(l + 1 , j )pb, (k;j): 1
(7)
This implies that player i = b,1 (k; j ), which is in position j on machine k , contributes (l + 1 , j )pb,1 (k;j) to the total costs of coalition S (K ). Note that this amount is independent of the other jobs that are scheduled on this machine. Now, we will define a permutation game (N; r) that arises from a m-machine sequencing situation (M; N; b0 ; p; ) with m machines and l jobs on each machine. For all i 2 N and for all j with (k , 1)l + 1 j kl, k 2 f1; :::; mg we define the square ml ml matrix A by
aij = [kl , j + 1]pi: 13
(8)
So, the rows of A correspond to the players i 2 N and the columns of A correspond to the positions in the processing order. The entry aij with (k ,1)l+1 j kl for some k 2 f1; :::; mg denotes the costs [kl , j + 1]pi of player i if it is processed on position j , (k , 1)l of machine k. We may assume without loss of generality that for all i 2 N job i is in position i of the processing order. Then the permutation game (N; r) that arises from an m-sequencing situation (M; N; b0; p; ) is now defined by
X r(S ) = max [aii , ai(i)] 2 S i2S
for all S N , where S is the set of permutations of coalition S and aij is given by (8). The relation between the m-sequencing game (M; w) and the permutation game (N; r) is expressed in the following Lemma. Lemma 2 Let (M; N; b0 ; p; ) be a m-sequencing situation in which i = 1 for all i 2 N . Let (M; w) be the corresponding m-machine game and let (N; r) be the corresponding permutation game. Then
w(K ) = r( S Nk (b0)) , k2K
X
k2K
v(Nk(b0))
for all K
M:
Proof. Consider K M , then for each schedule b 2 B S (K ) there exists a permutation b 2 S(K) that puts each job on the same machine and in the same position as b does. This permutation is defined as b(i) = (b1(i) , 1)l + b2(i) (9) for all i 2 N . Furthermore, each permutation b 2 B S(K). For all i 2 N we define
2 S(K) can be written as an admissible order
b1 (i) = k b2 (i) = (i) , (k , 1)l where k is such that (k , 1)l + 1 (i) kl. Hence, for each K M we have that
w (K ) = = = =
2 l 3 l X X X X X max 4 (l + 1 , j )pb , (k;j) , (l + 1 , j )pb, (k;j)5 , v(Nk (b0)) b2BS K k2K j =1 k2K j =1 k2K 2 l 3 XX X Xl X max 4 ab , (k;j);(k,1)l+j , ab, (k;j);(k,1)l+j 5 , v(Nk(b0)) b2BS K k2K j =1 k2K j =1 k 2K 2 3 X X X max 4 ai;b (i) , ai;b(i)5 , v(Nk (b0)) b2BS K i2S (K ) k2K i2S (K ) i X X h 0 max aii , ai(i) , v(Nk(b )) 2S K 0
(
)
(
)
(
)
(
)
1
1
1
0
1
0
i2S (K )
k2K
14
X = r(S (K )) , v(Nk (b0)) k 2K X S 0 = r( N (b )) , v(N (b0)); k2K
k
k
k2K
where the first equality holds by (7), the second equality by (8) and the third equality by (9). The fifth equality holds by the definition of a permutation game, and the last equality by the definition of S (K ). 2 In the next lemma we show that m-machine games are balanced. Lemma 3 Let (M; N; b0 ; p; ) be a m-sequencing situation in which i = 1 for all i 2 N and let (M; w) be the corresponding m-machine game. Then C (M; w) 6= ;. Proof. Let (N; r) be the permutation game that arises from the m-machine sequencing situation (M; N; p; ). Since (N; r) is balanced (cf. Tijs et al. (1984)), there exists an x 2 C (N; r). Define y 2 IRM by yk = x(Nk (b0)) , v(Nk (b0)) for all k
2 M . Then for K M we have X
k 2K
X x(Nk (b0)) , v(Nk(b0)) k2K k2K X S 0 r( Nk (b )) , v(Nk (b0))
yk =
X
k 2K
k2K
= w(K );
where the first equality follows from the definition of y , the inequality follows from x 2 C (N; r), and the second equality follows from Lemma 2. If K = M the inequality becomes an equality, which implies that y 2 C (M; w). 2 The proof of Theorem 3.3 is now a consequence of Lemma 3 and Theorem 3.1.2 Proof of Theorem 3.4. Note that an optimal schedule ^b(N ) of coalition N is established by first ordering the jobs of the players in N in a non- increasing order, i.e., i1 i2 ::: in where fi1; i2 ; :::; ing = N . Second, assign the jobs, after numbering the machines, in rotation to the machines:
Job of player i1 i2 ::: im j im+1 im+2 ::: i2m j ::: j in,r ::: in Machine 1 2 ::: m j 1 2 ::: m j ::: j 1 ::: r ::: m Then the proof is similar to the proof of Theorem 3.3. The only difference is the matrix that defines the matrix of the permutation game. Here, we define for all i 2 N and for all j with (k , 1)l + 1 j kl, k 2 f1; :::; mg the square ml ml matrix A by
aij = [j , (k , 1)l] i:2 15
Proof of Theorem 4.1. For the proof of incentive compatibility, take i 2 N , ; 2 A, where = (^ i; ,i ). Next, denote with b an optimal schedule for the m-machine sequencing situation (M; N; b; ; p) and with b an optimal schedule for the m-machine sequencing situation (M; N; b; ; p). We have to show that i(^ i; ,i) + i(^ i; ,1) i( ) + i ( ): Using (4) and the definition of i , this is equivalent with showing that
,
X
^i m 1
j :b (j )
X
2
,
^i + m
j : bb2 (j()j)>
+ Rewriting then gives
X 2
2
j : bb2 (j()j)<>b b2(i)(i) 2 2
X
2
^i + m m j ,
1 2
j : bb2 (j()j)>
^i) + m ( j ,
j : bb2 (j()j)>
j : bb2 (j()j)<>b b2(i)(i) 2 2
X
X
i2N
i( ) =
X
j : bb 2 (j()j)<>b b2(i)(i) 2 2
,m
2m
X i2N
i( ) =
m j 1
i2N j :b 2 (j )
Then
X
X
X
i m (^
,
0:
m j
1 2
1
X
m j ,
1 2
j : bb2 (j()j)>
m j :
1 2
2
X
m j
1 2
2
j :b (j )>b (i)
X
This inequality holds, since b 2 (j ) < b 2 (i) implies j j ^i. To check budget balancedness, take 2 A. Then
2
X
2
2
1
j :b (j )>b (i)
,
m j
1 2
1
X
,
m j
1 2
j :b (j )
X
X
^i + m
2
1
j :b (j )
1
j :b (j )
Rearranging terms yields
X
X
2
2
2
,
+
j : bb2 (j()j)<>b b2(i)(i) 2 2
X
^ i and b 2 (j ) > b 2 (i) implies
X
X
m j 1
i2N j :b 2 (j )>b 2 (i)
(, i + j ):
i2N j 2N :b 2 (j )
j ,
X
X
i2N j 2N :b 2 (j )>b 2 (i)
j
X X X X + i , j j 2N i2N :b (i)>b (j ) j 2N i2N :b (i)>b (j ) X X X X = j , j i2N j 2N :b (j )b (i) X X X X + j , j 2
2
= 0:
2
2
i2N j 2N :b 2 (j )>b 2 (i)
16
m j
1 2
, aj ) 0:
j 2N i2N :b 2 (i)>b 2 (j )
X
2
2
2
2
i2N j 2N :b 2 (j )
Hence, we have
P
i2N i ( ) = 0.
2
17
References [1] BORM P. AND HAMERS H. (1998): “On the Convexity of Sequencing Games with Due Dates," CentER Discussion Paper 9846, Tilburg University, The Netherlands. [2] CONWAY R., MAXWELL W., AND MILLER L. (1967): “Theory of Scheduling," AddisonWesley Publishing Company, London. [3] CURIEL I., PEDERZOLI G., AND TIJS S. (1989): “Sequencing Games," European Journal of Operational Research, 40, 344-351. [4] GREEN J., AND LAFFONT J. (1979): “Incentives in Public Decision Making" North-Holland, Amsterdam. [5] HAMERS H., BORM P., AND TIJS S. (1995): “On Games corresponding to Sequencing Situations with Ready Times," Mathematical Programmming, 70, 1-13. [6] NOUWELAND A. VAN DEN, KRABBENBORG M., AND POTTERS J. (1992): “Flowshops with a Dominant Machine," European Journal of Operational Research, 62, 38-46. [7] SHAPLEY L. (1971) , "Cores of Convex Games," International Journal of Game Theory, 1, 11-26. [8] SUIJS J. (1996) , "On Incentive Compatibility and Budget Balancedness in Public Decision Making," Economic Design, 2, 193-209. [9] TIJS S., PARTHASARATHY T., POTTERS J., AND RAJENDRA PRASSAD V. (1984): “Permutation Games: another Class of Totally Balanced Games," OR Spektrum, 6, 119-123.
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