Throughput Rate Optimization in High Multiplicity Sequencing Problems

Transcription

1 Throughput Rate Optimization in High Mutipicity Sequencing Probems Aexander Grigoriev Maastricht University, 6200 MD, Maastricht, The Netherands Joris van de Kundert Maastricht University, 6200 MD, Maastricht, The Netherands October 12, 2001 Abstract Mixed mode assemby systems assembe products (parts) of different types in certain prespecified quantities. A minima part set is a smaest possibe set of product type quantities, to be caed the mutipicities, in which the numbers of assembed products of the various types are in the desired ratios. It is common practice to repeatedy assembe minima part sets, and in such a way that the products of each of the minima part sets are assembed in the same sequence. Litte is known however regarding the resuting throughput rate, in particuar in comparison to the throughput rates attainabe by other input strategies. This paper investigates throughput and baancing issues in repetitive manufacturing environments. It considers sequencing probems that occur in this setting and how the repetition strategy infuences throughput. We mode the probems as a generaization of the traveing saesman probem and derive our resuts in this genera setting. Our anaysis uses we known concepts from scheduing theory and combinatoria optimization. Keywords: Manufacturing: performance, productivity; Production/scheduing: approximations, ine baancing, sequencing; Networks/graphs: traveing saesman.

2 1 Introduction Mass customization continues to pace new demands on manufacturers. These new demands concern manufacturing technoogy, as we as panning and scheduing issues. The customization is often achieved by offering a choice of (optiona) parts on a generic product. In manufacturing terms, this comes down to being abe to customizing the generic product during the assemby stage, using various components. This concept has become known as assembe to order. Together with ean manufacturing requirements, assembe to order techniques pose chaenging new panning and scheduing probems for manufacturers. In a mass customization environment, demand for products is not entirey handed at the customer eve. For severa stage of the production process, demand is rather deat with at the eve of product variations, for instance because the figures are (party) forecasts. The idea is that there is one product variation for each possibe choice of components (or severa cosey resembing choices combine into one variation). In this paper we refer to the variations as types. Further, we dea with a situation where the assemby process is executed on an assemby ine. As is customary in such settings, we assume that demand is specified at the eve of product types: expected number of products per type for the next panning period, e.g. a month. The question that we woud ike to answer is: how to sequence a products of the various types through the assemby process? Of course, this question has been studied before in severa different contexts. First of a, there is a ine of research deaing with so-caed assemby ine baancing probems (see e.g. Aigbedo and Monden (1997), Mitenburg (1989), Kubiak et a. (1997). This research originates from car assemby ine scheduing probems at Toyota. Mosty, these papers assume that each assemby step takes a same, constant, amount of time. The task might now be to sequence the cars in such a way that the output eves are at any time proportiona to the demand ratios. Aternativey, the sequencing probem might be to sequence the cars in such a way that the ines feeding the assemby ine have a baanced work oad. Another ine of research, having its roots in a variety of different manufacturing appications, aso deas with reated sequencing probem. These probems may again assume a inear production system, but the sequencing objective is now to optimize throughput rather than the the ine baance. Reated modes can be found in maintenance and data broadcast scheduing Zhang and van der Veen (1996), Aniy et a. (1998), Kenyon et a. (2000); Bar-Noy et a. (1998). The probem under investigation can be further defined and specified as foows. We denote by J = {1, 2,..., N} the set of product (job) types. For each product (job) type j J there is a nonnegative integer d j describing the demand of the j-th product for the next panning period, which specifies the number of type j jobs to be assembed. Further, we denote by c i,j the time interva required between inputting a type i job and a type j job, i, j J, into the system. These c i,j s can be considered as sequence dependent processing times in a sin- 2

3 ge machine scheduing probem, as sequence dependent switch over times, or as the waiting times between inputting parts into a permutation no-wait fowshop. Such a permutation no-wait fowshop system is often an adequate mode for an assemby system. Notice that this probem specification is quite genera. (For instance it contains scheduing probems in which the processing times are not sequence dependent as a specia case.) When d j = 1 for a j J, and interpreting the c i,j as the distance between a city i and a city j, the sequencing probem that minimizes the sum of the sequence dependent c i,j is the cassic traveing saesman probem (TSP). For reasons that wi become cear soon, we refer to scheduing probems in which d j = 1 for a job types as singe mutipicity probems. In the cassica, singe mutipicity, TSP there is an expicit input parameter c i,j for each pair of cities. Hence, the ength of the encoded input is O(N 2 max i,j {J J} c i,j ). In the sequencing probem that we investigate in this paper there are mutipicities d j, j J, of jobs having the same processing requirements. Therefore, in any natura and compact encoding the input is of ength O(N 2 max(max d j, max i,j {J J} c i,j )). Scheduing (and sequencing) probems in which the input is specified this way, have become known as high mutipicity scheduing probems. Other high mutipicity scheduing probems can be found for instance in Hochbaum and Shamir (1991), and in Cifford and Posner (1996). From a compexity perspective, high mutipicity scheduing probems have ead to severa characteristic questions. Let us demonstrate this by considering high mutipicity sequencing probems with a makespan minimization objective. First of a, note that expicity specifying a high mutipicity sequence of jobs requires an amount of space that is not poynomia in the input size. Therefore, it is not a priori cear that high mutipicity sequencing probem are in P-SPACE, et aone in NP. Hence, researchers have investigated whether optima sequences can be compacty encoded. In particuar, a question that has attracted attention are: Can poynomia agorithms for singe mutipicity probem be extended so as to sove in poynomia time, the high mutipicity version (see e.g. Agnetis (1997))? Another question that has been researched is: for probems that are known to be NP-Compete in the singe mutipicity version, is the high mutipicity version sovabe in poynomia time if the number of types is fixed? This question is especiay interesting since in many practica appications, the number of different types wi be sma compared to the numbers of mutipicities. In practica settings in which the above high mutipicity scheduing probems arises, makespan optimization is important but not necessariy the dominant objective function. In addition, managers and panners think it to be desirabe to have the output over time more or ess in proportion to the mutipicities of the types. Among the advantages of such a poicy are ow inventories and a smooth work oad. Consequenty, it has become standard practice to repeatedy execute a short assemby sequence in which the jobs are produced in the required reative quantities. The concept of minima part sets buids on this idea Hitz 3

4 (1979), Hitz (1980). A minima part set is constructed as foows. Let d 0 be the greatest common divisor of the set of mutipicities {d 1, d 2,..., d N }. Further et k j = d j /d 0, for j J. The minima part set (MPS) is described by the vector {k 1, k 2,..., k N }. A popuar approach to finding a good sequence is now to find a sequence for the MPS, that when repeatedy executed, yieds among a such sequences for MPSs the highest throughput rate. Aternativey, we sha formuate the probem in terms of finding an MPS sequence with shortest cyce time. In assemby or production systems, the approach described above ideay resuts in ow inventories, baanced workoads, and a simpe and stabe schedue, which is not too compicated to find. However, this approach may we yied sub-optima throughput rates. In this paper we investigate high mutipicity sequencing probems with a cyce time minimization objective. In particuar we are interested in the trade-off between ong run cyce time versus the ength of the sequence that is repeatedy executed. Indeed, a basic question that we wi address is whether for a given instance there exists a finite repetitive cycic poicy that yieds the optima throughput rate. The permutation no-wait fowshop scheduing probem F no wait C max in the representation scheme of Graham et a. (1979), is an important appication for the research presented in this paper. We therefore quicky review its (high mutipicity) compexity status. It is we known, see e.g. Wismer (1972), that the permutation no-wait fowshop scheduing probem can be modeed as a traveing saesman probem (TSP). For the case of two machines, the resuting TSP instances have a distance matrix with a specia structure, and, in the singe mutipicity case, the cyce time minimization probem can be soved in poynomia time by the agorithm of Gimore & Gomory Gimore and Gomory (1964). Agnetis in Agnetis (1997) shows that the high mutipicity version of the two machine no-wait fowshop scheduing probem can aso be soved in poynomia time, by an extension of the Gimore & Gomory agorithm. From Röck (1981) we know that for three and more machines the decision versions of permutation no-wait fowshop scheduing probems are NP-compete in the strong sense as we as the traveing saesman probem in genera. The idea of finding an optima sequence for an MPS is encountered in various other settings, such as the iterature on assemby ine baancing, or the iterature on maintenance scheduing. Another previousy investigated probem that fits in this framework is the the high mutipicity TSP (HMTSP), which is sometimes referred to as the traveing saesman probem with many visits to few cities. We derive severa resuts that can be appied to HMTSP. In this respect, our work extends earier work on the HMTSP, as it can be found in Cosmadakis and Papadimitriou (1984), Papadimitriou and Kaneakis (1980), Psaraftis (1980), Zhang and van der Veen (1996), Van de Kundert (1995). 4

5 2 Probem Statement and Formuations In this paper we consider a genera high mutipicity scheduing probem with sequence dependent processing times. In fact, we wi not be interested so much in the actua scheduing of tasks, i.e. specifying starting times, but rather in sequencing jobs, i.e. determining the order in which they are to be processed. More specificay, we wi be interested in what we ca the high mutipicity traveing saesman probem, since it is genera enough to serve as a mode for a variety of scheduing appications (see e.g. Remark 1 or Cosmadakis and Papadimitriou (1984); Zhang and van der Veen (1996)). Definition 2.1 The high mutipicity traveing saesman probem (HMTSP): P : min c i,j x i,j x i J (i) subject to x i,j = k j, j J; (ii) i J x i,j = k i, i J; (iii) x i,j 1, i J \J J J such that J ; (iv) x i,j Z +, i J, j J. (v) For a generic instance we wi refer to this formuation as P, and to the vaue of its optima soution as F (P ). Remark 1. Consider the high mutipicity formuation of the permutation nowait fowshop for a singe MPS with cyce time minimization objective. For each ordered pair of types (i, j) J J it is possibe to compute a number c i,j which is the time that must eapse between consecutivey sequenced products (jobs, parts) of type i and type j respectivey. That is ( ) c i,j = p 1,i + max m=1,...,m 1 0, m (p n,j p n+1,i ) It is we known (see e.g. Wismer (1972)) that using the parameters c i,j, i J, j J, the cassica, singe mutipicity, no-wait fowshop probem can be straightforwardy formuated as TSP. Indeed, etting the jobs correspond to cities, and etting c i,j, i J, j J, be the distance between cities i and j, the task is to find a cosed wak through a cities such that visiting every city exacty once and tota ength of which is minima overa such waks. The HMTSP is aso known as the traveing saesman probem with many visits to few cities. From its definition as an integer inear program we concude 5 n=1.

6 that the decision version of the HMTSP, is in NP (despite the exponentia number of constraints). Moreover, using Lenstra s agorithm Lenstra (1983) it can be soved in poynomia time when the number of cities (types) is fixed. In fact combinatoria, strongy poynomia, agorithms for the case where the number of cities is bounded from above by a constant have been deveoped by Cosmadakis, Kaneakis, Papadimitriou and Van de Kundert in Cosmadakis and Papadimitriou (1984); Papadimitriou and Kaneakis (1980); Van de Kundert (1995). We now expain how an (optima) sequence can be constructed from an (optima) soution x 0 to P (see for instance Zhang and van der Veen (1996); Eiset et a. (1995); Van de Kundert (1995)). Let [x 0 i,j] N N be an (optima) soution to (P ). We are going to describe a procedure ConvertToSequence that can be repeatedy executed to convert the soution and matrix [x 0 i,j] N N into a cosed wak that traverses arc (i, j), exacty x i,j times (i, j J), and which represents a high mutipicity sequence in which to visit the cities. We denote by G(J, A) the mutigraph in which the nodes correspond to the cities in the HMTSP, and each arc (i, j) occurs with mutipicity x 0 i,j. In the procedure ConvertToSequence we use the phrase simpe cyce, which is to be understood as a simpe cyce in G. More precisey, we define a cyce c = ((i 1, i 2 ), (i 2, i 3 ),..., (i t, i 1 )), i k J, k 1, 2,..., t to be a simpe cyce if it doesn t contain any subcyce, i.e. if i q i r for q, r 1,..., T and q r. Definition 2.2 Procedure ConvertToSequence Input: [x 0 i,j] N N. Output: A cosed wak, represented by a coection C of pairs (i, c), where i is an integer and c a simpe cyce. 1. C :=, n = Find a cyce c n = ((i 1, i 2 ), (i 2, i 3 ),..., (i t, i 1 )), i k J, k 1, 2,..., t, such that x n 1 i k,i k+1 > 0, k {1, 2,..., t 1} and x n 1 i t,i 1 > Let m n = min{x n 1 i,j for (i, j) c n and x n i,j := x n 1 i,j (i, j) c n }, C := C {m n, c n } and x n i,j := x n 1 i,j otherwise. m n 4. If [x n i,j] N N = [0] N N, output: C and stop, ese set n := n+1, and goto step 2. 6

7 The reader may verify that this agorithm can be read as an high mutipicity extension of the text book agorithm to find a euerian trai in a euerian graph in the foowing way. We construct a euerian wak by traversing the first edge i 1, i 2 of the first simpe cyce c 1. Then, we check whether city i 2 is the first city in any other cyce c. If this is is the case we continue by recursivey traversing c. Otherwise, or after having recursivey traversed c, we continue by traversing the second edge of c 1, i.e. (i 2, i 3 ). For city i 3, again we ook whether there is some simpe, and yet untraversed cyce c k that starts in i 3. If so, we start traversing c k, and otherwise, or after having recursivey traversed c k, we continue traversing c 1. When c 1 is competey traversed, i.e. after having traversed the arc (i t, i 1 ), and recursivey a cyces starting at i 1, we finish by traversing c 1 m 1 1 times without making recursive traversas. In genera, when during the construction of the euerian wak some simpe cyce c k is competey traversed, we traverse it m k 1 times, without making the recursive traversas. The graph G(J, A) is a euerian graph, and therefore must contain a euerian wak. For the correctness of the procedure ConvertToSequence we refer to Eiset et a. (1995)). A few words on its compexity are in order here. First of a, it is not hard to verify that step 2 can be executed in O(N) time. Further, x n i,j contains at east one more zero eement than x n 1 i,j, by definition of step 3. Hence steps 2 to 4 are executed O(N 2 ) times. This eaves the overa time compexity of ConvertToSequence to be O(N 3 ). In addition, this reasoning impies that the coection C consists of at most O(N 2 ) pairs (i, c). Since i and c are aso poynomiay bounded in the input size, the output of it ConvertToSequence is a poynomia encoding of an optima soution. Hence we have two compact encoding schemes of optima soutions: C, and x 0 i,j. In subsequent sections these compact encoding schemes wi be contrasted with not necessariy poynomia encoding schemes in which soutions are more expicitey specified. An exampe of such an encoding scheme is to disregard the mutipicities and competey specify the sequence in which the cities are to be visited. We now continue by considering some reaxations and extensions of Definition 2.1. For a generic instance we define T to be the probem that resuts from P by reaxing the subtour eimination constraints (iv): Definition 2.3 T : subject to min x c i,j x i,j i J x i,j = k j, j J; i J x i,j = k i, i J; x ij Z +, i J, j J. 7

8 We denote the vaue of the optima soution of T by F (T ). Probem T is a transportation probem, and therefore aways has an integra optima soution that can be found in poynomia time. Finay, we introduce a more genera probem formuation of the cyce time minimization probem. Since the research aims to find input sequences that yied maximum throughput, we propose a formuation that aows more genera input sequences, not just input sequences for MPSs: Definition 2.4 P () : subject to min x, 1 c i,j x i,j i J x i,j = k j, j J; i J x i,j = k i, i J; x i,j 1, i J \J J J such that J ; x i,j Z +, i J, j J; Z ++. Probem P () contains the decision variabe. Let (, x ) be some optima soution, then it encodes a sequence in which each city j J is visited i J x i,j = k j times. Hence the resuting sequence contains MPSs. Since the aim is to maximize throughput rate, this formuation minimizes the minimum cyce time achievabe by a sequence for the jobs in MPSs, over a natura numbers. Thus, in a scheduing context, and etting be infinity, the optima soution vaue specifies the maximum throughput rate attainabe whie producing a types in the prespecified ratios. We sha again refer to the objective function of P () by F (P ()), or F () for short. Notice that P () is not a inear program. However, for fixed it is, and as before P () can be soved in poynomia time when both and N are fixed Lenstra (1983). As before, we define T () to be the probem that resuts from P () after reaxing the subtour eimination constraints: 8

9 Definition 2.5 subject to F (T ()) = 1 min x c i,j x i,j i J x i,j = k j, j J; i J x i,j = k i, i J; x i,j Z +, i J, j J. Notice that by defining y i,j = x i,j /, T () can be rewritten as subject to F (T ()) = min x c i,j y i,j i J y i,j = k j, j J; i J y i,j = k i, i J; y i,j Z +, i J, j J. Hence, we concude that T () and T are identica transportation probems. By consequence for any natura number the vaue F (T ) yieds a ower bound for F (). 3 Genera Properties of Optima Soutions In this section we derive some basic properties of the probem P (). We derive some resuts on how the optima vaue F () decreases when increases. A of these resuts are based on the foowing inequaity: Theorem 3.1 For any natura number, the foowing inequaity hods F ( + 1) + 1 F () + 1 F (T ). (1) + 1 Proof. Let x +1 and x, Z ++, denote optima soutions for P (+1) and P () respectivey, and et x T denote an optima soution for T. Now, notice first that the vector x +x T is a feasibe soution for P (+1). Thus, for any number Z ++ 9

10 we have F ( + 1) = = = as required i J c i,j x +1 i,j c i,j x i,j + 1 c i,j x T i,j + 1 i J i J ( ) 1 c i,j x i,j + 1 c i,j x T i,j + 1 i J + 1 F () F (T ) i J Coroary 3.1 F () F ( + 1) hods for any Z ++. Proof. Using Theorem 3.1 we straightforwardy derive, F ( + 1) + 1 F () F (T ) = + 1 F () + 1 F (T ()) F () F () = F (), where the second inequaity foows from the fact that T () is obtained from P () by reaxation of the sub-tour eimination constraints. Theorem 3.2 F () > F ( + 1) if and ony if F () > F (T ). Proof. If F () > F (T ), then it must hod that F () = + 1 F () F () > + 1 F () F (T ) F ( + 1) by 3.1. Conversey, if F () = F (T ), it must hod that F ( + 1) F (T ( + 1)) = F (T ) = F (). 10

11 Combined with 3.1 this yieds that F ( + 1) = F (). By consequence we aso have the foowing coroary: Coroary 3.2 If there exists 0 such that F ( 0 ) = F (T ) then F () = F (T ) for any number 0. Hence, the question rises whether for a given instance there exists 0 such that F ( 0 ) = F (T )? Moreover, in case the answer is affirmative, one woud ike to efficienty compute or at east estimate a smaest number 0 such that the equaity hods. These questions wi be discussed and answered in the Sections 4 and 5. In case the answer is negative, there might sti be some ong run stabiization worthy of characterization (which in turn might be efficienty computabe). Pursuing this direction, we first consider the foowing theorem, which is an extension of Theorem 3.1. Theorem 3.3 For any fixed natura number and any natura number the foowing inequaity hods F () F ( ) + F (T ). (2) Proof. We sha prove by induction on from the basis. For any natura number it hods that F ( ) F ( )/ + ( ) F (T )/ = F ( ), and hence 2 hods for =. Now, suppose that for some number property 2 hods. We prove that based on this induction hypothesis the vaidity of this property for + 1 can be derived. From 3.1 we obtain, for every natura number, F ( + 1) F ()/( + 1) + F (T )/( + 1). Hence, by induction we have that F ( + 1) = and the proof is compete. + 1 F () F (T ) ( + 1 F ( ) + ) F (T ) + 1 F ( ) F (T ) F (T ) Finay, we prove that the vaue F () converges to F (T ) when goes to infinity. Theorem 3.4 If there exists a (finite) natura number such that F ( ) < then im F () = F (T ). (3) 11

12 Proof. Using Theorem 3.3, and etting > we derive Hence, F (T ) F () F ( ) + F (T ) = F (T ) + F ( ) F (T ). F (T ) = im F (T ) im {F (T ) + F ( ) F (T )} = F (T ). which yieds the desired resut. In Section 5 we obtain a much stronger resut which improves Theorems 3.1 and 3.3. Namey, we sha find that for a very genera cass of instances, and for a N 1, it hods that F () = F (N 1) (N 1)/ + F (T ) ( N + 1)/. We finish this section by demonstrating how optima soutions and objective function vaues change with. Exampe 3.1 Consider the foowing instance: Let J = {1, 2, 3}; k 1 = 1, k 2 = 1, k 3 = 1; c 1,3 = c 3,1 = a, (a > 1); c i,j = 1 for (i, j) / {(1, 3), (3, 1)}. The reader is encouraged to verify that For the transportation probem T it hods that F (T ) = 3, x T i,j = 0 for i j and x T i,j = 1 for i = j. The coection C as output by ConvertToSequencein this case is C = {(1, (1, 1)), (1, (2, 2)), )1, (3, 3))}. For the probem P (1) we have F (P ) = a + 2. An optima soution for this probem is x 1 i,j = 1 for (i, j) {(1, 2), (2, 3), (3, 1)} and x 1 i,j = 0 for any other arcs. In this case is : C = (1, ((1, 2), (2, 3), (3, 1))). For any natura number 2 we get F () = 3. An optima soution is specified by x 1,2 = x 2,3 = x 3,2 = x 2,1 = 1; x 1,1 = x 3,3 = 1; x 3,1 = x 1,3 = 0 and x 2,2 = 2. Further, C = ((1, (1, 2)(2, 3), (3, 2), (2, 1)), (( 1), (1, 1)), (( 2), (2, 2)), (( 1), (3, 3))). 12

13 In this exampe it hods that F () = F (T ), for 2. This exampe aso demonstrates that the ratio between F (P ) and F () can be extremey bad, namey F (P )/F () = (a + 2)/3, where a may be chosen arbitrariy. Exampe 3.2 Now, consider another instance which is identica to the one described in Exampe 3.1, but for the foowing arc engths: c 1,2 = c 2,1 = b, where 1 < b < a. It can be checked that in a cases discussed in Exampe 3.1 the same soutions are optima ones, but the objective function vaues are different. More specificay, For the transportation probem T it hods that F (T ) = 3, x T i,j = 0 for i j and x T i,j = 1 for i = j. The coection C as output by ConvertToSequence in this case is C = {(1, (1, 1)), (1, (2, 2)), )1, (3, 3))}. For the probem P (1) we have F (P ) = a + b + 1. An optima soution for this probem is x 1 i,j = 1 for (i, j) {(1, 2), (2, 3), (3, 1)} and x 1 i,j = 0 for any other arcs. In this case is : C = ((1, (1, 2)), (1, (2, 3)), (1, (3, 1))). For any natura number 2 we get F () = (3 + 2b 2)/ = 3 + (2b 2)/. An optima soution is again specified by x 1,2 = x 2,3 = x 3,2 = x 2,1 = 1; x 1,1 = x 3,3 = 1; x 3,1 = x 1,3 = 0 and x 2,2 = 2. Further, C = ((1, (1, 2)(2, 3), (3, 2), (2, 1)), (( 1), (1, 1)), (( 2), (2, 2)), (( 1), (3, 3))). In this exampe, F () stricty decreases when increases, and indeed there doesn t exist 0 such that F ( 0 ) = F (T ). Nevertheess, the optima soution dispays the same (stabe) behavior as in the previous exampe. 4 Optima Sequences of Bounded Length in the Stabe Case Definition 4.1 Consider an instance for which there exist a number 0 Z ++ such that F () > F (T ) for any < 0 and F () = F (T ) for any 0. Such an instance is caed stabe and 0 is caed the stabiization number. Theorem 4.1 For every stabe instance, the stabiization number 0 (N + 1) 2 /4. Proof. Consider a stabe instance and corresponding stabiization number 0. Let M = (J, A) be a compete directed mutigraph on the set of nodes J. In M et us consider a cosed wak W 0 = ((i 1, i 2 ), (i 2, i 3 ),..., (i t, i 1 )), i k J, k {1,..., t}, which represents an optima soution for the probem P ( 0 ). Ca to mind that any cosed wak which represents a feasibe soution of the probem P () passes through a cities j J exacty k j times. 13

14 In addition, et S = (s 1, s 2,..., s w ), s k A, k {1,..., w} be any cosed wak in the graph M. Reca that by definition, for a cosed wak it hods that there exists i, j, k such that s 1 = (i, j) and s w = (k, i). We say cosed wak S is compete if it passes through a vertices in G. More formay, we define: Definition 4.2 Given a mutigraph M(J, A) and a cosed wak S = (s 1, s 2,..., s w ), s k A, k {1,..., w} in M. We say S is compete if for every j J there exists k {1,..., w} and i J for which (i, j) = s k. Definition 4.3 Given a mutigraph M(J, A), cosed wak = S(s 1, s 2,..., s w ), s k A, k {1,..., w} in M, et S = (s i, s i+1,..., s j 1, s j ), where 1 i < j w be a cosed wak which is contained as a subsequence in S. We say S is a minima compete wak if, for every S = (s i, s i+1,..., s j 1, s j ), where 1 i < j w, the cosed wak S \ S = (s 1, s 2,..., s i 1, s j+1,..., s w ) is not compete. If cosed wak S is contained as a subsequence in S and S \ S is compete, we ca S ommissabe. Consider a minima compete wak W = ((j 1, j 2 ), (j 2, j 3 ),..., (j t, j t+1 )) where j 1 = j t+1 which is obtained from the compete cosed wak W 0 by removing ommissabe cosed waks from it as described above. We are going to prove that W contains at most (N + 1) 2 /4 arcs. To see this, et f be the number of vertices that are passed in W exacty once. More formay, f is the number of vertices j for which there is exacty one arc (j k, j k+1 ), k 1,..., t in W such that j k = j. First, we make cear that f 2. Consider an arbitrary cyce C that is contained as a subsequence in W. If no such cyce C exists, we are done since W is compete and hence f = N. Thus et us assume C exists. Then C (and aso W \ C) consists of at east two arcs, since otherwise it forms a oop, and therefore is ommissabe, which contradicts the minimaity of W. Now, et C 1 be a cyce that is contained as a subsequence in C with minimum number of arcs and et C 2 be defined anaogousy with respect to W \ C. There are two cases for C 1. Assume first that a vertices passed in the cyce C 1 are represented in W more than once. If a vertices passed in C 1 are passed aso in W \ C 1, C 1 is ommissabe contradicting the minimaity of W. Hence at east one vertex must be passed more than once in C 1. Thus C 1 contains a cyce. Again we have reached a contradiction since by definition of C 1 it doesn t contain any cyces contained in it as a subsequence. Thus, we concude that there is at east one vertex passed in C 1 which is passed in W ony once. Appying the same reasoning to C 2, we concude that f 2. Now et i and j be two vertices that are passed once in W and such that the part Q of the cosed wak from i to j doesn t contain any other vertex that is 14

15 passed once in W. (We wi say that i and j are neighboring.) It can be seen as foows that in Q no vertex is passed more than once. If there exist a vertex that is passed twice in Q we have found a cyce. By definition of i,j and Q, this cyce doesn t contain vertices that are passed once in W. Hence, the cyce must be ommissabe contradicting the minimaity of W. Thus a vertices that are passed in Q are passed once in Q. Now we know that in the wak there are f 2 vertices which are passed exacty once and that between pairs of neighboring such vertices, no other vertices occur more than once. This aows us to bound the tota number of vertices in W. Indeed, it must hod that the tota number of vertices and hence arcs in W does not exceed f +f(n f). Maximizing f +f(n f) over f = 2,..., N yieds f max = (N + 1)/2. It then foows that f max + f max (N f max ) = (N + 1) 2 /4 and therefore W has ength at most (N + 1) 2 /4. Using that the ength t of W is ess than or equa to (N + 1) 2 /4, we now finish the proof by constructing a soution to P (t) such that F (t) = F (T ). Since we consider a stabe instance, by definition F ( 0 ) = F (T ). Therefore the wak W 0 is aso an optima soution for the probem T ( 0 ). Let x 0 i,j be the number of times arc (i, j) is traveed in W 0, for a i, j J. Then, x 0 i,j/ 0 is a fractiona optima soution for probem T. Since the transportation probem has a totay unimoduar constraint matrix, the poyhedron of probem T is integra. By consequence, any fractiona optima soution can be represented as a convex combination of integer optima soutions. Hence, we have that for every arc (i, j) contained in the wak W 0, there exists an integer optima soution for T for which x i,j > 0. Now et us consider the wak W 0 and a corresponding minima compete cosed wak W. Reca that for every arc (i, j) W there exists a soution x i,j of the probem T which contains this arc. Let x be obtained by summing a x i,j, (i, j) W. Since W is a compete wak, x satisfies the subtour eimination constraints, and hence it is an optima soution to P (t). Since x is obtained by summing t soutions for probem T, x is a soution for P (t). Therefore F (t) = F (T ) and thus t 0. It then foows that 0 t (N + 1) 2 /4 as required. With Theorem 4.1 at hand, we are now abe to address a computationa compexity issue. Despite the fact that even for fixed the decision version of P () is a generaization of the TSP and therefore NP -compete, we have the foowing theorem: Theorem 4.2 The probem of deciding whether an instance is stabe can be soved in poynomia time. Proof. Firsty, we are going to check for each arc (i, j ) J J whether there exists an optima soution of T that contains this arc. This can be achieved by 15

16 soving a modification of T, where the modification consists of decreasing k i by one in the outfow contraint (iii) for city i, and decreasing k j by one in the infow constraint (ii) for city j. Let us ca this modified probem T i,j. Obviousy, T i,j is in turn a transportation probem that can be easiy soved in poynomia time. The aforementioned optima soution exists if and ony if F (T ) = F (T i,j ) + c i j. We subsequenty construct a directed mutigraph D = (J, A) as foows. For a (i, j ) such that F (T ) = F (T i,j ) + c i j, et j xi be obtained by sighty modifying an optima soution to T i,j. The modification consists of increasing j xi i j by one. Now, the arc set A of D is obtained as foows. For a (i, j), the mutipicity of arc (i, j) is defined to be the sum of the x i j ij over a (i j ) for which F (T ) = F (T i,j ) + c ij. Now, if D is strongy connected then it contains a Euerian tour ( see e.g. Eiset et a. (1995)), i.e. a cosed wak visiting every arc in this directed mutigraph. Ceary this tour visits every vertex as we and hence satisfies the subtour eimination constraints. But then this Euerian tour contains a soution to P () with vaue F (T )/ = F (T ). Conversey, suppose that D is not strongy connected. Thus, there exists a subset J J, J, such that D doesn t contain any arc (i, j) such that i J, j J \ J. Now, for contradiction, assume that there is some number such that F () = F (T ) and et x be an optima soution to P (). Define yi,j = x i,j/, (i, j) J J. Note, that y is an optima soution for T. Notice aso, that x is a feasibe soution to P () and hence satisfies the subtour eimination constraints. Then for J there must exist an arc (i, j) such that i J, j J \ J, and yi,j > 0. Then, in some optima soution z of the transportation probem T () it hods that zij > 0. By tota unimoduarity, there must exists an optima soution z 0 for T in which zi,j 0 is a positive integer. But then, by definition of A, it hods that (i, j) A. Since arc (i, j) connects J and J \ J we have arrived at a contradiction. Thus, we concude that an instance I is stabe if and ony if the directed mutigraph D is strongy connected. As described above, D can be constructed in poynomia time. Further, it is easy to check in poynomia time, whether D is strongy connected (see e.g. Ahuja and Orin (1994)), which competes the proof. We finish this section by giving a tight exampe for the bound derived in Theorem

17 Exampe 4.1 Consider the foowing instance. J = {1, 2, 3, 4, 5, 6}, and a a a +1 a a a a a a +1 a C = a a a a a +1 a a a a a +1 a where a > 1, k T = (1, 1, 1, 3, 3, 3). This instance has minima compete cosed wak and aso optima sequence [1, 4, 5, 6, 2, 5, 6, 4, 3, 6, 4, 5] : It is straightforward to generaize exampe 4, in particuar the distance matrix C to the case where J = {1,..., n} and a corresponding minima compete cosed wak W of ength (n+1) 2 /4 exists that entais the optimum soution vaue. et it be noted however that this doesn t mean that every optima soution necessariy resuts in a minima cosed wak of ength (n + 1) 2 /4. Indeed we sha see in the next Section, that a better bound can be obtained when using inear programming techniques. Nevertheess, if we ony encode soutions as sequences and consider resuting cosed waks, Theorem 4.1 is tight. Hence the section tite. Indeed, the next section goes to show that it pays to consider and encode soutions for this high mutipicity sequencing probem in a different way then by expicit sequences. 5 Optima Soutions of Bounded Length For further investigations it wi be convenient to define for the given vector δ Z N + the foowing transportation probem T (, δ) subject to F (T (, δ)) = min x c i,j x i,j i J x i,j = k j δ j, j J; i J x i,j = k i δ i, i J; x i,j Z +, i J, j J. Theorem 5.1 Consider an instance of the probem such that the foowing hods simutaneousy F (N 1) is finite; 17

18 The probem T (N 1, α) has a finite optima soution where the i-th component from the vector α Z N + represents the number of times city i is passed in a minima compete cosed wak obtained from an optima soution of the probem P (N 1) (see theorem 4.1). Then, for any natura number N 1 the foowing hods: and F ( + 1) = F () = F (N 1) N F () + 1 F (T ) (4) F (T ) N + 1 and the soution x = x N 1 + ( N + 1)x T is an optima soution of the probem P (). Proof. To prove the theorem we have to show that equation 4 hods. Equation 5 then foows by induction as in Theorem 3.3. To prove equation 4, we show that ( + 1)F ( + 1) F () + F (T ) for any natura number N 1. Together with Theorem 3.1 this yieds the desired resut. Consider an optima soution x 0 of the probem P (+1), N 1. As before, et W 0 be a corresponding cosed wak, and et W be a minima compete cosed wak obtained from W 0 by deeting ommissabe cyces. Let α be the vector such that α i is the number of times city i is passed in W. It can be seen that since W is minima and compete, 1 α i N 1 for a i J. Now consider a soution x of P () which is constructed as foows: x = x W + x T (,α) where x W ij (i J, j J) equas the number of times arc (i, j) occurs in W and x T (,α) is a soution of the probem T (, α). Since k i (N 1) k i N 1 α i for any index i J, we may concude that W is a feasibe compete tour for the probem P (). Thus, the soution x is feasibe for P () and the transportation probem T (, α) is correcty defined with stricty positive right hand-side coefficients k α. Since x is a feasibe soution of P (), we have F () + F (T ) c i,j x i,j + F (T (, α)) + F (T ) (i,j) W = (i,j) W c i,j x i,j + ( + 1) F (T ( + 1, α)) + F (T (, α)) + F (T ) ( + 1) F (T ( + 1, α)) = F ( + 1) + F (T (, α)) + F (T ) ( + 1) F (T ( + 1, α)). (5) 18

19 Hence, we can prove equation 4 by proving F (T (, α)) + F (T ) = ( + 1) F (T ( + 1, α)) for any N 1 (*). Since the poyhedron of the transportation probem is integer,we can assume without oss of generaity that F (T ), F (T (, α)) and F (T ( + 1, α)) are inear programs. Further, changing notation from x to x/ yieds that instead of proving that F (T (, α)) + F (T ) = ( + 1) F (T ( + 1, α)) it suffices to prove that F (T (1, α/)) + F (T ) = ( + 1) F (T (1, α/( + 1))). The reader is encourage to verify that the three inear programs T (1, α/), T, T (1, α/( + 1)) ony differ in the right hand sides. Let it be noted aso that T is by definition identica to T (1, 0), and hence F (T ) equas F (T (1, 0)). Let us denote the dua of T (1, α/) as DT (1, α/). It can be described as foows: ( DT (1, α )) : max (k i α i u,v )u i + ) (k j α j )v j i J subject to u i + v j c i,j, i J, j J. We simpy write DT for DT (1, 0), and indeed DT is the dua of T. Since the three probems T (1, α/), T, T (1, α/( + 1)) ony differ in the right hand sides, the three probems DT (1, α/), DT, DT (1, α/( + 1)) ony differ in the coefficients of the objective function. Now, suppose that despite the differences in the coefficients of the objective function, some optima soution (u, v ) of DT is aso optima for DT (1, α/) and DT (1, α/( + 1)). Then, by strong duaity, we have ( + 1) F (T (1, ( = ( + 1) = i J i J α + 1 )) (k i α i + 1 )u i + (k j α j + 1 )v j (( + 1) k i α i )u i + (( + 1) k j α j )v j ) = + ( k i α i )u i + ( k j α j )vj + k i u i + k j vj i J i J ( = (k i α i )u i + ) ( (k j α j )v j + k i u i + ) k j vj i J i J = F (T (1, α )) + F (T ) 19

20 which proves equation ( ) and hence equation 4 as required to prove the theorem. Thus, intuitivey, we have to show that for arge enough, or more specificay N 1, the differences in the coefficients of the objective function between the probems DT, DT (1, α/) and DT (1, α/( + 1)). are sma enough to yied a soution (u, v ) that is optima for a three of DT, DT (1, α/) and DT (1, α/( + 1)). Continuing in this intuitive fashion, such a soution (u, v ) can be found as foows. Consider the poyhedron of DT. Let us make some sma modifications to the objective function, and hence to the objective hyperpane. Let s say these modifications are obtained by adjusting the orthogona vector by a sma vector σ. It is cear that if optima face of the poyhedron is bounded in the σ-direction, then we preserve at east one optima point of this face. By the second condition of the theorem, F (T (N 1, α)) is finite. It then foows that in the direction α the optima face is bounded. Therefore with a sma adjustment of the orthogona vector in the α-direction, we keep at east one dua optima soution (u, v ) of T for the probems DT (1, α/) for any sufficienty big number Z +. More formay, the equation ( ) can be proved using the foowing theorem from sensitivity anaysis: Theorem 5.2 Partition Perturbation Theorem (see, e.g. Greenberg (1997)). Consider inear program LP (b) = min{cx x 0, Ax b} and its dua DLP (b) = max{πb π 0, πa c}. Suppose (x 0, π 0 ) is a pair of stricty compementary soutions for LP (b) and DLP (b) respectivey, and γ is an admissibe direction, i.e. LP (b + θγ) has an optima soution for some θ > 0. Define the differentia inear program: (b) = max{πγ π is optima soution of DLP(b)}. Then there exists θ > 0 such that the foowing hods true for θ [0, θ ]: the optima partition for LP (b + θγ) or, equivaenty, DLP (b + θγ) is the same as the optima partition for (b). By the conditions of Theorem 5.1 F (T (N 1, α)) and hence F (T (1, α/(n 1))) are finite and therefore α, α is an admissabe direction. Let (u, v ) be the optima soution of DT that is aso optima for (b) = max{(u, v) T, (α, α) (u, v) is an optima soution of DT}. Then, by the Partition Perturbation Theorem, there exists θ > 0 such that for any θ [0, θ ] the optima partition of T (1, θα) (and hence DT (1, θα)) is the same as the optima partition for T (and DT ). This in turn impies that the optima dua soution (u, v ) of DT is aso optima for DT (1, θα) for any θ [0, θ ]. By consequence, we can compete the proof by showing that θ 1/. To estimate θ for a genera inear program LP, the foowing approach is proposed in Monteiro and Mehrotra (1996). Consider the system of inear equations A B z = γ where A B denotes the submatrix of A whose coumns correspond to the indices from B = {n x n > 0}. In Monteiro and Mehrotra (1996), the authors ist the foowing two mutuay excusive cases: 20

21 If the system has no soution, that is γ is not in the range of A B, then there is not a number θ > 0 that preserves the optima partition. Assume that the system has a soution, say z, then θ = + if z 0 and θ min{x n /z n n B, z n > 0} otherwise. Since we have chosen the conditions of the theorem such that an admissabe direction exists, we are ceary in the second case. Thus, there exist a soution z of the system Bz = α 2 where α 2 Z 2N is the vector (α, α). Letting x be an optima soution to T, the vaue min{x i,j/zi,j x i,j > 0; zi,j > 0} bounds θ from above. Since the poyhedron of P is integra x i,j 1 if x i,j > 0. Hence, θ 1/Z, where Z = max{zi,j z i,j > 0}. Thus, it remains to bound max{zi,j z i,j > 0}. Let us denote by B the base matrix of the inear program T, which is obtained from the constraint matrix of T by eimination of a coumns corresponding to variabes x ij for which x ij = 0. The probem Bz = α 2 can be represented as the foowing probem on the graph. Given a graph G = (J, E) where E = {(i, j) J J x i,j > 0}. The probem consists in finding a circuation fow such that every city j J has infow and outfow equa to α j times. Negative components of the vector z are aowed, hence we may disregard the orientation of the x i,j in the edges. Consider the circuation fow z. Since every city j J of the graph D has infow and outfow α j -times, z i,j α j for a i, j. Hence, z i,j α j N 1 for a i, j J, and thus Z max α j. But then, θ 1/ max α j 1/(N 1), and hence for N 1 as stated in the theorem, θ 1/ and the proof is compete. Theorem 5.3 For every instance of P() with finite input data F () = F (N 1) N 1 + F (T ) N + 1 and the soution x = x N 1 + ( N + 1)x T is optima for P (). Proof. A conditions of the theorem 5.1 are satisfied when the input data are finite. Indeed, if the objective coefficients c i,j, i J, j J, and mutipicity coefficients k j, j J, are finite, then the vaues F (N 1) and F (T (N 1, α)) are finite. Theorem 5.4 If an instance is stabe then the stabiization number 0 is such that 0 N 1, and this bound is tight. Proof. According to Theorem 5.1, it hods for N that F () = ( 1) F ( 1) + F (T ) or equivaenty F ( 1) = ( F () F (T ))/( 1). By stabiity, we have that there exists such that F () = F (T ) Substituting F (T ) 21

22 for F (L) in the above equaity we get F ( 1) = F (T ). Since by Theorem 5.1 the equaity hods for for any N we have F (N 1) = F (T ) and immediatey 0 N 1. It remains to show that there exists instances for which 0 = N 1. Consider the foowing instance, which is a straightforward extension of Exampe 3.1. Exampe 5.1 Let J = {1, 2,..., N}, k i = 1 for any i J and the distance matrix is a a... a a 1 a 1 a... a a a... a 1 a a 1 a... a a 1 a 1 a... a a a 1 where a > 1. For the transportation probem F (T ) = N, x T i,j = 0 for i j and x T i,j = 1 for i = j. An optima coection C = {(1, (1, 1)), (1, (2, 2)),..., (1, (N, N))}. For = 1, an optima soution is given by x i,i+1 = 1 for i = 1,..., N 1, x N,0 = 1, and x i,j = 0, i, j J otherwise. Thus, the vaue of an optima soution is in this case F (1) = 1 + (N 2)a + 1 = 2 + (N 2)a. For = 2, an optima soution is given by x 1,2 = x 2,2 = x 2,1 = 1, x 1,3 = 1, x i,i+1 = 1 for i = 3,..., N 1, and x N,0 = 1, x i,i = 1, i = 3,..., N, and x i,j = 0, i, j J otherwise. Thus, in this case the vaue of an optima soution equas F (2) = ( (N 3)a (N 2))/2 = ((N 3)a + N + 3)/2, For any 1 < < N 1 an optima soution x i,j (i J, j J) is

23 where the * superscripts the -th row and -th coumn of the matrix. The vaue of this optima soution F () = = ( 1)2 + ( 1)( 1) (N ( + 1))a (N )( 1) ( 1)(N + 1) (N 1)a (where the equaity can be derived by adding a ( 1) terms and sighty rewriting the remainder). For = N 1 we get F (N 1) = N = F (T ). An optima soution is Thus, for this instance 0 N 1. We finish the proof by showing that 0 N 1. Indeed, for = N 2, which impies that N 3, (N 3)(2N 2) (N (N 2) 1)a F (N 2) = N 2 = 2N 2 8N a N 2 = 2(N 2)2 + a N 2 = 2(N 2) + a N 2 2(N 2) + 1 N 2 N + N N 2 N where the ast inequaity hods since N 3, which yieds 0 > N 2 as required. The previous instance aso proves a fina theorem of this section: Theorem 5.5 The bound of Theorem 5.1 is tight, i.e. there exist instances for which a conditions hod and F (N 2) > F (N 1). 23

24 6 Summary and Concusions This paper deas with a very genera high mutipicity sequencing probem, namey the high mutipicity traveing saesman probem (HMTSP). The research is motivated by the appicabiity of the HMTSP to a variety of scheduing probems. Latey, high mutipicity scheduing probems haven been widey studied, but very itte is known about the impact of scheduing the jobs as specified by a minima part set, as opposed to the more genera scheduing probem of scheduing jobs in non-minima, optima, part sets. This paper investigates how the cyce time, or ength of a HMTSP tour varies with the mutipicities in the (minima) part set. In particuar we investigate the behavior of the optima soution, and the optima soution vaue, when a quantities in the minima part set are mutipied by a factor. We show that the optima soution vaue decreases when increases. Moreover, we show that the ratio between the vaue of an optima soution for = 1 and for arbitrary can grow proportiona to. Further, we have investigated whether there exist finite for which the optima vaue cannot be improved upon. We have given a poynomia procedure that decides this probem in Section 4. In the same section we show that, if there exists finite for which the optima vaue is minima, then this can be bounded from above by (N +1) 2 /4. In fact, we have shown that, if we restrict our encoding schemes to sequences which expicity specify an order in which the cities are to be visited, then (N + 1) 2 /4 is a tight bound. However, in Section 5 we show that if the optima soution is given in terms of a soution to an integer program, this bound can be improved to be N 1. Section 5 aso shows that even if there is no finite for which the optima soution vaue is owest attainabe, the differences between the engths of cycic optima schedues for and + 1 are identica for a N 1. The bounds on estabish the importance of encoding schemes for soutions, especiay in the case of sequencing probems. Indeed, the resuts make cear that if the optima soution is given in terms of an expicit sequence, then the tightest bound on that can be derived is O(N 2 ), whereas this bound can be improved upon to be O(N) when using a more compact encoding scheme, which is based upon an integer programming formuation. The inear bound obtained in this paper appies to a high mutipicity scheduing probems that can be formuated as a HMTSP. It wi be interesting to see whether this resut, as we as the ratio anaysis, can be improved upon for specific probems. Finay, we conjecture that the bound of Theorem 5.1 can be improved to be (N 1)/ min i k i. 24

25 7 Acknowedgements The authors are gratefu to Yves Crama and Nadia Brauner for usefu discussions on high mutipicity issues. The authors thank Jos Sturm for providing references on sensitivity anaysis. References Agnetis, A No-wait fowshop scheduing with arge ot size. Annas of Operations Research. 70, Ahuja, T., Magnanti, F., and Orin, D Network Fows. Wiey Interscience, San Francisco. Aigbedo, H., and Monden, Y A parametric procedure for mutiecriterion sequence scheduing for just-in-time mixed-mode assemby ines. Internationa Journa of Production Research. 35, Aniy, S., Gass, C.A., and Hassin, R The scheduing of maintenance service. Discrete Appied Mathematics. 82, Bar-Noy, A., Bhatia, R., Naor, J.S., and Schiber, B Minimizing service and operation costs of periodic scheduing. In Proceedings of the 9th Annua ACM-SIAM Symposium on Discrete Agorithms (SODA), Cifford, J.J., and Posner, M.E Parae machine scheduing with high mutipicity. Working Paper The Ohio State University, Department of Industria, Weding and Systems Engineering. Cosmadakis, S., and Papadimitriou, C The traveing saesman probem with many visits to few cities. SIAM Journa of Computations. 13, Eiset, H.A., Gendreau, M., and Laporte, G Arc routing probems, part i: The chinese postman probem. Operations Research. 43, Gimore, P.C., and Gomory, R.E Sequencing a one-state variabe machine: A sovabe case of the traveing saesman probem. Journa of Operations Research Society of America. 12, Graham, R.L., Lawer, E.L., Lenstra, J.K., and Rinnooy Kan, A.H.G Optimization and approximation in deterministic sequencing and scheduing: a survey. Annas of Operations Research. 5, Greenberg, H.J RIM Sensitivity Anaysis from an Interior Soution. Word Wide Web, hgreenbe. 25