Variations in the efficiency of a mathematical programming solver according to the order of the constraints in the model

doi:10.3926/jiem.2008.v1n2.p4-15 ©© JIEM, 2008 – 01(02): 4-15 - ISSN: 2013-0953 Variations in the efficiency of a mathematical programming solver according to the order of the constraints in the model Rafael Pastor Universitat Politècnica de Catalunya (SPAIN) Received September 2008 Accepted November 2008 Abstract: It is well-known that the efficiency of mixed integer linear mathematical programming depends on the model (formulation) used. With the same mathematical programming solver, a given problem can be solved in a brief calculation time using one model but requires a long calculation time using another. In this paper a new, unexpected feature to be taken into account is presented: the order of the constraints in the model can change the calculation time of the solver considerably. For a test problem, the Response Time Variability Problem (RTVP), it is shown that the ILOG CPLEX 9.0 optimizer returns a ratio of 17.47 between the maximum and the minimum calculations time necessary to solve optimally 20 instances of the RTVP, according to the order of the constraints in the model. It is shown that the efficiency of the mixed integer linear mathematical programming depends not only on the model (formulation) used, but also on how the information is introduced into the solver. Keywords: mixed integer linear mathematical programming, response time variability problem, combinatorial optimization 1 Introduction Integer linear programming is a classical tool in practical operations research that can be applied to many problems (e.g. Salkin and Mathur, 1989) very effectively (e.g. in Corominas et al. 2008 it was applied to solve a real problem of a Variations in the efficiency of a mathematical programming solver according to the order of the constraints in the model R. Pastor 4 doi:10.3926/jiem.2008.v1n2.p4-15 ©© JIEM, 2008 – 01(02): 4-15 - ISSN: 2013-0953 motorcycle assembly line and in Pastor et al. 2008 it was applied to solve the case of a woodturning company). The technique is well-known and reliable but it must be handled carefully. It is known that its efficiency depends on the model (formulation) used: with the same mathematical programming solver, a given problem can be solved in a brief calculation time using one model but requires a long calculation time using another. Therefore, as stated by Billionnet (1999, p. 105): “Given a problem with a few dozen of variables one cannot be confident that integer programming will work until it has been tried on realistic instances”. Several techniques have been used to improve the efficiency of this tool. A standard technique is the elimination of symmetries: Margot (2007), for example, presents techniques for handling symmetries in integer linear programs in which variables can take integer values, which extends previous research that dealt exclusively with binary variables. Tightening the definition of the data and introducing redundant constraints have also provided good results: Corominas et al. (2006), for example, demonstrated the importance of modelling, as well as the huge impact that redundant constraints and the elimination of symmetries have on the effectiveness of MILPs for solving the Response Time Variability Problem (RTVP), an NP-hard scheduling problem (Corominas et al., 2007); the total computation time taken to solve 20 instances dropped from 38,603 to 398 seconds and its practical limit to obtaining optimal solutions was increased from 25 to around 40 units to be scheduled. This paper argues that the order of the constraints in a model can have a considerable effect on the time that a mathematical programming solver takes to solve a problem optimally. Lets us, for example, take three sets of constraints (A, B and C) of a problem to be solved. To introduce the sets of constraints in the mathematical programming solver in the order A-B-C, A-C-B, B-A-C, B-C-A, C-A-B and C-B-A is not indifferent and can cause its efficiency to vary considerably. This new, unexpected feature that must be taken into account in mathematical programming has not been presented previously (to the best of the authors’ knowledge). For an integer programming formulation of a test problem, RTVP, it is shown that the ILOG CPLEX 9.0 optimizer returns a ratio of 17.47 between the maximum and minimum calculation times needed to solve optimally 20 instances of the RTVP, according to the order of the constraints in the model: with one permutation of the Variations in the efficiency of a mathematical programming solver according to the order of the constraints in the model R. Pastor 5 doi:10.3926/jiem.2008.v1n2.p4-15 ©© JIEM, 2008 – 01(02): 4-15 - ISSN: 2013-0953 sets of constraints the solver takes only 335 seconds, whereas with another one it takes 5,851 seconds. It is shown that the efficiency of the mixed integer linear mathematical programming depends not only on the model (formulation) used, but also on how the information is introduced into the solver. The rest of the paper is organized as follows. Section 2 presents the RTVP. Section 3 describes the computational experiment carried out. Finally, Section 4 is devoted to the conclusions. 2 The test problem: the Response Time Variability Problem The Response Time Variability Problem (RTVP) occurs whenever products, clients or jobs need to be sequenced so as to minimize variability in the time between the instants at which they receive the necessary resources. It was recently defined in the literature and first presented by Corominas et al. (2007), who proved that it is NP-hard. This problem has a broad range of real-world applications. For example, it can be used in the automobile industry to sequence the models to be produced on a mixed-model assembly line (Monden, 1983). Other contexts in which the RTVP appears are the computer multi-threaded systems and network servers (Waldspurger and Weihl, 1995), the periodic machine maintenance problem (BarNoy et al., 2002), the scheduling of advertising slots for television (Brusco, 2008), the design of sales catalogs (problem introduced in Bollapragada et al., 2004) and the scheduling of waste collection (Herrmann, 2007). The abovementioned applications are examples of a very common situation in which a resource must be used successively by different units and it is important to schedule them in such a way that the different types of units share the resource in some fair manner. The RTVP proposes a measure of fairness: to minimize the variability of the distance (measured, for example, in number of slot times) between any two consecutive units of the same product (event, job or client); i.e., to have the distances between any two given consecutive units of the same product as constant as possible. The RTVP is formulated as follows. Let be the number of products, number of units of product and the total number of units Variations in the efficiency of a mathematical programming solver according to the order of the (...truncated)