Connecting special ordered inequalities and transformation and reformulation technique in multiple choice programming

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1 Computers & Operations Research 29 (2002) 1441}1446 Short communication Connecting special ordered inequalities and transformation and reformulation technique in multiple choice programming Edward Yu-Hsien Lin *, Dennis L. Bricker College of Management, Taipei University of Technology, Taipei 106, Taiwan Faculty of Administration, University of New Brunswick, Fredericton, N.B. Canada E3B 5A3 Department of Industrial Engineering, The University of Iowa, Iowa City, IA52242, USA Received 1 March 2000; received in revised form 1 June 2000 Abstract An article entitled: `A Note on Modeling Multiple Choice Requirements for Simple Mixed Integer Programming Solversa was published by Ogryczak (Comput. Oper. Res. 23 (1996) 199). In this article, Ogryczak proposed a reformulation technique called special ordered inequalities (SOI) to model the non-convex programming problems with special ordered sets (SOS) of variables. The SOI technique appears to be analogous to the reformulation technique introduced by Bricker (AIIE Trans. 9 (1977) 105) and is related to the reformulation and transformation technique (RTT) developed by Lin and Bricker (Eur. J. Oper. Res. 55(2) (1991) 228); Lin and Bricker (Eur. J. Oper. Res. 88 (1996) 182). Since none of this literature was cited in the references of Ogryczak (Comput. Oper. Res. 23 (1996) 199), we would like to use this note to di!erentiate SOI and RTT and to elaborate their connection. Scope and purpose In the context of non-convex programming, two major types of special ordered sets (SOS) of variables have been identi"ed and studied by researchers. SOS1 are sets of non-negative variables where, for each set, at most one of the variables can be non-zero in the "nal solution. The most common application of SOS1 is multiple choice programming (MCP) which can be found in the modeling of many integer programming problems in location, distribution, scheduling, etc. SOS2requires that, for each set, at most two of the variables can be non-zero in the "nal solution and, if they are, they must be adjacent. SOS2has been widely used in separable programming to model non-linear functions using sets of piece-wise linear functions. Bricker introduced an explicit reformulation technique for SOS in Lin and Bricker developed Research related to this paper is supported by the grant from the Natural Science and Engineering Research Council of Canada, Grant Number: OGP * Correspondence address: College of Management, Taipei University of Technology, Taipei 106, Taiwan. Tel.: # ; fax: # addresses: yhlin@unb.ca (E.Y.-H. Lin), dbricker@icaen.uiowa.edu (D.L. Bricker) /02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S (00)

2 1442 E.Y.-H. Lin, D.L. Bricker / Computers & Operations Research 29 (2002) 1441}1446 a reformulation and transformation technique (RTT) to implicitly compose the optimal Simplex tableau for MCP in They also elaborated upon it with a computational report in Without citing the work by Bricker, or that by Lin and Bricker, Ogryczak proposed an analogous reformulation technique called special ordered inequalities (SOI) for SOS in This note aims to elaborate upon the connection between SOI and RTT as the supplementary information for the future research in SOS Elsevier Science Ltd. All rights reserved. Keywords: Non-convex programming; Special ordered sets; Multiple choice programming 1. Introduction This note aims to connect the research result on special ordered sets (SOS) published by Ogryczak in Computers and Operations Research [1] with the earlier work by Bricker [2], and that by Lin and Bricker [3,4] which were not cited in [1]. Speci"cally, we elaborate the connection between the special ordered inequalities (SOI) proposed by Ogryczak, and the reformulation and transformation technique (RTT) introduced by Lin and Bricker. For the bene"t of the readers, we "rst provide a brief background on the SOS in non-convex programming. Special ordered sets are, in a broad de"nition, sets of variables that exist in a non-convex problem with some special requirements. Variables in SOS usually fall in mutually exclusive sets. Two major types of SOS have been identi"ed and studied by researchers. Special ordered sets type 1 (SOS1) are sets of non-negative variables where, for each set, at most one of the variables can be non-zero in the "nal solution. The `restricted SOS1a requires that, for each set, exactly one of the variables can be non-zero in the "nal solution. The most common application of SOS1 is multiple choice programming (MCP) that contains the restricted SOS1 with binary variables. A survey on MCP was conducted by Lin in 1994 [5]. Special ordered sets type 2(SOS2) are sets of non-negative variables where, for each set, at most two of the variables can be non-zero in the "nal solution and, if they are, they must be adjacent. SOS2have been widely used in separable programming to model non-linear functions using sets of piece-wise linear functions. Although not as commonly used as SOS1, successful applications of SOS2have been reported in the "elds of utility planning [6,7] and pollution control [8]. In an attempt to locate the global optimum of a non-convex problem with SOS, Beale and Tomlin [9] suggested that variables in each set of SOS be treated as unity. That is, during the branching phase of the branch-and-bound procedure, instead of dichotomizing the individual variables in the SOS as in the conventional method, each set of the SOS is partitioned into two subsets. Variables in each subset are then forced to vanish, respectively. Mixed integer programming (MIP) software capable of handling SOS in this manner is said to be equipped with a SOS solver. When a SOS solver is adopted, it is necessary to identify a partitioning point for every SOS so that two sub-problems can be generated to enter the candidate list. Beale and Tomlin proposed a weighted mean method (WMM) for this purpose [9]. However, the performance of the WMM depends on the proper weight assigned to each variable, usually through a user-de"ned reference row. Moreover, unlike the dichotomy of a single variable where penalties can be calculated, partitioning SOS creates di$culty in penalty calculation. Tomlin suggested that the `pseudopenaltya associated with the variables in a subset be estimated through a composite variable [10].

3 E.Y.-H. Lin, D.L. Bricker / Computers & Operations Research 29 (2002) 1441} Such a composite variable is formed by aggregating the rows associated with the variables in this subset from the optimal Simplex tableau of the LP relaxed problem. 2. Special ordered inequalities (SOI) In general, for a non-convex problem, the constraint that addresses the multiple-choice requirement associated with a particular special ordered set has the form of x #x #2#x "1 (1) x *0 and integer, j"1, 2,2, r. (2 ) The SOI proposed by Ogryczak [1] introduces a set of new variables y, j"1, 2,2, r, where y "x, y "y #x for j"2,3,2, r. Ogryczak noted that (1) and (2) can then be expressed in y variables as y "1, and y *y, j"1, 2,2, r!1, respectively, or simply put them into a set of variable upper bounding (VUB) constraints of 1*y *y *2*y *y *0. (3) Ogryczak conducted a computational experiment to test the e!ect of such reformulation using the simple MIP software without a SOS solver. Speci"cally, he solved 10 water quality management problems [11] (with problem size ranging from 41 to 161 variables accompanied by 5}20 multiple choice constraints) using MOMIP [12] on a DEC 5000/240 computer. Ogryczak reported a signi"cant reduction of the CPU time recorded during the branch-and-bound phase when every SOS constraint of type (1), and its implicit integrality and non-negativity requirements of (2) are reformulated into the associated VUB constraints of (3). 3. Reformulation and transformation technique (RTT) Based on Bricker's earlier work [2], the RTT was introduced by Lin and Bricker through a matrix operation [3]. In particular, it reformulates (1) by de"ning a set of transformation variables y as y"¹x, where y"[y, y,2, y ], x"[x, x,2, x ], and ¹ is a block diagonal matrix with its entries t "1, if i)j, and t "0, if otherwise. For example, when r"5, its associated transformation matrix ¹ would have the form of ¹" Technically, for every SOS in a non-convex problem P, a corresponding transformation matrix ¹ can be de"ned to reformulate P into P through y"¹x.

4 1444 E.Y.-H. Lin, D.L. Bricker / Computers & Operations Research 29 (2002) 1441}1446 In order to compare the e!ect of WMM and RTT using the SOS solver, Lin and Bricker conducted a computational experiment on IBM ES9000/320 in IBM's APL2language [13]. A total of 24 test problems (with the problem size ranging from 50 to 240 variables accompanied by 5}30 multiple choice constraints) are generated. These test problems are generated through a de"ned `coe$cient of tightnessa so that problems from loosely to tightly structured are covered. In view of the lack of an external reference row, variable subscript is used as the weight for that variable when P is solved through WMM (i.e., to avoid the sorting of variables for every SOS). By adopting the worst alternative heuristic [10] and calculating the improved penalty [14], Lin and Bricker solved these test problems using WMM and RTT, respectively, and recorded the CPU time for the whole operation (instead of just for the branch-and-bound phase as in Ogryczak's experiment). Based on the result of paired t-tests, Lin and Bricker found that the CPU time consumed by the whole operation using RTT seems to vary depending on the density of the overall transformation matrix (i.e., the one that integrates the matrix of ¹ for every SOS). In general, RTT outperforms WMM for the case where there is a high number of SOS with each set having relatively low number of variables. 4. Connecting SOI and RTT The SOI technique, in e!ect, rede"nes ¹ in RTT to be a block diagonal matrix with its entries t "1, if i*j, and t "0, if otherwise. For example, when r"5, the transformation matrix ¹ associated with SOI would have the form of ¹" In this case, partitioning the multiple-choice constraint (1) at any point can be carried out by dichotomizing its associated y variable. For instance, in SOI, partitioning (1) into x #x #2#x "0 and x #x #2#x "0, where r'k'1, is equivalent to setting y "0 and y "1, respectively. In the case of RTT, this is equivalent to setting y "1 and y "0, respectively. Branching strategies for binary variables in conventional branch-andbound procedure can then be directly applied on SOS. Thus, the conventional MIP solver can be applied on this set of y variables to preserve the advantage of SOS partitioning. That is, branching y variables is expected to result in a smaller and more balanced branching tree and thus accelerate the convergence of the branch-and-bound procedure. Ogryczak's computational experiment on SOI basically proves this point. However, one should note that reformulation of (1) and (2) from P to (3) in P on the machine does consume a certain amount of CPU time, particularly for larger-size problems with many SOS. Also the reformulated problem P will have relatively a larger size of Simplex tableau to maintain due to the added VUB constraints (i.e., a set of constraints (3) for every SOS in P ). For the latter case, Lin and Bricker developed a procedure to implicitly compose the optimal Simplex tableau of

5 E.Y.-H. Lin, D.L. Bricker / Computers & Operations Research 29 (2002) 1441} the LP relaxed P directly from the optimal Simplex tableau of the LP relaxed P [3]. This procedure allows one to calculate the true penalty associated with the SOS variables in a partitioned subset in P through a corresponding y variable in P. RTT also eliminates the requirement of the reference row as suggested by Tomlin [10]. Such a reference row is normally not readily available in every non-convex problem. In conclusion, SOI o!ers an explicit reformulation of SOS and illustrates that, as far as the branch-and-bound phase is concerned, the problem can be solved more e$ciently using simple MIP software without the SOS solver. On the other hand, RTT, in addition to o!ering the analogous reformulation, further proposes an implicit procedure to solve P. Therefore, it can serve as an alternative to WMM in a MIP software with the SOS solver. References [1] Ogryczak W. A note on modeling multiple choice requirements for simple mixed integer programming solvers. Computers and Operations Research 1996;23:199}205. [2] Bricker DL. Reformulation of special ordered sets for implicit enumeration algorithms with applications in non-convex separable programming. AIIE Transactions 1977;9:105}203. [3] Lin EYH, Bricker DL. On the calculation of true and pseudo penalties in multiple choice integer programming. European Journal of Operational Research 1991;55(2):228}36. [4] Lin EYH, Bricker DL. Computational comparison on the partitioning strategies in multiple choice integer programming. European Journal of Operational Research 1996;88:182}202. [5] Lin EYH. Multiple choice programming: A state-of-the-art review. International Transactions in Operational Research 1994;1(4):409}21. [6] Sullivan J. The application of mathematical programming methods to oil and gas "eld development planning. Mathematical Programming 1988;42:189}200. [7] Creegan JB, Monforte FA. Southern California Gas Company uses special ordered sets to model regulatory guidelines. Interfaces 1990;20:28}42. [8] Tomlin JA. Special ordered sets and an application to gas supply operations planning. Mathematical Programming 1988;42:69}84. [9] Beale EML, Tomlin JA. Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. In: Lawrence J, editor. Proceedings of the Fifth International Conference on Operational Research. London: Tavistock Publications, p. 447}54. [10] Tomlin JA. Branch-and-bound methods for integer and non-convex programming. In: Abadie J, editor. Integer and Nonlinear Programming. Amsterdam: North-Holland, p. 437}50. [11] Berkemer R, Makowski M, Watkins D. A prototype of a decision support system for water quality management in central and eastern Europe. WP IIASA, Luxemburg. [12] Ogryczak W, Zorychata K. Modular optimizer for mixed integer programming - MOMIP Version 1.1. WP IIASA, Laxenburg, [13] IBM APL2 Development Department. APL2 Version 2, Programming Language Reference (SH ). IBM Corporation, San Jose, CA, [14] Armstrong RD, Sinha P. Improved penalty calculation for a mixed integer branch-and-bound algorithm. Mathematical Programming 1974;6:212}23. Edward Yu-Hsien Lin is a Professor and the Dean of the College of Management at the National Taipei University of Technology in Taiwan who used to teach at the University of New Brunswick in Canada for 16 years. He received his B.S. in Industrial Engineering from Tunghai University, Taiwan; M.S. and Ph.D. in Industrial Engineering from The University of Iowa. Dr. Lin's research interests lie mainly in integer programming and combinatorial optimization. His research papers have appeared in a variety of international journals.

6 1446 E.Y.-H. Lin, D.L. Bricker / Computers & Operations Research 29 (2002) 1441}1446 Dennis Bricker is an Associate Professor of Industrial Engineering at The University of Iowa. He received B.S. and M.S. degrees in Mathematics from the University of Illinois at Urbana-Champaign, after which he received M.S. and Ph.D. degrees in Industrial Engineering and Management Science from Northwestern University. Dr. Bricker's research, which has been published in a wide variety of technical journals, is primarily in optimization.