Improved Computer - Based Planning Techniques

This is Part II of a two-part series. Pan I showed how pure and generalized network models, and advances in methods of solving them. have resulted in dramatic cost savings for OR/MS practitioners. This paper focuses on network related formulation (NETFORM) models, which encompass an even wider variety of applications. We show how NETFORM has enabled the efficient solution of problems in scheduling, production, distribution, and other areas Ihal were too large or difficult to be handled by previously applied techniques, including mixed integer programming. Introduction All too frequently Management Science practitioners discover that their major problems do not fit into straightforward linear programming frameworks, but instead involve nonlinearities expressed through discrete (integer programming) relationships. When this occurs, the inappropriateness of LP models typically causes a good deal of consternation. The history of discrete optimization abounds with situations where problems that would have been easy to solve as linear programs turned into computational monstrosities with the addition of discrete conditions. •Panl appe-Mcd \nlnierfaces. Vol. 8, No, 4. pp. 16—25. This two-pan paper is a condensed version of an original paper delivered at the SHARE XLVIII Conference in Houston. March 6-11. 1977. The full version of this paper may be obtained by writing Fred Glover or Darwin KUngman. tThis research was partly supported by Project NR047-172. ONR Contract NOOOI4.78-C-O222 and Projeci NR047-C2I. ONR Contracts NOO0i4-75-C-06l6 and N00014-75-C0569 with Ihe Center for Cybernetic Studies, The University of Texas at Auslin. NETWORKS/GRAPHS — PLANNING 12 INTERFACES August 1979 Recently, it has been discovered that a wide variety of discrete optimization problems have major network or network-related components. This invites the use of network models and solution techniques to replace the unsatisfactory approaches previously used. Moreover, advances in network-related formulation (or NETFORM) models have provided the ability to capture a still larger range of problems by means of representations that have an inherent network structure, making network solution technology applicable to these problems as well. The major purpose of this paper (Part II) is to present real-world applications that demonstrate the practical value of NETFORM models and the cost savings that result from their use. This new NETFORM modeling technology, as noted in Part I of this paper, has several attractive features which lead to: (1) improved communication between practitioners and modelers because ofthe pictorial aspect ofthe models, (2) improved insight into the problem, making it possible to "see" where critical relationships lie and to interpret solutions obtained, and (3) an ability to solve many discrete optimization problems far more efficiently than in the past, including problems once believed to be too large or too complex to be solved within reasonable time limits. The capacity of NETFORM models to render complex problems more solvable derives in large part from the significant advances in pure and generalized network solution methods, particularly in their efficient computer implementation. Part I of this paper discussed fundamental model concepts that led to real-world applications ofthe NETFORM modeling technique, and reported computational comparisons of network computer codes with a state-of-the-art commercial LP code. Some NETFORM building blocks The uses of arc multipliers described in Part I represent just a part of their full range of application. Introducing the requirement that flows on particular arcs must occur in integer (whole number) amounts makes it possible to model a much larger variety of applications, including problems such as assigning personnel to jobs where staffing requirements vary by time period, scheduling production on machines and assembly lines, scheduling payments on accounts subject to discrete payment levels, and determining optimal sizing and placement of electrical power stations. The power of NETFORM techniques, based upon the imposition of integer requirements in generalized networks, is illustrated by the fact that they enable the modeler to represent any 0-J LP problem as an integer generalized network (GN) problem [3], [4]. Other NETFORM model components include arcs with "all-ornone" flows and side constraints. We will illustrate both the integer GN and the "all-or-none" model techniques in this paper. These techniques can also accommodate mixed integer 0-1 LP problems where the continuous part of the problem is a transportation, transshipment, or generalized network problem itself. An illustration in [7] shows how contemporary financial capital allocation problems can be modeled as integer GN problems. Other important real-world applications with this type of "mixed" structure include a variety of plant location models, energy models, physical distribution models, and dynamic production/distribution scheduling models. The NETFORM representation is able to express rigorously all of the elements of these problems in a pictorial form. Consequently, it effectively replaces the INTERFACES August 1979 13 obscure and unilluminating algebraic representation by an equivalent, but much easier to understand, pictorial representation. But the advantages ofthe NETFORM approach do not end here. We have found that its underlying network-related structures can also be exploited by special solution methods that are substantially more efficient than the methods previously developed for the algebraic representations. Figure 1 illustrates a useful modeling device commonly employed in the NETFORM approach. The relevant arc data are depicted by the same conventions employed in Part I. In particular, lower and upper bounds on the flow across an arc appear in parentheses, costs appear in squares, and multipliers appear in half ellipses. (Occasionally, we omit one or another of these notational components of an arc when it is not important to the discussion.) In addition, we introduce the convention that an asterisk above an arc indicates that its flow must be integer valued. FIGURE 1. Ship Scheduling. Ship Node Ship/Schedule Nodes Port Nodet