Using Binary Variables to Represent Logical Conditions in Optimization Models

Abstract

Logical conditions that link different elements of a business decision are very common in managerial practice. For example, a firm can ship only to and from warehouses that are open; patients needing MRIs can only get service in clinics that have MRI equipment; regarding an old power plant, one can decide to close it or retrofit it, but one obviously cannot retrofit a closed plant. The list goes on. In quantitative modeling of such situations, a natural step is to use IF statements such as IF(a warehouse in city N is open, then we can ship to/from it; otherwise no shipments can be made in/out of a warehouse in N). In optimization models, however, IF statements lead to non-linearity with all the associated challenges. Fortunately, nearly all logical conditions can be modeled linearly using binary variables. This note describes some helpful modeling techniques for doing that. Excerpt UVA-QA-0786 Apr. 11, 2012 Using binary variables to represent logical conditions in optimization MODELS Logical conditions that link different elements of a business decision are very common in managerial practice. For example, a firm can ship only to and from warehouses that are open; patients needing MRIs can only get service in clinics that have MRI equipment; regarding an old power plant, one can decide to close it or retrofit it, but one obviously cannot retrofit a closed plant. The list goes on. In quantitative modeling of such situations, a natural step is to use IF statements such as IF(a warehouse in city N is open, then we can ship to/from it; otherwise no shipments can be made in/out of a warehouse in N). In optimization models, however, IF statements lead to non-linearity with all the associated challenges. Fortunately, nearly all logical conditions can be modeled linearly using binary variables. This note describes some helpful modeling techniques for doing that. What Are Binary Variables? A binary variable is one that can take two values: 1 (which traditionally stood for “True”) or 0 (which stood for “False”). The actual meanings for 1 and 0 can be context-specific: “yes/no,” “open/closed,” “have MRI/does not have MRI,” “good/bad,” “small/large,” and so on without restriction. Binary representations are at the heart of computing and, as surprising as it may be, any digital object—number, text, this electronic file, digital image, sound recording, video, and everything else you see, hear, or do on your computer/smartphone/tablet—is wholly composed of zeros and ones. Logical conditions in optimization modeling are no different. . . .