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Unformatted text preview: 18 Chapter 2 Getting Started with Spreadsheets Before describing a linear program, we introduce two preliminary definitions. A linear expression appears below; its variables are A, B, and C, and the dependence on these vari ables is linear. A 3.4 B + 2 C Alinear constrain t requires a linear expression to bear one of three relations to a number that are indicated below: 2 A 3 B = 6 and A 3 .4 B + 2 C 2 and C O . Thus, in a linear constraint, the linear expression can be equal to the number, less than or equal to the number, or greater than or equal to the number. A Linear program maximi zes or minimizes a linear expression, subject to finitely many linear constraints. What distinguishes a linear program from classical mathematics is the inequal ities. Commodities exist in nonnegative quantities. Yet the requirement that variables be nonnegative played a minor rOle in mathematics before the advent of linear program ming. Are Linear Programs Important? Yes! A staggering dive rsity of problems can be posed as linear pro grams. Linear programs are now used routinely in industry and government for long-range planning and for the control of day-to-day operations. Linear programming influences the management of forests, the operation of oil refineries, the scheduling of airlines, the planning for future energy needs, and many other activities. Several chapters of this text are devoted to linear programs and their generalizations. The term linear programming encompasses the art of formulating problems as linear programs and the science of solving them. Linear programming may be the most significant development in applicable mathematics in the twentieth century. 2.2. WHAT CAN YOU LEARN FROM TifiS CHAPTER? This chapter prepares you for the rest of the text by introducing the following three subjects: Linear progr ams. The fundamenta ls of Excel. A spreadshe et add-in called Solver, whose uses include the computation of solutions to linear progr ams. These subjects complement each other. The spreadsheet operations that help us solve lin ear programs have many other uses, even in chapters that have nothing to do with linear programs. Solver will be particularly handy. Why? Spreadsheet functions cannot solve equa tions; Solver can. 2.3. A RESOU RCE ALLOCATION PROBLEM The efficient allocation of resources is important in many fields, including engineering, management, and economics. Engineers face the problem of designing and operating sys tems in ways that maximize quality and minimize cost. Managers have to grapple with the problem of organizing activities in a way that m aximizes profit. For economists the effi cient allocation of scarce resources is the central issue. 2.3. A Resource Allocation Problem 19 Resource allocation issues take many forms. Listed here are three types of resource al location problems that may concern a manufacturing facility: Produce a required set of outputs as inexpensively as possible....
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- Fall '09