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Chapter 02

# Chapter 02 - CHAPTER 2 LINEAR PROGRAMMING MODELS GRAPHICAL...

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CHAPTER 2 LINEAR PROGRAMMING MODELS: GRAPHICAL AND COMPUTER METHODS Note: Permission to use the computer program GLP for all LP graphical solution screenshots in this chapter granted by its author, Jeffrey H. Moore, Graduate School of Business, Stanford University. Software copyrighted by Board of Trustees of the Leland Stanford Junior University. All rights reserved. SOLUTIONS TO DISCUSSION QUESTIONS 2-1. The requirements for an LP problem are listed in Section 2.2. It is also assumed that conditions of certainty exist; that is, coefficients in the objective function and constraints are known with certainty and do not change during the period being studied. Another basic assumption that mathematically sophisticated students should be made aware of is proportionality in the objective function and constraints. For example, if one product uses 5 hours of a machine resource, then making 10 of that product uses 50 hours of machine time. LP also assumes additivity. This means that the total of all activities equals the sum of each individual activity. For example, if the objective function is to maximize Profit = 6X 1 + 4X 2 , and if X 1 = X 2 = 1, the profit contributions of 6 and 4 must add up to produce a sum of 10. 2-2. If we consider the feasible region of an LP problem to be continuous (i.e., we accept non-integer solutions as valid), there will be an infinite number of feasible combinations of decision variable values (unless of course, only a single solution satisfies all the constraints). In most cases, only one of these feasible solutions yields the optimal solution. 2-3. A problem can have alternative optimal solutions if the level profit or level cost line runs parallel to one of the problem’s binding constraints (refer to Section 2.6 in the chapter). 2-4. A problem can be unbounded if one or more constraints are missing, such that the objective value can be made infinitely larger or smaller without violating any constraints (refer to Section 2.6 in the chapter). 2-5. This question involves the student using a little originality to develop his or her own LP constraints that fit the three conditions of (1) unbounded solution, (2) infeasibility, and (3) redundant constraints. These conditions are discussed in Section 2.6, but each student’s graphical displays should be different. 2-6. The manager’s statement indeed has merit if he/she understood the deterministic nature of LP input data. LP assumes that data pertaining to demand, supply, materials, costs, and resources are known with certainty and are constant during the time period being analyzed. If the firm operates in a very unstable environment (for example, prices and availability of raw materials change daily, or even hourly), the LP model’s results may be too sensitive and volatile to be trusted. The application of sensitivity analysis might, however, be useful to determine whether LP would still be a good approximating tool in decision making in this environment.

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