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Krajewski IN Supplement E - Supplement E Linear Programming...

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Supplement E Linear Programming A. Basic Concepts 1. Linear programming is an optimization process with several characteristics a. A single objective function states mathematically what is being maximized or minimized. b. Decision variables represent choices that the decision maker can control. Solving the problem yields their optimal values based on the assumption that decision variables are continuous. c. Constraints are limitations that restrict the permissible choices for the decision variables, which can be expressed mathematically in one of three ways: , =, or constraints. d. The feasible region includes all of the combinations of the decision variables which satisfy the given constraints. Usually an infinite number of possible solutions. e. A parameter , also known as a coefficient or given constant, is a value that the decision maker cannot control and that does not change when the solution is implemented. f. Assume parameters are known with certainty, and without doubt. g. The objective function and constraints are assumed to be linear , which implies proportionality and additivity—there can be no products or powers of decision variables. h. We assume the model to exhibit nonnegativity, which means that the decision variables must be positive or zero. 2. Formulating a problem a. Step 1: Define the decision variables. What must be decided? Define each decision variable specifically, remembering that the definitions must be equally useful in the constraints. b. Step 2: Write out the objective function. What is to be maximized or minimized? Write out an objective function to make what is being optimized a linear function of the decision variables. c. Step 3: Write out the constraints. What limits the values of the decision variables? E-1
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E-2 Supplement E: Linear Programming Identify the constraints and the parameters for each decision variable in them. To be formally correct, also write out the nonnegativity constraints. d. As a consistency check, make sure the same unit of measure is being used on both sides of each constraint and the objective function. 3. Problem Formulation. Use Application E.1: Crandon Manufacturing . The Crandon Manufacturing Company produces two principal product lines. One is a portable circular saw, and the other is a precision table saw. Two basic operations are crucial to the output of these saws: fabrication and assembly. The maximum fabrication capacity is 4000 hours per month; each circular saw requires 2 hours, and each table saw requires 1 hour. The maximum assembly capacity is 5000 hours per month; each circular saw requires 1 hour, and each table saw requires 2 hours. The marketing department estimates that the maximum market demand next year is 3500 saws per month for both products.
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