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Unformatted text preview: Optimization Models Draft of August 26, 2005 II. Elementary Linear Optimization Models Robert Fourer Department of Industrial Engineering and Management Sciences Northwestern University Evanston, Illinois 602083119, U.S.A. (847) 4913151 [email protected] http://www.iems.northwestern.edu/ ˜ 4er/ Copyright c 1989–2005 Robert Fourer A–36 Optimization Models — § 3.0 Draft of August 26, 2005 A–37 3. MinimumCost Input Models The diet formulation in our introductory chapters is only one representative of a broad class of input optimization models. We now give a more general definition of the minimumcost input selection problem and the associated op timization model. We observe that this model corresponds closely to the diet model introduced in the preceding chapters, but that other interpretations of the inputs and outputs give rise to useful blending, scheduling, cutting, and packing models. 3.1 The input optimization problem Imagine a process of some sort that combines a variety of inputs. From each unit of a particular input, the process derives fixed amounts of each of several outputs, with the amount of each output produced being proportional to the amount of each input consumed. Each input also has a cost per unit, again proportional to the amount. Our goal is to choose an amount of each input to be purchased, within specified limits, so that the total outputs thereby produced meet specified requirements at the least total cost. We build a mathematical model of this problem (shown in Figure 3–1) much as we did for the diet problem in Chapter 1. We start by defining the funda mental sets of entities: inputs I and outputs O . We then say that each unit of an input i ∈ I can be obtained at a cost c i , and contributes amounts a oi of the outputs o ∈ O . The number of units of input i ∈ I consumed must lie between l i and u i , while the number of units of output o ∈ O produced must be at least b o and at most d o . The decision variables are the amounts of the inputs to be purchased, so we may write them as x i for each i ∈ I . Then since the cost of input i purchased is proportional to the amount of input i purchased, the cost is given by c i x i , and the objective is to minimize the total of these costs, ∑ i ∈I c i x i . Similarly since the amount of output o derived from input i is proportional to the amount of o purchased, the amount of o derived from input i is given by a oi x i . The total of output o derived from all inputs is thus ∑ i ∈I a oi x i , and the constraints say that this amount lies between the requirements, b o ≤ ∑ i ∈I a oi x i ≤ d o , for each output o ∈ O . An analogous description in the AMPL modeling language is given by Figure 3–2. The transcription from mathematics to AMPL here proceeds in the same general way as it did for the diet model in Chapter 1 (from Figure 1–3 to Figure 1–4). The set J is denoted INPUTS , the variables x i become InBuy[i] , and the limits l...
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 Optimization, Shift work, Optimization Models

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