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# chap1_2011 - Chapter 1 Problem Solving with Mathematical...

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Chapter 1 Problem Solving with Mathematical Models

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Mathematical Model A mathematical model is the collection of variables and relationships needed to describe pertinent features of such a problem. Operations Research (OR) [1.1] Is the study of how to form mathematical models of complex engineering and management problems and how to analyze them to gain insight about possible solutions.
EXAMPLE 1.1: Mortimer Middleman Mortimer Middleman--friends call him MM--operates a modest wholesale diamond business. Several times each year MM travels to Antwerp, Belgium, to replenish his diamond supply on the international market. The wholesale price there averages approximately \$700 per carat, but Antwerp market rules require him to buy at least 100 carats each trip. Mortimer and his staff then resell the diamonds to jewelers at a profit of \$200 per carat. Each of the Antwerp trips requires 1 week, including the time for Mortimer to get ready, and costs approximately \$2000.

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EXAMPLE 1.1: Mortimer Middleman Customer demand values in Figure 1.1(a) show that business has been good. Over the past year, customers have come in to order an average of 55 carats per week. 0 2 4 6 8 10 12 Demand
EXAMPLE 1.1: Mortimer Middleman Part (c) of Figure 1.1 illustrates Mortimer’s problem. Weekly levels of on-hand diamond inventory have varied widely, [Figure 1.1(c)] 0 2 4 6 8 10 12

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EXAMPLE 1.1: Mortimer Middleman depending on the ups and downs in sales and the pattern of MM’s replenishment trips [Figure 1.1(b)]. 0 2 4 6 8 10 12 Repl.
EXAMPLE 1.1: Mortimer Middleman Sometimes Mortimer believes that he is holding too much inventory. The hundreds of carats of diamonds on hand during some weeks add to his insurance costs and tie up capital that he could otherwise invest. MM has estimated that these holding costs total 0.5 % of wholesale value per week (i.e., 0.005 x \$700 = \$3.50 per carat per week). At other times, diamond sales—and Mortimer’s \$200 per carat profit—have been lost because customer demand exceeded available stock [see Figure 1.1(d)]. When a customer calls, MM must either fill the order on the spot or lose the sale.

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EXAMPLE 1.1: Mortimer Middleman 0 2 4 6 8 10 Lost_sales [Figure 1.1(d)].
EXAMPLE 1.1: Mortimer Middleman Adding this all up for the past year, MM estimates holding costs of \$38,409, unrealized profits from lost sales of \$31,600, and resupply travel costs of \$24,000, making the annual total \$94,009. Can he do better?

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1.2 Optimization and the Operations Research Process Modeling Analysis Inference Assessment
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## This note was uploaded on 12/03/2011 for the course ESI 6316 taught by Professor Staff during the Summer '11 term at FIU.

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chap1_2011 - Chapter 1 Problem Solving with Mathematical...

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