Handout12 - ENGRI 1101 Engineering Applications of OR Fall...

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Unformatted text preview: ENGRI 1101 Engineering Applications of OR Fall ’08 Handout 12 Lectures on Linear Programming Problems 1 Motivating Example Manufacturing planning problem A fine tableware manufacturing company produces a line of stylish wine glasses, which come in two sizes: 6 oz. and 10 oz. The manager of the facility that produces these glasses is trying to decide how many glasses of each type to produce for sale during the upcoming holiday season. Each case of 6 oz. glasses manufactured and sold at full price brings in a net profit of $5.00, while each case of 10 oz. glasses manufactured and sold at full price brings in a net profit of $4.50. In deciding on the production plan, the manager is facing the following considerations: • Based on prior years, an estimate of the demand for the glasses at their full (non-discounted) retail price has been obtained: 800 cases of 6 oz. glasses and 1,000 cases of 10 oz. glasses. To maintain the exclusive image of the brand, the company does not want to produce more than the estimated demand figures suggest they can sell at full price. • The glasses are produced on the same machines, but a case of 6 oz. glasses requires more time to manufacture (more detailed work is required). In particular, each case of 6 oz. glass requires 6 hours of machine time, while each case of 10 oz. glasses requires 4 hours of time. During the production cycle, a limited amount — 6,000 hours — of machine time will be available. • To better balance the production process, the manager would like to restrict the proportion of larger glasses produced. In particular, 6 oz. glasses should constitute at least 40% of the total amount produced. The goal is to maximize the profit generated by the glasses, while complying with all the considerations described above. Assuming that the full-price demand estimates above are accurate, what is the best production plan the manager can propose? Output (decision variables): • x 1 — number of cases (in 100s) of 6 oz. glasses to manufacture • x 2 — number of cases (in 100s) of 10 oz. glasses to manufacture Constraints: • Do not exceed available machine hours: 6 x 1 + 4 x 2 ≤ 60 • At least 40% of cases are 6 oz. glasses:- 3 x 1 + 2 x 2 ≤ • Do not exceed demand for each type of glasses (and nonnegativity): 0 ≤ x 1 ≤ 8 and 0 ≤ x 2 ≤ 10 Objective function: • Maximize net profit: max5 x 1 + 4 . 5 x 2 (in $100’s) 1 Putting it all together, we get the following mathematical programming problem: max 5 x 1 + 4 . 5 x 2 s . t . 6 x 1 + 4 x 2 ≤ 60- 3 x 1 + 2 x 2 ≤ ≤ x 1 ≤ 8 ≤ x 2 ≤ 10 2 LP defined The above mathematical programming problem is, in fact, of a very special kind: it’s called a linear program- ming problem, because the objective function is a linear function of the variables, and all the constraints are expressed as linear inequalities on the variables.expressed as linear inequalities on the variables....
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This note was uploaded on 11/12/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell.

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Handout12 - ENGRI 1101 Engineering Applications of OR Fall...

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