Slides16_col - Lecture 16 Linear Programming Models Example...

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1 Lecture 16 Linear Programming Models Example I This is a production planning model. A furniture manufacturer produces four types of items: tables, chairs, desks, bookcases . Production of these items consumes the following resources: pine boards, oak boards, labor . Resources are consumed at the following rates: Each table yields $12 profit; each chair, $5; each desk, $15; each bookcase, $10. Assume that for the current planning horizon (say, one week) we have available 1500 ft. pine, 1000 ft. oak, and 20 employees who will each work 40 hours. Also assume demand for the coming week is at least 40 tables, 130 chairs, 30 desks and that all items produced will be sold. 5 ft. pine, 1 . .. 9 . .. 12 . .. 2 ft. oak, 3 . .. 4 . .. 1 . .. 3 hrs. labor 2 . .. 5 . .. 10 . .. per table; per chair; per desk; per bookcase.
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2 Example I Question: Which production schedule will maximize profit? Model: constraints: 5T + 1C + 9D + 12B 1500 2T + 3C + 4D + 1B 1000 3T + 2C + 5D + 10B 800. variables: T = #tables, C = #chairs, D = #desks, B = #bookcases. T 40, C 130, D 30, B 0. profit: 12T + 5C + 15D + 10B. (meet demand) (don’t exceed available resources) Example I We thus like to determine values for T,C,D,B in order to: s.t. 5T + 1C + 9D + 12B 1500 2T + 3C + 4D + 1B 1000 3T + 2C + 5D + 10B 800 T 40, C 130, D 30, B 0. max 12T + 5C + 15D + 10B • This is a so-called linear programming (LP) problem . Linear means that the changes in the production level produce proportional changes in resource consumption, and similar for profit; programming means “planning” over some fixed horizon, as opposed to “computer programming” (though large models require computers to solve these problems).
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3 Example I Note that we must optimize . I.e., we must choose the best production schedule among a set of feasible (or allowable) alternatives specified by problem constraints . In the example, the profit function 12T+5C+15D+10B, ranks the set of feasible alternatives; since the objective for this problem is to choose the best production schedule according to this ranking, this function is called the objective function . Many linear programming problems are of this form, dealing with production, allocation or distribution of goods or resources. History of Linear Programming “Linear programming can be viewed as part of a great revolutionary development which has given mankind the ability to state general goals and to lay out a path of detailed decisions to take in order to ‘best’ achieve its goals when faced with practical situations of great complexity.” George Dantzig 1914-2005 1. Recognizing (as a result of my wartime years as a practical program planner) that most practical planning relations could be reformulated as a system
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Slides16_col - Lecture 16 Linear Programming Models Example...

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