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Linear Programming

Linear Programming. Introduction George B Dantzig developed LP in 1947. It is a problem solving approach designed to help managers/decision makers in planning relative trade-off in resource allocations.

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Linear Programming

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  1. Linear Programming • Introduction George B Dantzig developed LP in 1947. It is a problem solving approach designed to help managers/decision makers in planning relative trade-off in resource allocations. The word “programming” in LP should not be confused with computer programming it simply means ‘choosing a course of action’.

  2. Common Terms in LP • Optimization • Objective Function • Constraints – Inequality/Equality • RHS values • Non-negativity • Binding Constraints • Redundant Constraints • Sensitivity Analysis

  3. Solution Techniques • Algebraic • Graphic • Simplex Algorithm • Spreadsheet approach [Solver]

  4. Four Common Elements in LP 1. We always Maximize or minimize a problem. 2. All resources have some constraints/limits. 3. There are always choices that can be made. 4. All relationships are assumed to be linear.

  5. Steps in Problem Formulation • Understand the Problem • Identify the Variables • Define the objective Function • Formulate/write out the Constraints • Solve and check.

  6. Formulating LP Problems The product-mix problem at Shader Electronics • Two products • Shader X-pod, a portable music player • ShaderBlueBerry, an internet-connected color telephone • Determine the mix of products that will produce the maximum profit

  7. Hours Required to Produce 1 Unit X-pods BlueBerrys Available Hours Department (X1)(X2) This Week Electronic 4 3 240 Assembly 2 1 100 Profit per unit $7 $5 Formulating LP Problems Decision Variables: X1 = number of X-pods to be produced X2 = number of BlueBerrys to be produced

  8. 4 + 3 £ 240 X1 (Electronic ) 2 + 1 £ 100 (Assembly ) , + 5 <= 0 X1 X2 X2 7 Formulating LP Problems Objective: Maximize: X1 ST: Constraints: X2 X2 X1

  9. 120 100 80 60 40 20 0 Assembly Number of BlueBerrys Electronic 20 40 60 80 100 Number of X-Pods Formulating LP Problems

  10. LP Applications Scheduling school buses to minimize total distance traveled Allocating police patrol units to high crime areas in order to minimize response time to 911 calls Scheduling tellers at banks so that needs are met during each hour of the day while minimizing the total cost of labor

  11. LP Applications Selecting the product mix in a factory to make best use of machine- and labor-hours available while maximizing the firm’s profit Picking blends of raw materials in feed mills to produce finished feed combinations at minimum costs Determining the distribution system that will minimize total shipping cost

  12. LP Applications Developing a production schedule that will satisfy future demands for a firm’s product and at the same time minimize total production and inventory costs Allocating space for a tenant mix in a new shopping mall so as to maximize revenues to the leasing company

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