Linear Programming Introduction George B Dantzig developed LP in 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’. 1

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

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

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. 4

Steps in Problem Formulation 1.Understand the Problem 2.Identify the Variables 3.Define the objective Function 4.Formulate/write out the Constraints 5.Solve and check. 5

Formulating LP Problems The product-mix problem at Shader Electronics  Two products 1.Shader X-pod, a portable music player 2.Shader BlueBerry, an internet- connected color telephone  Determine the mix of products that will produce the maximum profit

Formulating LP Problems X-podsBlueBerrysAvailable Hours Department(X 1 )(X 2 )This Week Hours Required to Produce 1 Unit Electronic43240 Assembly21100 Profit per unit$7$5 Decision Variables: X 1 = number of X-pods to be produced X 2 = number of BlueBerrys to be produced

Formulating LP Problems 8 Objective: Maximize: ST: Constraints: ) (Assembly    X2X2  )(Electronic  X1X1  X2X2 X1X1  X1X1 X1X1 X2X2 X2X2

Formulating LP Problems Number of BlueBerrys Number of X-Pods Assembly Electronic

LP Applications 1.Scheduling school buses to minimize total distance traveled 2.Allocating police patrol units to high crime areas in order to minimize response time to 911 calls 3.Scheduling tellers at banks so that needs are met during each hour of the day while minimizing the total cost of labor

LP Applications 4.Selecting the product mix in a factory to make best use of machine- and labor- hours available while maximizing the firm’s profit 5.Picking blends of raw materials in feed mills to produce finished feed combinations at minimum costs 6.Determining the distribution system that will minimize total shipping cost

LP Applications 7.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 8.Allocating space for a tenant mix in a new shopping mall so as to maximize revenues to the leasing company