Operations Management Linear Programming Module B Power Point

Outline ¨ Requirements of a Linear Programming Problem ¨ Formulating Linear Programming Problems ¨ Shader Electronics example ¨ Graphical Solution to a Linear Programming Problem ¨ Graphical representation of Constraints ¨ Iso-Profit Line Solution Method ¨ Corner-Point Solution Method Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 2 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Outline - continued ¨ Sensitivity Analysis ¨ Solving Minimization Problems ¨ Linear Programming Applications ¨ Production Mix Example ¨ Diet Problem Example ¨ Production Scheduling Example ¨ Labor Scheduling Example ¨ The Simplex Method of LP Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 3 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Learning Objectives When you complete this chapter, you should be able to : ¨ Identify or Define: ¨ ¨ ¨ Objective function Constraints Feasible region Iso-profit/iso-cost methods Corner-point solution Shadow price Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 4 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Learning Objectives continued When you complete this chapter, you should be able to : ¨ Describe or Explain: How to formulate linear models ¨ Graphical method of linear programming ¨ How to interpret sensitivity analysis ¨ Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 5 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

What is Linear Programming? ¨ Mathematical technique ¨ Not computer programming ¨ Allocates scarce resources to achieve an objective ¨ Pioneered by George Dantzig in World War II ¨ Developed workable solution in 1947 ¨ Called Simplex Method Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 6 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Examples of Successful LP Applications ¨ Scheduling school busses to minimize total distance traveled when carrying students ¨ Allocating police patrol units to high crime areas in order to minimize response time to 911 calls ¨ Scheduling tellers at banks to that needs are met during each hour of the day while minimizing the total 7 cost of labor Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Examples of Successful LP Applications - continued ¨ Picking blends of raw materials in feed mills to produce finished feed combinations at minimum costs ¨ Selecting the product mix in a factory to make best use of machineand labor-hours available while maximizing the firm’s profit ¨ Allocating space for a tenant mix in a new shopping mall so as to maximize revenues to the leasing company 8 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Requirements of a Linear Programming Problem 1 Must seek to maximize or minimize some quantity (the objective function) 2 Presence of restrictions or constraints - limits ability to achieve objective 3 Must be alternative courses of action from which to choose 4 Objectives and constraints must be expressible as linear equations or 9 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Formulating Linear Programming Problems ¨ Assume: ¨ You wish to produce two products (1) Walkman AM/FM/Cassette and (2) Watch. TV ¨ Walkman takes 4 hours of electronic work and 2 hours assembly ¨ Watch-TV takes 3 hours electronic work and 1 hour assembly ¨ There are 240 hours of electronic work time and 100 hours of assembly time available 10 ¨ Profit on a Walkman is $7; profit on a Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Formulating Linear Programming Problems continued ¨ Let: ¨ X 1 = number of Walkmans ¨ X 2 = number of Watch-TVs ¨ Then: + 3 X 2 240 electronics constraint ¨ 2 X 1 + 1 X 2 100 assembly constraint ¨ 7 X 1 + 5 X 2 = profit maximize profit ¨ 4 X 1 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 11 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Graphical Solution Method ¨ Draw graph with vertical & horizontal axes (1 st quadrant only) ¨ Plot constraints as lines, then as planes ¨ Use (X 1, 0), (0, X 2) for line ¨ Find feasible region ¨ Find optimal solution Corner point method ¨ Iso-profit line method ¨ Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 12 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Shader Electronic Company Problem Hours Required to Produce 1 Unit Department X 1 X 2 Available Hours Walkmans Watch-TV’s This Week Electronic 4 3 240 Assembly 2 1 Profit/unit $7 $5 100 Constraints: 4 x 1 + 3 x 2 240 (Hours of Electronic Time) 2 x 1 + 1 x 2 100 (Hours of Assembly Time) Objective: Maximize: 7 x 1 + 5 x 2 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 13 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Constraints Electronics 120 (Constraint A) 100 Assembly (Constraint B) 80 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 14 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Feasible Region Electronics (Constraint A) 120 100 Assembly (Constraint B) 80 60 40 Feasible Region 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 15 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Iso-Profit Lines Electronics (Constraint A) Assembly (Constraint B) 120 100 80 7*X 1 + 5*X 2 = 420 60 7*X 40 1 +5 *X 2 20 =2 10 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 16 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Solution Electronics (Constraint A) 120 ISO-Profit Line 100 80 Assembly (Constraint B) Solution Point (X 1=30, X 2=40) 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 17 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Number of Watch-TVs (X 2) Shader Electronic Company Solution Corner Point Solution Electronics (Constraint A) Assembly (Constraint B) 120 100 Possible Corner Point Solution Optimal solution 80 60 40 20 0 0 10 20 30 40 50 60 70 80 Number of Walkmans (X 1) Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 18 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Formulation of Solution ¨ Decision variables X 1 = tons of BW chemical produced ¨ X 2 = tons of color chemical produced ¨ ¨ Objective ¨ Minimize Z = 2500 X 1 + 3000 X 2 ¨ Constraints X 1 30 (BW); X 2 20 (Color) ¨ X 1 + X 2 60 (Total tonnage) ¨ X 1 0; X 2 0 (Non-negativity) ¨ Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 19 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Simplex Steps for Maximization 1 Choose the variable with the greatest positive Cj- Zj to enter the solution 2 Determine the row to be replaced by selecting that one with the smallest (non-negative) quantity-to-pivot column ratio 3 Calculate the new values for the pivot row 4 Calculate the new values for the other row(s) 5 Calculate the Cj and Cj-Zj values for this 20 tableau. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Sensitivity Analysis ¨ Projects how much a solution might change if there were changes in variables or input data. ¨ Shadow price (dual) - value of one additional unit of a resource Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 21 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Minimization Example BW: $2, 500 You’re an analyst for a manufacturing division of Kodak, which makes BW & color chemicals. cost per month At least 30 tons of BW and at least 20 tons of color must be made each month. The total chemicals made must be at least 60 tons. How many tons of each chemical should Color: $ 3, 000 be made to minimize costs? manufacturing cost per © 1995 Corel Corp. month Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 22 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Graphical Solution 80 Find values for X 1 + X 2 � 60. BW 60 X 1 30, X 2 20. Total Feasible Region Tons, 40 Color Chemical 20 (X 2) Color 0 0 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 20 40 60 80 Tons, BW Chemical (X 1) 23 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Optimal Solution: Corner Point Method 80 Find corner points. BW 60 Total Tons, 40 Color Chemical 20 Feasible Region B Color A 0 0 Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 20 40 60 Tons, BW Chemical 24 80 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458

Simplex Steps for Minimization 1 Choose the variable with the greatest negative Cj- Zj to enter the solution 2 Determine the row to be replaced by selecting that one with the smallest (nonnegative) quantity-to-pivot column ratio 3 Calculate the new values for the pivot row 4 Calculate the new values for the other row(s) 5 Calculate the Cj and Cj-Zj values for this tableau. If there any Cj-Zj numbers less than zero, return to step 1. Power. Point presentation to accompany Operations Management, 6 E (Heizer & Render) 25 © 2001 by Prentice Hall, Inc. , Upper Saddle River, N. J. 07458