1. © 2008 Prentice-Hall, Inc. Chapter 7 To accompany Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna Power Point slides created by Jeff Heyl Linear Programming Models: Graphical and Computer Methods © 2009 Prentice-Hall, Inc.

2. © 2009 Prentice-Hall, Inc. 7 – 2 Introduction Many management decisions involve trying to make the most effective use of limited resources Machinery, labor, money, time, warehouse space, raw materials Linear programmingLP Linear programming (LP) is a widely used mathematical modeling technique designed to help managers in planning and decision making relative to resource allocation mathematical programming Belongs to the broader field of mathematical programming programming In this sense, programming refers to modeling and solving a problem mathematically

3. © 2009 Prentice-Hall, Inc. 7 – 3 Requirements of a Linear Programming Problem LP has been applied in many areas over the past 50 years All LP problems have 4 properties in common maximizeminimize objective function 1.All problems seek to maximize or minimize some quantity (the objective function) constraints 2.The presence of restrictions or constraints that limit the degree to which we can pursue our objective 3.There must be alternative courses of action to choose from linearinequalities 4.The objective and constraints in problems must be expressed in terms of linear equations or inequalities

4. © 2009 Prentice-Hall, Inc. 7 – 4 LP Properties and Assumptions PROPERTIES OF LINEAR PROGRAMS 1. One objective function 2. One or more constraints 3. Alternative courses of action 4. Objective function and constraints are linear ASSUMPTIONS OF LP 1. Certainty 2. Proportionality 3. Additivity 4. Divisibility 5. Nonnegative variables Table 7.1

5. © 2009 Prentice-Hall, Inc. 7 – 5 Basic Assumptions of LP certainty We assume conditions of certainty exist and numbers in the objective and constraints are known with certainty and do not change during the period being studied proportionality We assume proportionality exists in the objective and constraints additivity We assume additivity in that the total of all activities equals the sum of the individual activities divisibility We assume divisibility in that solutions need not be whole numbers nonnegative All answers or variables are nonnegative

6. © 2009 Prentice-Hall, Inc. 7 – 6 Formulating LP Problems Formulating a linear program involves developing a mathematical model to represent the managerial problem The steps in formulating a linear program are 1.Completely understand the managerial problem being faced 2.Identify the objective and constraints 3.Define the decision variables 4.Use the decision variables to write mathematical expressions for the objective function and the constraints

7. © 2009 Prentice-Hall, Inc. 7 – 7 Formulating LP Problems product mix problem One of the most common LP applications is the product mix problem Two or more products are produced using limited resources such as personnel, machines, and raw materials The profit that the firm seeks to maximize is based on the profit contribution per unit of each product The company would like to determine how many units of each product it should produce so as to maximize overall profit given its limited resources

8. © 2009 Prentice-Hall, Inc. 7 – 8 Flair Furniture Company The Flair Furniture Company produces inexpensive tables and chairs Processes are similar in that both require a certain amount of hours of carpentry work and in the painting and varnishing department Each table takes 4 hours of carpentry and 2 hours of painting and varnishing Each chair requires 3 of carpentry and 1 hour of painting and varnishing There are 240 hours of carpentry time available and 100 hours of painting and varnishing Each table yields a profit of $70 and each chair a profit of $50

9. © 2009 Prentice-Hall, Inc. 7 – 9 Flair Furniture Company The company wants to determine the best combination of tables and chairs to produce to reach the maximum profit HOURS REQUIRED TO PRODUCE 1 UNIT DEPARTMENT ( T ) TABLES ( C ) CHAIRS AVAILABLE HOURS THIS WEEK Carpentry43240 Painting and varnishing21100 Profit per unit$70$50 Table 7.2

10. © 2009 Prentice-Hall, Inc. 7 – 10 Flair Furniture Company The objective is to Maximize profit The constraints are 1.The hours of carpentry time used cannot exceed 240 hours per week 2.The hours of painting and varnishing time used cannot exceed 100 hours per week The decision variables representing the actual decisions we will make are T = number of tables to be produced per week C = number of chairs to be produced per week

11. © 2009 Prentice-Hall, Inc. 7 – 11 Flair Furniture Company We create the LP objective function in terms of T and C Maximize profit = $70 T + $50 C Develop mathematical relationships for the two constraints For carpentry, total time used is (4 hours per table)(Number of tables produced) + (3 hours per chair)(Number of chairs produced) We know that Carpentry time used ≤ Carpentry time available 4 T + 3 C ≤ 240 (hours of carpentry time )

12. © 2009 Prentice-Hall, Inc. 7 – 12 Flair Furniture Company Similarly Painting and varnishing time used ≤ Painting and varnishing time available 2 T + 1 C ≤ 100 (hours of painting and varnishing time) This means that each table produced requires two hours of painting and varnishing time Both of these constraints restrict production capacity and affect total profit

13. © 2009 Prentice-Hall, Inc. 7 – 13 Flair Furniture Company The values for T and C must be nonnegative T ≥ 0 (number of tables produced is greater than or equal to 0) C ≥ 0 (number of chairs produced is greater than or equal to 0) The complete problem stated mathematically Maximize profit = $70 T + $50 C subject to 4 T + 3 C ≤240 (carpentry constraint) 2 T + 1 C ≤100 (painting and varnishing constraint) T, C ≥0 (nonnegativity constraint)

14. © 2009 Prentice-Hall, Inc. 7 – 14 Cycle Trends is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from aluminum and steel alloys. The anticipated unit profits are $10 for the Deluxe and $15 for the Professional. The number of pounds of each alloy needed per frame is summarized on the table. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly. How many Deluxe and Professional frames should Cycle Trends produce each week? Pounds of each alloy needed per frame 1- Example: LP Formulation Aluminum Alloy Steel Alloy Deluxe 2 3 Professional 4 2

15. © 2009 Prentice-Hall, Inc. 7 – 15 Montana Wood Products manufacturers two- high quality products, tables and chairs. Its profit is $15 per chair and $21 per table. Weekly production is constrained by available labor and wood. Each chair requires 4 labor hours and 8 board feet of wood while each table requires 3 labor hours and 12 board feet of wood. Available wood is 2400 board feet and available labor is 920 hours. Management also requires at least 40 tables and at least 4 chairs be produced for every table produced. To maximize profits, how many chairs and tables should be produced? 2- Example: LP Formulation

16. © 2009 Prentice-Hall, Inc. 7 – 16 The Sureset Concrete Company produces concrete. Two ingredients in concrete are sand (costs $6 per ton) and gravel (costs $8 per ton). Sand and gravel together must make up exactly 75% of the weight of the concrete. Also, no more than 40% of the concrete can be sand and at least 30% of the concrete be gravel. Each day 2000 tons of concrete are produced. To minimize costs, how many tons of gravel and sand should be purchased each day? 3- Example: LP Formulation

17. © 2009 Prentice-Hall, Inc. 7 – 17 A company produces two products that are processed on two assembly lines. Assembly line 1 has 100 available hours, and assembly line 2 has 42 available hours. Each product requires 10 hours of processing time on line 1, while on line 2 product 1 requires 7 hours and product 2 requires 3 hours. The profit for product 1 is $6 per unit, and the profit for product 2 is $4 per unit. Formulate a linear programming model for this problem. 4- Example: LP Formulation

18. © 2009 Prentice-Hall, Inc. 7 – 18 A California grower has a 50-acre farm on which to plant strawberries and tomatoes. The grower has available 300 hours of labor per week and 800 tons of fertilizer, and he has contracted for shipping space for a maximum of 26 acres' worth of strawberries and 37 acres' worth of tomatoes. An acre of strawberries requires 10 hours of labor and 8 tons of fertilizer, whereas an acre of tomatoes requires 3 hours of labor and 20 tons of fertilizer. The profit from an acre of strawberries is $400, and the profit from an acre of tomatoes is $300. The farmer wants to know the number of acres of strawberries and tomatoes to plant to maximize profit. Formulate a linear programming model for this problem. 5- Example: LP Formulation