Linear Programming

1. Chapter 13 Supplement Linear Programming Operations Management - 5th Edition Roberta Russell & Bernard W. Taylor, III

2. Lecture Outline • What is LP? • Where is LP used? • LP Assumptions • Model Formulation • Examples • Solving

3. Linear Programming (LP) A model consisting of linear relationships representing a firm’s objective and resource constraints LP is a mathematical modeling technique used to determine a level of operational activity in order to achieve an objective, subject to restrictions called constraints

4. Example 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

5. Example 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

6. Example 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

7. Common Elements to LP • Decision variables • Should completely describe the decisions to be made by the decision maker (DM) • Objective Function (OF) • DM wants to maximize or minimize some function of the decision variables • Constraints • Restrictions on resources such as time, money, labor, etc.

8. LP Assumptions • OF and constraints must be linear • Proportionality • Contribution of each decision variable is proportional to the value of the decision variable • Additivity • Contribution of any variable is independent of values of other decision variables

9. LP Assumptions, cont’d. • Divisibility • Allow both integer and non-integer (real numbers) • Certainty • All coefficients are known with certainty • We are dealing with a deterministic world

10. LP Model Formulation • Data • Input to the model – given in the problem • Decision variables • Mathematical symbols representing levels of activity of an operation • The quantities to be determined

11. LP Model Formulation, cont’d. • Objective function (OF) • The quantity to be optimized • A linear relationship reflecting the objective of an operation • Most frequent objective of business firms is to maximize profit • Most frequent objective of individual operational units (such as a production or packaging department) is to minimize cost

12. LP Model Formulation, cont’d. • Constraint • A linear relationship representing a restriction on decision making • Binding relationships • Attach a word description to each set of constraints • Include bounds on variables

13. RESOURCE REQUIREMENTS Labor Clay Revenue PRODUCT (hr/unit) (lb/unit) ($/unit) Bowl 1 4 40 Mug 2 3 50 There are 40 hours of labor and 120 pounds of clay available each day Formulate this problem as a LP model and solve LP Formulation: Example

14. LP Formulation: Example • Variables • b = number of bowls to produce • m = number of mugs to produce • z = total revenue • Objective Function • Max z = 40b + 50m • Constraints • b + 2m< 40 (labor constraint) • 4b + 3m< 120 (clay constraint) • b, m> 0 (non-negativity)

15. LP Formulation: Solution Solution is: b = 24 bowls m= 8 mugs z = $1,360

16. Bowls and Mugs Solved • Use OMTools > Linear Programming

17. Another Example • Joe’s Woodcarving, Inc. manufactures two types of wooden toys: soldiers and trains. • Unlimited supply of raw material, but only 100 finishing hours and 80 carpentry hours • Demand for trains unlimited, but at most 40 soldiers can be sold each week

18. Wooden Toys Example • Variables • ??? • OF • ??? • Constraints • ???

19. Wooden Toys Solved

20. RESOURCE REQUIREMENTS Labor Cotton Profit PRODUCT (hr/unit) (lb/unit) ($/unit) Corduroy 3.2 7.5 $3.10 Denim 3.0 5.0 $2.25 There are 3000 hours of labor and 6500 pounds of cotton available each month. There is a maximum demand of 510 yards of corduroy each month, but no limit on denim. Formulate this problem as a LP model and solve it. Example – Problem #S13-1a

21. #S13-1a Let c = yards of corduroy to produce d = yards of denim to produce z = total profit Maximize z = 3.1 c + 2.25 d Subject to 7.5c+ 5d 6500 (cotton constraint) 3.2c + 3d< 3000 (labor constraint) c510 (demand constraint) c, d 0 (non-negativity)

22. Mixed Nuts • Crazy Joe makes two blends of mixed nuts: party mix and regular mix. • Crazy Joe has 10 lbs of cashews and 24 lbs of peanuts • Crazy Joe wants to maximize revenue. Please help him.

23. Ah, Nuts Formulation • Let • p = lbs of party mix to make • r = lbs of regular mix to make • z = total revenue • Max z = 6p + 4r • Subject to • 0.6p + 0.9r< 24 (peanut constraint) • 0.4p + 0.1r< 10 (cashew constraint) • p, r> 0 (non-negativity constraints)

24. Blending • Determines “recipe” requirements to come up with an end product • Examples • Diet • Gasoline

25. Jack Sprat • A well-known nursery rhyme goes “Jack Sprat could eat no fat. His wife (Jill) could eat no lean …” Suppose Jack needs to have at least one pound of lean meat per day, while Jill needs at least 0.4 lbs of fat per day. They can buy either beef or pork with the following attributes: • How much of each meat product should they buy to meet their daily requirements and minimize costs?

26. Jack Sprat Formulation • Let • ??? • Min ??? • Subject to • ??? • Non-negativity

27. Sensitivity Analysis • How sensitive the results are to parameter changes • Change in the value of coefficients • Change in a right-hand-side value of a constraint • Trial-and-error approach • Analytic postoptimality method

28. Sensitivity Report Program B.1

29. Changes in Resources • The right-hand-side values of constraint equations may change as resource availability changes • The shadow price of a constraint is the change in the value of the objective function resulting from a one-unit change in the right-hand-side value of the constraint

30. Changes in Resources • Shadow prices are often explained as answering the question “How much would you pay for one additional unit of a resource?” • Shadow prices are only valid over a particular range of changes in right-hand-side values • Sensitivity reports provide the upper and lower limits of this range

31. Multiple Optimal Solutions • Often, real world problems can have more than one optimal solution • When would this happen? • What does the graph have to look like? • Do want to have “ties”?

32. No Solutions • Can we ever have a problem without a feasible solution? • When would this happen? • What would the graph look like? • Does this mean we did something wrong?