Part 2

1. Linear Programming Part 2 Operations Management - 5th Edition

2. Lecture Outline • Remind me what LP is • What types of problems can we solve with LP? • Formulation review • More examples

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. 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.

5. 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

6. 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

7. Types of Problems • Supplement 10 • SCM: transportation and transshipment models • Chapter 13 • Aggregate Planning • Supplement 13 • Product Mix • Blending

8. LP Model Formulation – NPS Format • Indices • Data • Decision Variables • Objective Function • Constraints • Bounds • Non-negativity

9. LP Model Formulation – Variable-by-Variable • Let • varName1 = # of products to produce • Max varName2 = OF • Subject to • Constraints (brief description)

10. 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 in both NPS and variable-by-variable format. Solve the LP. Example – Problem #S13-1a

11. #S13-1a – NPS Format • Indices • p = products {c, d} • Data • PROFITp = $ profit per yard of p made • LABORp = # of hours to produce a yard of p • COTTONp = lbs of cotton for a yard of p • TOTLABOR = total hours available • TOTCOTTON = total lbs of cotton available • DEMANDp = max demand for product p

12. #S13-1a – NPS Format • Variables • nump = yards of p to produce • totprofit = total profit • Objective Function • Max totprofit = • Constraints (labor constraint) (cotton constraint) (demand constraints) (non-negativity)

13. #S13-1a – Variable-by-variable Format Let numc = yards of corduroy to produce numd = yards of denim to produce Maximize totprofit = 3.1 numc + 2.25 numd Subject to 7.5numc + 5numd6500 (cotton constraint) 3.2numc + 3numd≤ 3000 (labor constraint) numc 510 (demand constraint) numc , numd0

14. 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.

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

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

17. 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?

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