OPSM 301 Operations Management

1. Koç University OPSM 301 Operations Management Class 10: Introduction to Linear Programming Zeynep Aksin zaksin@ku.edu.tr

2. Announcements • Assignment 2 due on Monday • Midterm 1 next Wednesday • On Monday October 31, OPSM 301 class will be held in the computer lab SOS 180 • Graded class participation activity • Will show how to use Excel Solver to solve linear programs • You will need this for assignment 3

3. A Kristen’s like example R1 R2 R3 2 min/unit 10 min/unit 6 min/unit • Flow time T = 2+10+6 = 18 min. • System cycle time 1/R= 10 min. • Throughput rate R= 6 units / hour • Utilizations: R1: 2/10=20% • R2=100% (bottleneck) • R3=6/10=60%

4. Tools: Gantt Chart Gantt charts show the time at which different activities are performed, as well as the sequence of activities 1 2 3 4 activities Resources time

5. Three Workers R3 6 min/unit R2 10 min/unit R1 2 min/unit W2 W3 W1 R3 1 2 3 4 5 R2 1 2 3 4 5 R1 1 2 3 4 5 10 20 30 40 50 60

6. Three Workers Throughput time for a rush order of 1 unit System cycle time R3 1 2 3 4 5 R2 1 2 3 4 5 Throughput time for an order of 5 units R1 1 2 3 4 5 10 20 30 40 50 60

7. Two Workers W2 W1 R3 1 2 3 4 5 R2 1 2 3 4 5 R1 1 2 3 4 5 1 2 3 4 5 10 20 30 40 50 60

8. Continue Kristen’s Cookie story.. • The business matures • Demand information is available • You and your roommate decide to focus on chocolate chip or oatmeal raisin cookies

9. Product Mix Decisions: Kristen Cookies offers 2 products Sale Price of Chocolate Chip Cookies: $5.00/dozen Cost of Materials: $2.50/dozen Sale Price of Oatmeal Raisin Cookies: $5.50/dozen Cost of Materials: $2.40/dozen Maximum weekly demand of Chocolate Chip Cookies: 100 dozen Maximum weekly demand of Oatmeal Raisin Cookies: 50 dozen Total weekly operating expense $270

10. Product Mix Decisions Total time available in week: 20 hrs

11. Product Mix Decisions Margin per dozen Chocolate Chip cookies = $2.50 Margin per dozen Oatmeal Raisin cookies = $3.10 Margin per oven minute from Chocolate Chip cookies = $2.50 / 10 = $ 0.250 Margin per oven minute from Oatmeal Raisin cookies = $3.10 / 15 = $ 0.207

12. Baking only one type • If I bake only chocolate chip: • In 20 hours I can bake 120 dozen • At a margin of 2.50 I will make 120*2.5=300 • But my demand is only 100 dozen! • If I bake only oatmeal raisin: • In 20 hours I can bake 80 dozen • At a margin of 3.10 I will make 80*3.10=248 • But my demand is only 50 dozen! • What about a mix of chocolate chip and oatmeal raisin? What is the best product mix?

13. Linear programming

14. Announcement • Linear programming: Appendix A from another book-copy in course pack • Skip graphical solution, skip sensitivity analysis for now • You can use examples done in class, example A1, solved problem 1, Problem 3 as a study set (and all other problems if you like)

15. Introduction • We all face decision about how to use limited resources such as: • time • money • workers/manpower

16. Mathematical Programming... • find the optimal, or most efficient, way of using limited resources to achieve objectives. • Optimization

17. Example Applications • OPSM: Product mix problem-how much of each product should be produced given resource constraints to maximize profits • Finance: Construct a portfolio of securities that maximizes return while keeping "risk" below a predetermined level • Marketing: Develop an advertising strategy to maximize exposure of potential customers while staying within a predetermined budget

18. Components of Linear Programming • A specified objectiveor a single goal, such as the maximization of profit, minimization of machine idle time etc. • Decision variables represent choices available to the decision maker in terms of amounts of either inputs or outputs • Constraintsare limitations which restrict the alternatives available to decision makers

19. Conditions for Applicability of Linear Programming • Resources must be limited • There must be an objective function • There must be linearity in the constraints and in the objective function • Resources and products must be homogeneous • Decision variables must be divisible and non-negative

20. Components of Linear Programming • There are three types of constraints: • (=<) An upper limit on the amount of some scarce resource • (>=) A lower bound that must be achieved in the final solution • (=) An exact specification of what a decision variable should be equal to in the final solution • Parameters are fixed and given values which determine the relationships between the decision variables of the problem

21. LP for Optimal Product Mix Selection xcc: Dozens of chocolate chip cookies sold. xor: Dozens of oatmeal raisin cookies sold. Max 2.5 xcc + 3.1 xor subject to 8 xcc + 5 xor < 1200 10 xcc + 15 xor < 1200 4 xcc + 4 xor < 1200 xcc < 100 xor < 50 TechnologyConstraints Market Constraints

22. Solving the LP using Excel Solver Optimal product-mix Optimal Profit Constraint not binding in optimal solution

23. Reading the variable information • The optimal solution for Kristen’s is to produce, 100 dozen chocolate chip and 13.33 dozen oatmeal raisin resulting in an optimal profit of $291.33. (This is the maximum possible profit attainable with the current resources)

24. Follow me using the file on the network drive • Go to STORAGE • E:\COURSES\UGRADS\OPSM301\SHARE • Copy KristensLPexample.xls to your desktop • Open the spreadsheet and click on first worksheet

25. How Solver Views the Model • Target cell - the cell in the spreadsheet that represents the objective function • Changing cells - the cells in the spreadsheet representing the decision variables • Constraint cells - the cells in the spreadsheet representing the LHS formulas on the constraints

26. Goals For Spreadsheet Design • Communication - A spreadsheet's primary business purpose is that of communicating information to managers. • Reliability - The output a spreadsheet generates should be correct and consistent. • Auditability - A manager should be able to retrace the steps followed to generate the different outputs from the model in order to understand the model and verify results. • Modifiability - A well-designed spreadsheet should be easy to change or enhance in order to meet dynamic user requirements.

27. Lets consider a slightly different version • Unit profits from Aqua-Spas is $325 • Available hours of labor is 1500 • Make the appropriate changes in your spreadsheet and resolve.

28. Aqua-Spa Hydro-Lux Pumps 1 1 Labor 9 hours 6 hours Tubing 12 feet 16 feet Unit Profit $350 $300 An Example LP Problem Blue Ridge Hot Tubs produces two types of hot tubs: Aqua-Spas & Hydro-Luxes. Find profit maximizing product-mix. There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available.

29. 5 Steps In Formulating LP Models: 1. Understand the problem 2. Identify the decision variables: X1=number of Aqua-Spas to produce X2=number of Hydro-Luxes to produce 3. State the objective function as a linear combination of the decision variables: MAX: Profit = 350X1 + 300X2

30. 5 Steps In Formulating LP Models(continued) 4. State the constraints as linear combinations of the decision variables. 1X1 + 1X2 <= 200 } pumps 9X1 + 6X2 <= 1566 } labor 12X1 + 16X2 <= 2880 } tubing 5. Identify any upper or lower bounds on the decision variables. X1 >= 0 X2 >= 0

31. Summary of the LP Model for Blue Ridge Hot Tubs MAX: 350X1 + 300X2 S.T.: 1X1 + 1X2 <= 200 9X1 + 6X2 <= 1566 12X1 + 16X2 <= 2880 X1 >= 0 X2 >= 0

32. Solving LP Problems:An Intuitive Approach • Idea: Each Aqua-Spa (X1) generates the highest unit profit ($350), so let’s make as many of them as possible! • How many would that be? • Let X2 = 0 • 1st constraint: 1X1 <= 200 • 2nd constraint: 9X1 <=1566 or X1 <=174 • 3rd constraint: 12X1 <= 2880 or X1 <= 240 • If X2=0, the maximum value of X1 is 174 and the total profit is $350*174 + $300*0 = $60,900 • This solution is feasible, but is it optimal? • No!

33. The Steps in Implementing an LP Model in a Spreadsheet 1. Organize the data for the model on the spreadsheet. 2. Reserve separate cells in the spreadsheet to represent each decision variable in the model. 3. Create a formula in a cell in the spreadsheet that corresponds to the objective function. 4. For each constraint, create a formula in a separate cell in the spreadsheet that corresponds to the left-hand side (LHS) of the constraint.

34. Let’s Implement a Model for the Blue Ridge Hot Tubs Example... MAX: 350X1 + 300X2 } profit S.T.: 1X1 + 1X2 <= 200 } pumps 9X1 + 6X2 <= 1566 } labor 12X1 + 16X2 <= 2880 } tubing X1, X2 >= 0 } nonnegativity

35. Preparing Excel • You need the Solver add-in • First check whether you have this add-in • Click on the DATA tab • Check if you have Solver under Analysis (far right) • If not • Click on the Office Button (far left top) • Click on Excel Options (bottom of dialogue box) • Select Add-Ins from menu on the left • Add Solver add-in from the right menu

36. In-class exercise • Prepare a spreadsheet for the Blue Ridge Hot Tubs product mix problem we just formulated • Solve the LP using solver • Save the file with your name_lastname in E:\COURSES\UGRADS\OPSM301\HOMEWORK • Consider the following changes • Unit profits from Aqua-Spas is $325 • Available hours of labor is 1500 • Make the appropriate changes in your spreadsheet and resolve.

37. Implementing the Model