OPSM 301 Operations Management Class 10: Introduction to Linear Programming Koç University Zeynep Aksin

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

Product Mix Decisions Total time available in week:20 hrs

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 = $ Margin per oven minute from Oatmeal Raisin cookies = $3.10 / 15 = $ 0.207

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?

Linear programming

Introduction  We all face decision about how to use limited resources such as: –time –money –workers/manpower

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

Characteristics of Optimization Problems  Decisions  Constraints  Objectives

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

Solving the LP using Excel Solver Number to make Total profit Unit Profits ConstraintsValueRHS (constraint) You Oven Room Mate Market cc10100 Market or Constraint not binding in optimal solution Optimal Profit Optimal product-mix

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

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. Aqua-SpaHydro-Lux Pumps11 Labor 9 hours6 hours Tubing12 feet16 feet Unit Profit$350$300

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

5 Steps In Formulating LP Models (continued) 4. State the constraints as linear combinations of the decision variables. 1X 1 + 1X 2 <= 200} pumps 9X 1 + 6X 2 <= 1566} labor 12X X 2 <= 2880} tubing 5. Identify any upper or lower bounds on the decision variables. X 1 >= 0 X 2 >= 0

Summary of the LP Model for Blue Ridge Hot Tubs MAX: 350X X 2 S.T.:1X 1 + 1X 2 <= 200 9X 1 + 6X 2 <= X X 2 <= 2880 X 1 >= 0 X 2 >= 0

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

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.

Let’s Implement a Model for the Blue Ridge Hot Tubs Example... MAX: 350X X 2 } profit S.T.:1X 1 + 1X 2 <= 200} pumps 9X 1 + 6X 2 <= 1566} labor 12X X 2 <= 2880} tubing X 1, X 2 >= 0} nonnegativity

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

Follow me using the file on the network drive  Go to STORAGE F:\COURSES\UGRADS\OPSM301\SHARE  Copy SolverQ1.xls to your desktop  Open the spreadsheet and click on first worksheet

Implementing the Model

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

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.

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.