Presentation on theme: "OPSM 301 Operations Management"— Presentation transcript:
1 OPSM 301 Operations Management Koç UniversityOPSM 301 Operations ManagementClass 10:Introduction to Linear ProgrammingZeynep Aksin
2 Assignment 2 due on Monday Midterm 1 next Wednesday AnnouncementsAssignment 2 due on MondayMidterm 1 next WednesdayOn Monday October 31, OPSM 301 class will be held in the computer lab SOS 180Graded class participation activityWill show how to use Excel Solver to solve linear programsYou will need this for assignment 3
3 A Kristen’s like example 2 min/unit10 min/unit6 min/unitFlow time T = = 18 min.System cycle time 1/R= 10 min.Throughput rate R= 6 units / hourUtilizations: R1: 2/10=20%R2=100% (bottleneck)R3=6/10=60%
4 Tools: Gantt ChartGantt charts show the time at which different activities are performed, as well as the sequence of activities1234activitiesResourcestime
8 Continue Kristen’s Cookie story.. The business maturesDemand information is availableYou 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/dozenCost of Materials: $2.50/dozenSale Price of Oatmeal Raisin Cookies: $5.50/dozenCost of Materials: $2.40/dozenMaximum weekly demand ofChocolate Chip Cookies: 100 dozenOatmeal Raisin Cookies: 50 dozenTotal weekly operating expense $270
10 Product Mix DecisionsTotal time available in week: 20 hrs
11 Margin per dozen Chocolate Chip cookies = $2.50 Product Mix DecisionsMargin per dozen Chocolate Chip cookies = $2.50Margin per dozen Oatmeal Raisin cookies = $3.10Margin per oven minute from Chocolate Chip cookies = $2.50 / 10 = $ 0.250Margin 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 dozenAt a margin of 2.50 I will make 120*2.5=300But my demand is only 100 dozen!If I bake only oatmeal raisin:In 20 hours I can bake 80 dozenAt a margin of 3.10 I will make 80*3.10=248But my demand is only 50 dozen!What about a mix of chocolate chip and oatmeal raisin? What is the best product mix?
14 AnnouncementLinear programming: Appendix A from another book-copy in course packSkip graphical solution, skip sensitivity analysis for nowYou 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 We all face decision about how to use limited resources such as: IntroductionWe all face decision about how to use limited resources such as:timemoneyworkers/manpower
16 Mathematical Programming... find the optimal, or most efficient, way of using limited resources to achieve objectives.Optimization
17 Example ApplicationsOPSM: Product mix problem-how much of each product should be produced given resource constraints to maximize profitsFinance: Construct a portfolio of securities that maximizes return while keeping "risk" below a predetermined levelMarketing: Develop an advertising strategy to maximize exposure of potential customers while staying within a predetermined budget
18 Components of Linear Programming A specified objective or 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 outputsConstraints are limitations which restrict the alternatives available to decision makers
19 Conditions for Applicability of Linear Programming Resources must be limitedThere must be an objective functionThere must be linearity in the constraints and in the objective functionResources and products must be homogeneousDecision 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 solutionParameters 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 xorsubject to8 xcc xor < 120010 xcc xor < 12004 xcc xor < 1200xcc < 100xor <Technology ConstraintsMarket Constraints
22 Solving the LP using Excel Solver Number to make100Total profitUnit Profits2.53.1ConstraintsValueRHS (constraint)You851200Oven1015Room Mate4Market cc1Market or50Optimal product-mixOptimal ProfitConstraint notbinding in optimalsolution
23 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)
24 Follow me using the file on the network drive Go to STORAGEE:\COURSES\UGRADS\OPSM301\SHARECopy KristensLPexample.xls to your desktopOpen the spreadsheet and click on first worksheet
25 How Solver Views the Model Target cell - the cell in the spreadsheet that represents the objective functionChanging cells - the cells in the spreadsheet representing the decision variablesConstraint 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 $325Available hours of labor is 1500Make the appropriate changes in your spreadsheet and resolve.
28 An Example LP ProblemBlue Ridge Hot Tubs produces two types of hot tubs: Aqua-Spas & Hydro-Luxes. Find profit maximizing product-mix.Aqua-Spa Hydro-LuxPumps 1 1Labor 9 hours 6 hoursTubing 12 feet 16 feetUnit Profit $350 $300There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available.
29 5 Steps In Formulating LP Models: 1. Understand the problem2. Identify the decision variables:X1=number of Aqua-Spas to produceX2=number of Hydro-Luxes to produce3. State the objective function as a linear combination of the decision variables:MAX: Profit = 350X X2
30 5 Steps In Formulating LP Models (continued) 4. State the constraints as linear combinations of the decision variables.1X1 + 1X2 <= 200 } pumps9X1 + 6X2 <= 1566 } labor12X1 + 16X2 <= 2880 } tubing5. Identify any upper or lower bounds on the decision variables.X1 >= 0X2 >= 0
31 Summary of the LP Model for Blue Ridge Hot Tubs MAX: 350X X2S.T.: 1X1 + 1X2 <= 2009X1 + 6X2 <= 156612X1 + 16X2 <= 2880X1 >= 0X2 >= 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 = 01st constraint: 1X1 <= 2002nd constraint: 9X1 <= or X1 <=1743rd constraint: 12X1 <= or X1 <= 240If X2=0, the maximum value of X1 is 174 and the total profit is $350*174 + $300*0 = $60,900This 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: 350X X2 } profitS.T.: 1X1 + 1X2 <= 200 } pumps9X1 + 6X2 <= 1566 } labor12X1 + 16X2 <= 2880 } tubingX1, X2 >= 0 } nonnegativity
35 Preparing Excel You need the Solver add-in First check whether you have this add-inClick on the DATA tabCheck if you have Solver under Analysis (far right)If notClick on the Office Button (far left top)Click on Excel Options (bottom of dialogue box)Select Add-Ins from menu on the leftAdd Solver add-in from the right menu
36 Solve the LP using solver In-class exercisePrepare a spreadsheet for the Blue Ridge Hot Tubs product mix problem we just formulatedSolve the LP using solverSave the file with your name_lastname in E:\COURSES\UGRADS\OPSM301\HOMEWORKConsider the following changesUnit profits from Aqua-Spas is $325Available hours of labor is 1500Make the appropriate changes in your spreadsheet and resolve.