Presentation on theme: "Operations Management Dr. Ron Lembke"— Presentation transcript:
1 Operations Management Dr. Ron Lembke Linear ProgrammingOperations ManagementDr. Ron Lembke
2 Motivating ExampleSuppose you are an entrepreneur making plans to make a killing over the summer by traveling across the country selling products you design and manufacture yourself. To be more straightforward, you plan to follow the Dead all summer, selling t-shirts.
4 ExampleYou are really good with tie-dye, so you earn a profit of $25 for each t-shirt.The sweatshirt screen-printed sweatshirt makes a profit of $20.You have 4 days before you leave, and you want to figure out how many of each to make before you head out for the summer.You plan to work 14 hours a day on this. It takes you 30 minutes per tie dye, and 15 minutes to make a sweatshirt.
5 ExampleYou have a limited amount of space in the van. Being an engineer at heart, you figure:If you cram everything in the van, you have 40 cubit feet of space in the van.A tightly packed t-shirt takes 0.2 ft3A tightly packed sweatshirt takes 0.5 ft3.How many of each should you make?
6 Summary 14 hrs / day Van: 40.0 ft3 4 days Tshirt: 0.2 ft3 30 min / tshirtSshirt: 0.5 ft3 15 min / SshirtHow many should we make of each?
7 Linear ProgrammingWhat we have just done is called “Linear Programming.”Has nothing to do with computer programmingInvented in WWII to optimize military “programs.”“Linear” because no x3, cosines, x*y, etc.
8 Standard Form Max 3x + 4y s.t. x + y <= 10 x + 2y <= 12 Linear programs are written the following way:Max 3x + 4ys.t. x + y <= 10x + 2y <= 12x >= 0y >= 0
9 Standard Form Max 3x + 4y s.t. x + y <= 10 x + 2y <= 12 Linear programs are written the following way:Objective CoefficientsObjectiveFunctionMax 3x + 4ys.t. x + y <= 10x + 2y <= 12x >= 0y >= 0ConstraintsRHS (right hand side)Non-negativityConstraintsLHS (left hand side)inequalities
10 Example 2 mp3 - 4 hrs electronics work - 2 hrs assembly time DVD - 3 hrs assembly time- 1 hrs assembly timeHours available: 240 (elect) 100 (assy)Profit / unit: mp3 $7, DVD $5X1 = number of mp3 players to makeX2 = number of DVD players to make
11 Standard Form Max 7x1 + 5x2 s.t. 4x1 + 3x2 <= 240 electronicsassembly
20 Mathematical Solution Obviously, graphical solution is slowWe can prove that an optimal solution always exists at the intersection of constraints.Why not just go directly to the places where the constraints intersect?
24 Constraint Intersections 80204060100Find profits of each point.(0, 80)$400X2mp3(30,40)$410(50, 0)$350(0, 0)$0X1DVD players
25 Do we have to do this?Obviously, this is not much fun: slow and tediousYes, you have to know how to do this to solve a two-variable problem.We won’t solve every problem this way.
26 Constraint Intersections Start at (0,0), or some other easy feasible point.Find a profitable direction to go along an edgeGo until you hit a corner, find profits of point.If new is better, repeat, otherwise, stop.100Good news:Excel can dothis for us.8060mp3X24020X1DVD players
27 Minimization Example Min 8x1 + 12x2 s.t. 5x1 + 2x2 ≥ 20 4x1 + 3x2 ≥ 24
28 Minimization Example Min 8x1 + 12x2 s.t. 5x1 + 2x2 ≥ 20 4x1 + 3x2 ≥ 24 If x1=0, 2x2=20, x2=10 (0,10)If x2=0, 5x1=20, x1=4 (4,0)4x1 + 3x2 =24If x1=0, 3x2=24, x2=8 (0,8)If x2=0, 4x1=24, x1=6 (6,0)x2= 2If x1=0, x2=2No matter what x1 is, x2=2
34 Formulating in ExcelWrite the LP out on paper, with all constraints and the objective function.Decide on cells to represent variables.Enter coefficients of each variable in each constraint in a block of cells.Compute amount of each constraint being used by current solution.
35 Current solutionAmount of eachconstraint usedby current solution
36 Formulating in Excel5. Place inequalities in sheet, so you remember <=, >=6. Enter amount of each constraint7. Enter objective coefficients8. Calculate value of objective function9. Make sure you have plenty of labels.10. Widen columns for readability.
37 RHS of constraints, Inequality signs. Objective Function value of current solutionRHS of constraints,Inequality signs.
38 Solving in ExcelAll we have so far is a big ‘what if” tool. We need to tell the LP Solver that this is an LP that it can solve.Choose ‘Solver’ from ‘Tools’ menu
43 Solving in Excel Choose ‘Solver’ from ‘Tools’ menu Tell Solver what is the objective function, and which are variables.Tell Solver to minimize or maximizeAdd constraints:Click ‘Add’, enter LHS, RHS, choose inequalityClick ‘Add’ if you need to do more, or click ‘Ok’ if this is the last one.Add rest of constraints
48 Assuming LinearYou have to tell Solver that the model is Linear. Click ‘options,’ and make sure the ‘Assume Linear Model’ box is checked.On this box, checking “assume non-negative” means you don’t need to actually add the non-negativity constraints manually.Solve the LP: Click ‘Solve.’ Look at Results.
49 Solution is Found When a solution has been found, this box comes up. You can choose between keeping the solution and goingback to your original solution.Highlight the reports that you want to look at.
50 SolutionAfter clicking on the reports you want generated, they will be generated on new worksheets.You will return to the workbook page you were at when you called up Solver.It will show the optimal solution that was found.
52 Answer Report Gives optimal and initial values of objective function Gives optimal and initial values of variablesTells amount of ‘slack’ between LHS and RHS of each constraint, tells whether constraint is binding.
54 Sensitivity Report Variables: Final value of each variable Reduced cost: how much objective changes if current solution is changedObjective coefficient (from problem)
55 Sensitivity Report Variables: Allowable increase: Allowable decrease How much the objective coefficient can go up before the optimal solution changes.Allowable decreaseHow much the objective coefficient can go down before optimal solution changes.Up to , Down to
56 Sensitivity Report Constraints Final Value (LHS) Shadow price: how much objective would change if RHS increased by 1.0Allowable increase, decrease: how wide a range of values of RHS shadow price is good for.