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Linear Programming Operations Management Dr. Ron Lembke Motivating Example Suppose 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. Example You 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. Example You 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 ft3 A tightly packed sweatshirt takes 0.5 ft3. How many of each should you make? Summary Van: Tshirt: Sshirt: 40.0 ft3 0.2 ft3 0.5 ft3 14 4 30 15 hrs / day days min / tshirt min / Sshirt How many should we make of each? Linear Programming What we have just done is called “Linear Programming.” Has nothing to do with computer programming Invented in WWII to optimize military “programs.” “Linear” because no x3, cosines, x*y, etc. Standard Form Linear programs are written the following way: Max 3x + 4y s.t. x + y <= 10 x + 2y <= 12 x >= 0 y >= 0 Standard Form Linear programs are written the following way: Objective Coefficients Objective Function Constraints Non-negativity Constraints Max 3x + 4y s.t. x + y <= 10 x + 2y <= 12 x >= 0 y >= 0 LHS (left hand side) RHS (right hand side) inequalities Example 2 - 4 hrs electronics work - 2 hrs assembly time DVD - 3 hrs assembly time - 1 hrs assembly time Hours available: 240 (elect) 100 (assy) Profit / unit: mp3 $7, DVD $5 X1 = number of mp3 players to make X2 = number of DVD players to make mp3 Standard Form Max 7x1 + 5x2 s.t. 4x1 + 3x2 2x1 + 1x2 x1 x2 <= <= >= >= 240 100 0 0 electronics assembly Graphical Solution 100 80 X2 60 mp3 40 20 0 0 20 40 X1 DVD players 60 80 Graphical Solution X1 = 0, X2 = 80 100 80 X2 60 mp3 40 X1 = 60, X2 = 0 20 0 0 20 40 X1 DVD players 60 80 Graphical Solution X1 = 0, X2 = 100 100 80 X2 60 mp3 40 X1 = 50, X2 = 0 20 0 0 20 40 X1 DVD players 60 80 Graphical Solution 100 80 Feasible Region – Satisfies all constraints X2 60 mp3 40 20 0 0 20 40 X1 DVD players 60 80 Isoprofit Lnes 100 80 X2 60 Isoprofit Line: mp3 $7X1 + $5X2 = $210 (0, 42) 40 20 0 0 20 40 (30,0) X1 DVD players 60 80 Isoprofit Lines 100 80 X2 60 $280 mp3 40 $210 20 0 0 20 40 X1 DVD players 60 80 Isoprofit Lines 100 80 $350 X2 60 $280 mp3 40 $210 20 0 0 20 40 X1 DVD players 60 80 Isoprofit Lines 100 (0, 82) 80 $7X1 + $5X2 = $410 X2 60 mp3 40 20 (58.6, 0) 0 0 20 40 X1 DVD players 60 80 Mathematical Solution Obviously, graphical solution is slow We 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? Constraint Intersections 100 X1 = 0 and 4X1 + 3X2 <= 240 So X2 = 80 (0, 80) 80 X2 60 mp3 40 4X1 + 3X2 <= 240 20 0 (0, 0) 0 20 40 X1 DVD players 60 80 Constraint Intersections 100 (0, 80) 80 X2 60 mp3 X2 = 0 and 2X1 + 1X2 <= 100 So X1 = 50 40 20 0 (0, 0) 0 20 40 (50, 0) 60 X1 DVD players 80 Constraint Intersections 4X1+ 3X2 <= 240 2X1 + 1X2 <= 100 – multiply by -2 100 4X1+ 3X2 <= 240 -4X1 -2X2 <= -200 add rows together (0, 80) 80 0X1+ 1X2 <= 40 X2 = 40 substitute into #2 X2 60 2X1+ 40 <= 100 So X1 = 30 mp3 40 20 0 (0, 0) 0 20 40 (50, 0) 60 X1 DVD players 80 Constraint Intersections 100 Find profits of each point. (0, 80) 80 $400 X2 60 mp3 (30,40) $410 40 20 (50, 0) $350 0 (0, 0) $0 0 20 40 X1 DVD players 60 80 Do we have to do this? Obviously, this is not much fun: slow and tedious Yes, you have to know how to do this to solve a two-variable problem. We won’t solve every problem this way. Constraint Intersections Start at (0,0), or some other easy feasible point. 1. Find a profitable direction to go along an edge 2. Go until you hit a corner, find profits of point. 3. If new is better, repeat, otherwise, stop. 100 Good news: Excel can do this for us. 80 60 mp3 X2 40 20 0 0 20 40 60 X1 DVD players 80 Formulating in Excel 1. 2. 3. 4. Write 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. Current solution Amount of each constraint used by current solution Formulating in Excel 5. Place inequalities in sheet, so you remember <=, >= 6. Enter amount of each constraint 7. Enter objective coefficients 8. Calculate value of objective function 9. Make sure you have plenty of labels. 10. Widen columns for readability. Objective Function value of current solution RHS of constraints, Inequality signs. Solving in Excel All 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 Solving in Excel 1. 2. 3. Choose ‘Solver’ from ‘Tools’ menu Tell Solver what is the objective function, and which are variables. Tell Solver to minimize or maximize Solving in Excel 1. 2. 3. 4. Choose ‘Solver’ from ‘Tools’ menu Tell Solver what is the objective function, and which are variables. Tell Solver to minimize or maximize Add constraints: 5. Click ‘Add’, enter LHS, RHS, choose inequality Click ‘Add’ if you need to do more, or click ‘Ok’ if this is the last one. Add rest of constraints Add Constraint Dialog Box Constraints Added Assuming Linear 6. You have to tell Solver that the model is Linear. Click ‘options,’ and make sure the ‘Assume Linear Model’ box is checked. Assume Linear Assuming Linear 6. 7. You 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 nonnegativity constraints manually. Solve the LP: Click ‘Solve.’ Look at Results. Solution is Found When a solution has been found, this box comes up. You can choose between keeping the solution and going back to your original solution. Highlight the reports that you want to look at. Solution After 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. Optimal Solution Answer Report Gives optimal and initial values of objective function Gives optimal and initial values of variables Tells amount of ‘slack’ between LHS and RHS of each constraint, tells whether constraint is binding. Answer Report Microsoft Excel 11.0 Answer Report Worksheet: [lp_sony.xls]Sheet1 Report Created: 1/14/2004 3:30:08 PM Target Cell (Max) Cell Name $E$2 Objective Actual Original Value 12 Final Value 410 Adjustable Cells Cell Name $C$4 Variables DVD $D$4 Variables mp3 Original Value 1 1 Final Value 30 40 Constraints Cell Name $E$6 Electronics Actual $E$7 Assembly Actual $E$8 DVD Non-Neg Actual Cell Value Formula 240 $E$6<=$G$6 100 $E$7<=$G$7 30 $E$8>=$G$8 Status Binding Binding Not Binding Slack 0 0 30 Sensitivity Report Variables: Final value of each variable Reduced cost: how much objective changes if current solution is changed Objective coefficient (from problem) Sensitivity Report Variables: Allowable increase: How much the objective coefficient can go up before the optimal solution changes. Allowable decrease How much the objective coefficient can go down before optimal solution changes. Up to 24.667, Down to 23.333 Sensitivity Report Constraints Final Value (LHS) Shadow price: how much objective would change if RHS increased by 1.0 Allowable increase, decrease: how wide a range of values of RHS shadow price is good for. Sensitivity Report Microsoft Excel 11.0 Sensitivity Report Worksheet: [lp_sony.xls]Sheet1 Report Created: 1/14/2004 3:30:08 PM Adjustable Cells Cell Name $C$4 Variables DVD $D$4 Variables mp3 Final Reduced Value Cost 30 0 40 0 Objective Coefficient 7 5 Allowable Increase 3 0.25 Final Value 240 100 Constraint R.H. Side 240 100 Allowable Increase 60 20 Constraints Cell Name $E$6 Electronics Actual $E$7 Assembly Actual Shadow Price 1.5 0.5 Limits Report Tells ranges of values over which the maximum and minimum objective values can be found. Rarely useful Limits Report Microsoft Excel 11.0 Limits Report Worksheet: [lp_sony.xls]Limits Report 2 Report Created: 1/14/2004 3:30:08 PM Target Cell Name $E$2 Objective Actual Adjustable Cell Name $C$4 Variables DVD $D$4 Variables mp3 Value 410 Value 30 40 Lower Target Limit Result 2.377E-12 200 2.017E-11 210 Upper Target Limit Result 30 410 40.00 410.000