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Linear Programming Operations Management Dr. Ron Tibben-Lembke

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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.

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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.

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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 ft 3 A tightly packed sweatshirt takes 0.5 ft 3. How many of each should you make?

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Summary 14hrs / day Van:40.0 ft 3 4days Tshirt:0.2 ft 3 30min / tshirt Sshirt:0.5 ft 3 15min / Sshirt How many should we make of each?

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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 x 3, cosines, x*y, etc.

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Standard Form Linear programs are written the following way: Max3x+4y s.t.x+y<=10 x+2y<=12 x>=0 y>=0

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Standard Form Linear programs are written the following way: Max3x+4y s.t.x+y<=10 x+2y<=12 x>=0 y>=0 Objective Function Constraints LHS (left hand side) RHS (right hand side) inequalities Non-negativity Constraints Objective Coefficients

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Example 2 mp3- 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

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Standard Form Max7x 1 +5x 2 s.t. 4x 1 +3x 2 <=240 2x 1 +1x 2 <=100 x 1 >=0 x 2 >=0 electronics assembly

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Graphical Solution 020406080 80 20 40 60 0 100 DVD players mp3 X2X2 X1X1

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Graphical Solution 020406080 80 20 40 60 0 100 DVD players mp3 X 1 = 0, X 2 = 80 X 1 = 60, X 2 = 0 Electronics Constraint X2X2 X1X1

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Graphical Solution 020406080 80 20 40 60 0 100 DVD players mp3 X 1 = 0, X 2 = 100 X 1 = 50, X 2 = 0 Assembly Constraint X2X2 X1X1

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Graphical Solution 020406080 80 20 40 60 0 100 DVD players mp3 Assembly Constraint Electronics Constraint Feasible Region – Satisfies all constraints X2X2 X1X1

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020406080 80 20 40 60 0 100 DVD players mp3 Isoprofit Line: $7X 1 + $5X 2 = $210 (0, 42) (30,0) Isoprofit Lnes X2X2 X1X1

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Isoprofit Lines 020406080 80 20 40 60 0 100 DVD players mp3 $210 $280 X2X2 X1X1

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Isoprofit Lines 020406080 80 20 40 60 0 100 DVD players mp3 $210 $280 $350 X2X2 X1X1

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Isoprofit Lines 020406080 80 20 40 60 0 100 DVD players mp3 (0, 82) (58.6, 0) $7X 1 + $5X 2 = $410 X2X2 X1X1

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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?

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Constraint Intersections 020406080 80 20 40 60 0 100 DVD players mp3 X 1 = 0 and 4X 1 + 3X 2 <= 240 So X 2 = 80 X2X2 X1X1 4X 1 + 3X 2 <= 240 (0, 0) (0, 80)

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Constraint Intersections 020406080 80 20 40 60 0 100 DVD players mp3 X 2 = 0 and 2X 1 + 1X 2 <= 100 So X 1 = 50 X2X2 X1X1 (0, 0) (0, 80) (50, 0)

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Constraint Intersections 020406080 80 20 40 60 0 100 DVD players mp3 4X 1 + 3X 2 <= 240 2X 1 + 1X 2 <= 100 – multiply by -2 X2X2 X1X1 (0, 0) (0, 80) (50, 0) 4X 1 + 3X 2 <= 240 -4X 1 -2X 2 <= -200 add rows together 0X 1 + 1X 2 <= 40 X 2 = 40 substitute into #2 2 X 1 + 40 <= 100 So X 1 = 30

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Constraint Intersections 020406080 80 20 40 60 0 100 DVD players mp3 X2X2 X1X1 (0, 0) $0 (0, 80) $400 (50, 0) $350 (30,40) $410 Find profits of each point.

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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.

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Constraint Intersections 020406080 80 20 40 60 0 100 DVD players mp3 X2X2 X1X1 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. Good news: Excel can do this for us.

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Formulating in Excel 1. Write the LP out on paper, with all constraints and the objective function. 2. Decide on cells to represent variables. 3. Enter coefficients of each variable in each constraint in a block of cells. 4. Compute amount of each constraint being used by current solution.

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Amount of each constraint used by current solution Current solution

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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.

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RHS of constraints, Inequality signs. Objective Function value of current solution

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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

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Solving in Excel 1. Choose ‘Solver’ from ‘Tools’ menu 2. Tell Solver what is the objective function, and which are variables. 3. Tell Solver to minimize or maximize

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Solving in Excel 1. Choose ‘Solver’ from ‘Tools’ menu 2. Tell Solver what is the objective function, and which are variables. 3. Tell Solver to minimize or maximize 4. Add constraints: Click ‘Add’, enter LHS, RHS, choose inequality Click ‘Add’ if you need to do more, or click ‘Ok’ if this is the last one. 5. Add rest of constraints

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Add Constraint Dialog Box

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Constraints Added

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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.

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Assume Linear

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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. On this box, checking “assume non-negative” means you don’t need to actually add the non- negativity constraints manually. 7. Solve the LP: Click ‘Solve.’ Look at Results.

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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.

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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.

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Optimal Solution

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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.

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Answer Report

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Sensitivity Report Variables: Final value of each variable Reduced cost: how much objective changes if current solution is changed Objective coefficient (from problem)

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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

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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.

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Sensitivity Report

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Limits Report Tells ranges of values over which the maximum and minimum objective values can be found. Rarely useful

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Limits Report

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