1. Operations Management Dr. Ron Lembke
Linear Programming
Operations Management
Dr. Ron Lembke

2. 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.

4. 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.

5. 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?

6. Summary 14 hrs / day Van: 40.0 ft3 4 days
Tshirt: 0.2 ft3 30 min / tshirt
Sshirt: 0.5 ft3 15 min / Sshirt
How many should we make of each?

7. 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.

8. Standard Form Max 3x + 4y s.t. x + y <= 10 x + 2y <= 12
Linear programs are written the following way:
Max 3x + 4y
s.t. x + y <= 10
x + 2y <= 12
x >= 0
y >= 0

9. Standard Form Max 3x + 4y s.t. x + y <= 10 x + 2y <= 12
Linear programs are written the following way:
Objective Coefficients
Objective
Function
Max 3x + 4y
s.t. x + y <= 10
x + 2y <= 12
x >= 0
y >= 0
Constraints
RHS (right hand side)
Non-negativity
Constraints
LHS (left hand side)
inequalities

10. 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

11. Standard Form Max 7x1 + 5x2 s.t. 4x1 + 3x2 <= 240
electronics
assembly

12. Graphical Solution 80 20 40 60
100 X2
mp3 X1
DVD players

13. Graphical Solution 100 80 60 40 20 0 20 40 60 80 X1 = 0, X2 = 80 X2 80 20 40 60
100 X2
mp3
X1 = 60, X2 = 0
Electronics Constraint X1
DVD players

14. Graphical Solution 100 80 60 40 20 0 20 40 60 80 X1 = 0, X2 = 100 X2 80 20 40 60
100
Assembly Constraint X2
mp3
X1 = 50, X2 = 0 X1
DVD players

15. Graphical Solution 80 20 40 60
100
Assembly Constraint X2
Feasible Region – Satisfies all constraints
mp3
Electronics Constraint X1
DVD players

16. Isoprofit Lnes 100 80 60 40 20 0 20 40 60 80 X2 $7X1 + $5X2 = $210 80 20 40 60
100 X2
Isoprofit Line:
mp3
$7X1 + $5X2 = $210
(0, 42)
(30,0) X1
DVD players

17. Isoprofit Lines 100 80 60 $280 40 $210 20 0 20 40 60 80 X2 X1 mp3 80 20 40 60
100 X2
$280
mp3
$210 X1
DVD players

18. Isoprofit Lines 100 80 $350 60 $280 40 $210 20 0 20 40 60 80 X2 X1 mp3 80 20 40 60
100
$350 X2
$280
mp3
$210 X1
DVD players

19. Isoprofit Lines 80 20 40 60
100
(0, 82)
$7X1 + $5X2 = $410 X2
mp3
(58.6, 0) X1
DVD players

20. 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?

21. Constraint Intersections 80 20 40 60
100
X1 = 0 and 4X1 + 3X2 <= 240
So X2 = 80
(0, 80) X2
mp3
4X1 + 3X2 <= 240
(0, 0) X1
DVD players

22. Constraint Intersections 80 20 40 60
100
(0, 80) X2
mp3
X2 = 0 and 2X1 + 1X2 <= 100
So X1 = 50
(0, 0)
(50, 0) X1
DVD players

23. Constraint Intersections
4X1+ 3X2 <= 240
2X1 + 1X2 <= 100 – multiply by -2 80 20 40 60
100
4X1+ 3X2 <= 240
-4X1 -2X2 <= add rows together
(0, 80)
0X1+ 1X2 <= 40 X2 = 40 substitute into #2 X2
2X1+ 40 <= 100 So X1 = 30
mp3
(0, 0)
(50, 0) X1
DVD players

24. Constraint Intersections 80 20 40 60
100
Find profits of each point.
(0, 80)
$400 X2
mp3
(30,40)
$410
(50, 0)
$350
(0, 0) $0 X1
DVD players

25. 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.

26. Constraint Intersections
Start at (0,0), or some other easy feasible point.
Find a profitable direction to go along an edge
Go until you hit a corner, find profits of point.
If new is better, repeat, otherwise, stop.
100
Good news:
Excel can do
this for us. 80 60
mp3 X2 40 20 X1
DVD 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 =24
If x1=0, 3x2=24, x2=8 (0,8)
If x2=0, 4x1=24, x1=6 (6,0)
x2= 2
If x1=0, x2=2
No matter what x1 is, x2=2

29. Graphical Solution 10 8 6 4 2 0 2 4 6 8 X2 X1 5x1 + 2x2 =20 8 2 4 6 10
5x1 + 2x2 =20 X2
4x1 + 3x2 =24
x2=2 X1

30. [5x1+ 2x2 =20]*3
[4x1 +3x2 =24]*2
15x1+ 6x2 = 60
8x1 +6x2 = 48
(0,10) 8 2 4 6 10 -
7x1 = 12
5x1 + 2x2 =20
x1 = 12/7= 1.71
5x1+2x2 =20
5* x2 =20
2x2 = 11.45
x2 = 5.725
(1.71,5.73) X2
(1.71,5.73)
4x1 +3x2 =24
x2 =2 X1

31. 4x1 +3x2 =24
x2 =2
4x1 +3*2 =24
4x1 =18
x1=18/4 = 4.5
(4.5,2)
(0,10) 8 2 4 6 10
5x1 + 2x2 =20 X2
(1.71,5.73)
4x1 +3x2 =24
x2 =2
(4.5,2) X1

32. Lowest Cost
(0,10)
Z=8x1 +12x2
8*0 + 12*10 = 120 8 2 4 6 10
5x1 + 2x2 =20
Z=8x1 +12x2
8* *5.73 = 82.44 X2
(1.71,5.73)
4x1 +3x2 =24
Z=8x1 +12x2
8* *2 = 60
x2 =2
(4.5,2) X1

33. Profit Line 10 8 6 4 2 0 2 4 6 8 10 12 X2 X1 Z=8x1 +12x2 Try 8*12 = 96 X1

34. Formulating in Excel
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.

35. Current solution
Amount of each
constraint used
by current solution

36. 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.

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

38. 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

39. Click “Data” then “Solver”

40. If Solver Doesn’t Appear

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

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 maximize
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.
Add rest of constraints

44. Add Constraint Dialog Box

45. Constraints Added

46. Assuming Linear
You have to tell Solver that the model is Linear. Click ‘options,’ and make sure the ‘Assume Linear Model’ box is checked.

47. Assume Linear

48. Assuming Linear
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.
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 going
back to your original solution.
Highlight the reports that you want to look at.

50. 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.

51. Optimal Solution

52. 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.

53. Answer Report

54. Sensitivity Report Variables: Final value of each variable
Reduced cost: how much objective changes if current solution is changed
Objective 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 decrease
How 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 1.0
Allowable increase, decrease: how wide a range of values of RHS shadow price is good for.

57. Sensitivity Report

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

59. Limits Report