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

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?

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:

Max 3x + 4y

s.t. x + y<= 10

x + 2y<= 12

x>= 0

y>= 0

ObjectiveFunction

Constraints

LHS (left hand side)

RHS (right hand side)

inequalities

Non-negativityConstraints

Objective Coefficients

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 $5X1 = number of mp3 players to makeX2 = number of DVD players to make

Standard Form

Max 7x1 + 5x2

s.t. 4x1 + 3x2 <=240

2x1 + 1x2 <=100

x1 >=0

x2 >=0

electronics

assembly

Graphical Solution

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3X2

X1

Graphical Solution

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3

X1 = 0, X2 = 80

X1 = 60, X2 = 0

Electronics Constraint

X2

X1

Graphical Solution

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3

X1 = 0, X2 = 100

X1 = 50, X2 = 0

Assembly Constraint

X2

X1

Graphical Solution

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3

Assembly Constraint

Electronics Constraint

Feasible Region – Satisfies all constraintsX2

X1

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3Isoprofit Line:

$7X1 + $5X2 = $210

(0, 42)

(30,0)

Isoprofit Lnes

X2

X1

Isoprofit Lines

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3

$210

$280X2

X1

Isoprofit Lines

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3

$210

$280

$350

X2

X1

Isoprofit Lines

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3

(0, 82)

(58.6, 0)

$7X1 + $5X2 = $410

X2

X1

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

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3

X1 = 0 and 4X1 + 3X2 <= 240So X2 = 80

X2

X1

4X1 + 3X2 <= 240

(0, 0)

(0, 80)

Constraint Intersections

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3X2 = 0 and 2X1 + 1X2 <= 100So X1 = 50

X2

X1

(0, 0)

(0, 80)

(50, 0)

Constraint Intersections

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3

4X1+ 3X2 <= 2402X1 + 1X2 <= 100 – multiply by -2

X2

X1

(0, 0)

(0, 80)

(50, 0)

4X1+ 3X2 <= 240-4X1 -2X2 <= -200 add rows together

0X1+ 1X2 <= 40 X2 = 40 substitute into #2

2X1+ 40 <= 100 So X1 = 30

Constraint Intersections

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3X2

X1

(0, 0)$0

(0, 80)$400

(50, 0)$350

(30,40)$410

Find profits of each point.

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

0 20 40 60 80

80

20

40

60

0

100

DVD players

mp3X2

X1

Start at (0,0), or some other easy feasible point.1. Find a profitable direction to go along an edge2. 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.

Minimization Example

Min 8x1 + 12x2

s.t. 5x1 + 2x2 ≥20

4x1 + 3x2 ≥ 24

x2 ≥ 2

x1 , x2 ≥ 0

Minimization ExampleMin 8x1 +

12x2

s.t. 5x1 +2x2 ≥ 20

4x1 +3x2 ≥ 24

x2 ≥ 2

x1 , x2

≥ 0

5x1 + 2x2 =20

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=2No matter what x1 is, x2=2

Graphical Solution

0 2 4 6 8

8

2

4

6

0

10

5x1 +

2x2 =20

X2

X1

4x1 +

3x2 =24

x2=2

0 2 4 6 8

8

2

4

6

0

10

5x1 +

2x2 =20

X2

X1

4x1 +3x

2 =24

x2 =2

(0,10)[5x1+ 2x2 =20]*3

[4x1 +3x2 =24]*2

15x1+ 6x2 = 60

8x1 +6x2 = 48- 7x1 = 12

x1 = 12/7= 1.71

5x1+2x2 =20

5*1.71 + 2x2 =20

2x2 = 11.45

x2 = 5.725

(1.71,5.73)

(1.71,5.73)

0 2 4 6 8

8

2

4

6

0

10

5x1 +

2x2 =20

X2

X1

4x1 +3x

2 =24

x2 =2

(0,10)

(1.71,5.73)

4x1 +3x2 =24

x2 =2

4x1 +3*2 =24

4x1 =18

x1=18/4 = 4.5

(4.5,2)

(4.5,2)

0 2 4 6 8

8

2

4

6

0

10

5x1 +

2x2 =20

X2

X1

4x1 +3x

2 =24

x2 =2

(0,10)

(1.71,5.73)

Z=8x1 +12x2

8*0 + 12*10 = 120

(4.5,2)

Z=8x1 +12x2

8*1.71 + 12*5.73 = 82.44

Z=8x1 +12x2

8*4.5+ 12*2 = 60

Lowest Cost

Profit Line

0 2 4 6 8 10 12

8

2

4

6

0

10

5x1 +

2x2 =20

X2

X1

4x1 +

3x2 =24

x2=2

Z=8x1 +12x2

Try 8*12 = 96x1=0

12x2=96, x2=8

x2=0

8x1=96, x1=12

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.

Amount of eachconstraint used by current solution

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.

RHS of constraints,Inequality signs.

Objective Functionvalue of current solution

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

Click “Data” then “Solver”

If Solver Doesn’t Appear

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

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

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. 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.

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.

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 ReportMicrosoft Excel 11.0 Answer ReportWorksheet: [lp_sony.xls]Sheet1Report Created: 1/14/2004 3:30:08 PM

Target Cell (Max)Cell Name Original Value Final Value

$E$2 Objective Actual 12 410

Adjustable CellsCell Name Original Value Final Value

$C$4 Variables DVD 1 30$D$4 Variables mp3 1 40

ConstraintsCell Name Cell Value Formula Status Slack

$E$6 Electronics Actual 240 $E$6<=$G$6 Binding 0$E$7 Assembly Actual 100 $E$7<=$G$7 Binding 0$E$8 DVD Non-Neg Actual 30 $E$8>=$G$8 Not Binding 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 decreaseHow 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 ReportWorksheet: [lp_sony.xls]Sheet1Report Created: 1/14/2004 3:30:08 PM

Adjustable CellsFinal Reduced Objective Allowable

Cell Name Value Cost Coefficient Increase$C$4 Variables DVD 30 0 7 3$D$4 Variables mp3 40 0 5 0.25

ConstraintsFinal Shadow Constraint Allowable

Cell Name Value Price R.H. Side Increase$E$6 Electronics Actual 240 1.5 240 60$E$7 Assembly Actual 100 0.5 100 20

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 ReportWorksheet: [lp_sony.xls]Limits Report 2Report Created: 1/14/2004 3:30:08 PM

TargetCell Name Value

$E$2 Objective Actual 410

Adjustable Lower Target Upper TargetCell Name Value Limit Result Limit Result

$C$4 Variables DVD 30 2.377E-12 200 30 410$D$4 Variables mp3 40 2.017E-11 210 40.00 410.000