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Linear Programming Operations Management Dr. Ron Lembke
Linear Programming
Operations ManagementDr. 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 / dayVan: 40.0 ft3 4 daysTshirt: 0.2 ft3 30 min / tshirtSshirt: 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 + 4ys.t. x + y
<= 10x + 2y
<= 12x
>= 0y
>= 0
Standard Form Linear programs are written the following way:
Max 3x + 4ys.t. x + y
<= 10x + 2y
<= 12x
>= 0y
>= 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 <=100x1 >=0
x2 >=0
electronicsassembly
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.
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 constraint7. Enter objective coefficients8. Calculate value of objective function9. 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
Solving in Excel
1. Choose ‘Solver’ from ‘Tools’ menu2. 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’ menu2. Tell Solver what is the objective function, and
which are variables.3. Tell Solver to minimize or maximize4. 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$8 Not Binding 30
Sensitivity Report
Variables: Final value of each variable Reduced cost: how much objective
changes if current solution is changed Objective coefficient (from problem)
Sensitivity Report
Variables: Allowable increase:
How much the objective coefficient can go up before the optimal solution changes.
Allowable decreaseHow much the objective coefficient can go
down before optimal solution changes. Up to 24.667, Down to 23.333
Sensitivity Report
Constraints Final Value (LHS) Shadow price: how much objective would
change if RHS increased by 1.0 Allowable increase, decrease: how wide a
range of values of RHS shadow price is good for.
Sensitivity ReportMicrosoft Excel 11.0 Sensitivity ReportWorksheet: [lp_sony.xls]Sheet1Report Created: 1/14/2004 3:30:08 PM
Adjustable CellsFinal Reduced Objective Allowable
Cell Name Value Cost Coefficient Increase$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
