Operations
Management
Linear Programming
Module B - Part 2
B-1
Problem B.23
1. Gross Distributors packages and distributes industrial
supplies. A standard shipment can be packaged in a class A
container, a class K container, or a class T container. The
profit from using each type of container is: $8 for each class
A container, $6 for each class K container, and $14 for each
class T container. The amount of packing material required
by each A, K and T container is 2, 1 and 3 lbs., respectively.
The amount of packing time required by each A, K, and T
container is 2, 6, and 4 hours, respectively. There is 120 lbs
of packing material available each week. Six packers must
be employed full time (40 hours per week each). Determine
how many containers to pack each week.
B-2
Problem B.23
Container
A
K
T
Profit
$8
$6
$14
Amount available
Packing
material (lbs.)
2
1
3
Packing
time (hrs.)
2
6
4
120
=240
B-3
Problem B.23
xi = Number of class i containers to pack each
week. i=A, K, T
Maximize: 8xA +
6xK + 14xT
2xA + xK +
2xA + 6xK +
3xT  120 (lbs.)
4xT = 240 (hours)
xA, xK, xT  0
B-4
Linear Programming Solutions
Unique Optimal Solution.
Multiple Optimal Solutions.
Infeasible (no solution).
x + y  800
x
 1000
x, y  0
 Unbounded (infinite solution).
Maximize 3x + 2y
x + y  1000
B-5
Computer Solutions
Optimal values of decision variables and
objective function.
Sensitivity information for objective function
coefficients.
Sensitivity information for RHS (right-hand
side) of constraints and shadow price.
B-6
Computer Solutions
Enter data from formulation in Excel.
1 row for the coefficients of objective.
 1 row for coefficients & RHS of each constraint.
 1 final row for solution (decision variable) values.

Select Solver from the Tools Menu.
B-7
Computer Solutions - Spreadsheet
B-8
Computer Solutions - Spreadsheet
B-9
Computer Solutions - Spreadsheet
B-10
Computer Solutions - Solver
B-11
Computer Solutions - Solver
B-12
Computer Solutions - Solver Parameters
B-13
Computer Solutions
Set Target Cell: to value of objective function.

E3
Equal To: Max or Min
By Changing Cells: = Sol’n values (decision
variable values).

B7:D7
Subject to the Constraints:
Click Add to add each constraint:
 LHS =,  ,  RHS

B-14
Computer Solutions - Adding Constraints
Cell Reference: LHS location
Select sign : <=, =, >=
Constraint: RHS location
B-15
Computer Solutions - Adding Constraints
1st constraint.
Click Add.
Repeat for second constraint.
B-16
Computer Solutions
Click Options to set up Solver for LP.
B-17
Computer Solutions - Solver Options
 Check ‘on’ Assume Linear Model and
Assume Non-Negative.
B-18
Computer Solutions
Click Solve to find the optimal solution.
B-19
Computer Solutions - Solver Results
B-20
Computer Solutions - Optimal Solution
Optimal solution is to use:

0 A containers

17.14 K containers

34.29 T containers
Maximum profit is $583 per week.

Actually $582.857… in Excel values are rounded.
B-21
Computer Solutions
Optimal solution is to use:

0 class A containers.

17.14 class K containers.

34.29 class T containers.
Maximum profit is $582.857 per week.
Select Answer and Sensitivity Reports and
click OK.

New pages appear in Excel.
B-22
Computer Solution - Answer Report
Microsoft Excel 8.0e Answer Report
Worksheet: [probb.23.xls]Sheet1
Report Created: 1/31/01 9:53:27 PM
Target Cell (Max)
Cell
Name
$E$3 Objective LHS
Original Value Final Value
28 582.8571429
Adjustable Cells
Cell
Name
Original Value Final Value
$B$7 Sol'n values A cont.
1
0
$C$7 Sol'n values K cont.
1 17.14285714
$D$7 Sol'n values T cont
1 34.28571429
Constraints
Cell
Name
$E$4 lbs. LHS
$E$5 hours LHS
Cell Value
Formula
Status Slack
120 $E$4<=$F$4 Binding
0
240 $E$5=$F$5 Binding
0
B-23
Sensitivity Analysis
Projects how much a solution will change if
there are changes in variables or input data.
Shadow price (dual) - Value of one additional
unit of a resource.
B-24
Computer Solution - Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Worksheet: [probb.23.xls]Sheet1
Report Created: 1/31/01 9:53:27 PM
Adjustable Cells
Cell
$B$7
$C$7
$D$7
Final
Value
Reduced
Objective
Allowable Allowable
Name
Cost
Coefficient
Increase
Decrease
Sol'n values A cont.
0 -1.142857143
8 1.142857143
1E+30
Sol'n values K cont. 17.14285714
0
6
8
1E+30
Sol'n values T cont 34.28571429
0
14
1E+30
1.6
Constraints
Cell
Name
$E$4 lbs. LHS
$E$5 hours LHS
Final
Value
120
240
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
B-25
Allowable Allowable
Increase
Decrease
60
80
480
80
Computer Solution - Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Worksheet: [probb.23.xls]Sheet1
Report Created: 1/31/01 9:53:27 PM
Adjustable Cells
Cell
$B$7
$C$7
$D$7
Final
Value
Reduced
Objective
Allowable Allowable
Name
Cost
Coefficient
Increase
Decrease
Sol'n values A cont.
0 -1.142857143
8 1.142857143
1E+30
Sol'n values K cont. 17.14285714
0
6
8
1E+30
Sol'n values T cont 34.28571429
0
14
1E+30
1.6
Optimal solution:
0
class A containers
17.14285… class K containers
34.28571… class T containers
Profit = 0(8) + 17.14285(6) + 34.28571(14) = $582.857
B-26
Computer Solution - Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Worksheet: [probb.23.xls]Sheet1
Report Created: 1/31/01 9:53:27 PM
Adjustable Cells
Cell
$B$7
$C$7
$D$7
Final
Value
Reduced
Objective
Allowable Allowable
Name
Cost
Coefficient
Increase
Decrease
Sol'n values A cont.
0 -1.142857143
8 1.142857143
1E+30
Sol'n values K cont. 17.14285714
0
6
8
1E+30
Sol'n values T cont 34.28571429
0
14
1E+30
1.6
Constraints
Cell
Name
$E$4 lbs. LHS
$E$5 hours LHS
Final
Value
120
240
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
B-27
Allowable Allowable
Increase
Decrease
60
80
480
80
Sensitivity for Objective Coefficients
Objective
Allowable Allowable
Coefficient
Increase
Decrease
8 1.142857143
1E+30
6
8
1E+30
14
1E+30
1.6
 As long as coefficients are in range indicated, then
current solution is still optimal, but profit may
change!
 Current solution is optimal as long as:
Coefficient of xA is between -infinity and 9.142857
Coefficient of xK is between -infinity and 14
Coefficient of xT is between 12.4 and infinity
B-28
Sensitivity for Objective Coefficients
Objective
Allowable Allowable
Coefficient
Increase
Decrease
8 1.142857143
1E+30
6
8
1E+30
14
1E+30
1.6
 If profit for class K container was 12 (not 6), what
is optimal solution?
B-29
Sensitivity for Objective Coefficients
Objective
Allowable Allowable
Coefficient
Increase
Decrease
8 1.142857143
1E+30
6
8
1E+30
14
1E+30
1.6
 If profit for class K container was 12 (not 6), what
is optimal solution?
 xA=0, xK=17.14, xT=34.29 (same as before)
 profit = 685.71 (more than before!)
B-30
Sensitivity for Objective Coefficients
Objective
Allowable Allowable
Coefficient
Increase
Decrease
8 1.142857143
1E+30
6
8
1E+30
14
1E+30
1.6
 If profit for class K container was 16 (not 6), what
is optimal solution?
B-31
Sensitivity for Objective Coefficients
Objective
Allowable Allowable
Coefficient
Increase
Decrease
8 1.142857143
1E+30
6
8
1E+30
14
1E+30
1.6
 If profit for class K container was 16 (not 6), what
is optimal solution?
 Different!
 Resolve problem to get solution.
B-32
Computer Solution - Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Worksheet: [probb.23.xls]Sheet1
Report Created: 1/31/01 9:53:27 PM
Adjustable Cells
Cell
$B$7
$C$7
$D$7
Final
Value
Reduced
Objective
Allowable Allowable
Name
Cost
Coefficient
Increase
Decrease
Sol'n values A cont.
0 -1.142857143
8 1.142857143
1E+30
Sol'n values K cont. 17.14285714
0
6
8
1E+30
Sol'n values T cont 34.28571429
0
14
1E+30
1.6
Constraints
Cell
Name
$E$4 lbs. LHS
$E$5 hours LHS
Final
Value
120
240
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
B-33
Allowable Allowable
Increase
Decrease
60
80
480
80
Sensitivity for RHS values
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
Allowable Allowable
Increase
Decrease
60
80
480
80
 Shadow price is change in objective value for each unit
change in RHS as long as change in RHS is within range.
 Each additional lb. of packing material will increase profit
by $4.2857... for up to 60 additional lbs.
 Each additional hour of packing time will increase profit
by $0.2857... for up to 480 additional hours.
B-34
Sensitivity for RHS values
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
Allowable Allowable
Increase
Decrease
60
80
480
80
 Suppose you can buy 50 more lbs. of packing
material for $250. Should you buy it?
B-35
Sensitivity for RHS values
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
Allowable Allowable
Increase
Decrease
60
80
480
80
 Suppose you can buy 50 more lbs. of packing
material for $250. Should you buy it?
 NO. $250 for 50 lbs. is $5 per lb.
Profit increase is only $4.2857 per lb.
B-36
Sensitivity for RHS values
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
Allowable Allowable
Increase
Decrease
60
80
480
80
 How much would you pay for 50 more lbs. of
packing material?
B-37
Sensitivity for RHS values
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
Allowable Allowable
Increase
Decrease
60
80
480
80
 How much would you pay for 50 more lbs. of
packing material?
 $214.28
50 lbs.  $4.2857/lb. = $214.2857...
B-38
Sensitivity for RHS values
Shadow
Constraint
Price
R.H. Side
4.285714286
120
0.285714286
240
Allowable Allowable
Increase
Decrease
60
80
480
80
 If change in RHS is outside range (from allowable
increase or decrease), then we can not tell how the
objective value will change.
B-39
Extensions of Linear Programming
 Integer programming (IP): Some or all variables are
restricted to integer values.

Allows “if…then” constraints.

Much harder to solve (more computer time).
 Nonlinear programming: Some constraints or
objective are nonlinear functions.

Allows wider range of situations to be modeled.

Much harder to solve (more computer time).
B-40
Integer Programming
{
1
{
0
x1 
x2
1 if we build a factory in St. Louis
0 otherwise.
if we build a factory in Chicago
otherwise.
We will build one factory in Chicago or St. Louis.
x1 + x 2  1
We will build one factory in either Chicago or St. Louis.
x1 + x 2 = 1
If we build in Chicago, then we will not build in St. Louis.
x2  1 - x1
B-41
Harder Formulation Example
You are creating an investment portfolio from 4
investment options: stocks, real estate, T-bills
(Treasury-bills), and cash. Stocks have an annual rate
of return of 12% and a risk measure of 5. Real estate
has an annual rate of return of 10% and a risk measure
of 8. T-bills have an annual rate of return of 5% and a
risk measure of 1. Cash has an annual rate of return of
0% and a risk measure of 0. The average risk of the
portfolio can not exceed 5. At least 15% of the portfolio
must be in cash. Formulate an LP to maximize the
annual rate of return of the portfolio.
B-42
Another Formulation Example
A business operates 24 hours a day and employees
work 8 hour shifts. Shifts may begin at midnight, 4 am,
8 am, noon, 4 pm or 8 pm. The number of employees
needed in each 4 hour period of the day to serve
demand is in the table below. Formulate an LP to
minimize the number of employees to satisfy the
demand.
Midnight
- 4 am
3
4 am 8 am
6
8 am noon
13
Noon 4 pm
15
B-43
4 pm - 8 pm 8 pm midnight
12
9