Lecture 1
Linear Programming
© Copyright 2004, Alan Marshall
1
Agenda
>Math Programming
>Linear Programming
• Introduction
• Exercise: Lego Enterprises
• Terminology, Definitions
• Possible Outcomes
• Sensitivity Analysis
© Copyright 2004, Alan Marshall
2
Math Programming
>Deals with resource allocation to
maximize or minimize an objective
subject to certain constraints
>Types:
• Linear, Integer, Mixed, Nonlinear, Goal
>Relatively easy to solve using modern
computing technology (potentially too
easy!)
© Copyright 2004, Alan Marshall
3
Our Focus
>Linear, Integer (& Mixed
Linear/Integer)
>Recognizing when
linear/integer/mixed programming is
appropriate
>Developing basic models
>Computer solution
• Excel
>Interpreting results
© Copyright 2004, Alan Marshall
4
Lego Enterprises
>Table profit is $16; Chair profit is $10
>Table design
• 2 large blocks (side by side)
• 2 small blocks (stacked under, centered)
>Chair design
• 1 large block (seat)
• 2 small blocks (back, bottom)
>Objective: select product mix to
maximize profits using available
resources
© Copyright 2004, Alan Marshall
5
Understanding Lego Problem
> Formulate as LP
• Decision Variables, Objective Function,
Constraints
>Graph
• Constraints, Objective function
>Find solution
© Copyright 2004, Alan Marshall
6
LP Formulation
>Decision Variables
• T = # of tables
• C = # of chairs
>Objective
• Maximize profit =
>Constraints
• For large blocks:
• For small blocks:
© Copyright 2004, Alan Marshall
7
LP Formulation
>Decision Variables
• T = # of tables
• C = # of chairs
>Objective
• Maximize profit: Z = 16T + 10C
>Constraints
• For large blocks:
• For small blocks:
© Copyright 2004, Alan Marshall
8
LP Formulation
>Decision Variables
• T = # of tables
• C = # of chairs
>Objective
• Maximize profit: Z = 16T + 10C
>Constraints
• For large blocks: 2T + 1C < 6
• For small blocks:
© Copyright 2004, Alan Marshall
9
LP Formulation
>Decision Variables
• T = # of tables
• C = # of chairs
>Objective
• Maximize profit: Z = 16T + 10C
>Constraints
• For large blocks: 2T + 1C < 6
• For small blocks: 2T + 2C < 8
© Copyright 2004, Alan Marshall
10
Graphing Lego Example
>Draw quadrant & axes
• use T on x-axis and C on y-axis
>Add constraint lines
• Find intercepts: set T to zero and solve
for C, set C to zero and solve for T
>Add profit equation
• Select reasonable value
>Move profit equation outwards, as far
as feasible
© Copyright 2004, Alan Marshall
11
Graphing Lego Example
> Draw quadrant &
axes
Lego Problem
8
• use T on x-axis and
C on y-axis
7
6
Chairs
5
4
3
2
1
0
0
1
2
3
Tables
© Copyright 2004, Alan Marshall
4
5
12
Graphing Lego Example
> Add constraint lines
Lego Problem
• Find intercepts: set
T to zero and solve
for C, set C to zero
and solve for T
8
7
6
> Large:
Chairs
5
• Tables: Max = 3
• Chairs: Max = 6
4
3
2
1
0
0
1
2
3
Tables
© Copyright 2004, Alan Marshall
4
5
13
Graphing Lego Example
> Add constraint lines
Lego Problem
• Find intercepts: set
T to zero and solve
for C, set C to zero
and solve for T
8
7
6
> Large:
Chairs
5
• Tables: Max = 3
• Chairs: Max = 6
4
3
2
1
0
0
1
2
3
Tables
© Copyright 2004, Alan Marshall
4
5
14
Graphing Lego Example
> Add constraint lines
Lego Problem
• Find intercepts: set
T to zero and solve
for C, set C to zero
and solve for T
8
7
6
> Small:
Chairs
5
• Tables: Max = 4
• Chairs: Max = 4
4
3
2
1
0
0
1
2
3
Tables
© Copyright 2004, Alan Marshall
4
5
15
Graphing Lego Example
> Add constraint lines
Lego Problem
• Find intercepts: set
T to zero and solve
for C, set C to zero
and solve for T
8
7
6
> Small:
Chairs
5
• Tables: Max = 4
• Chairs: Max = 4
4
3
2
1
0
0
1
2
3
Tables
© Copyright 2004, Alan Marshall
4
5
16
Graphing Lego Example
> Add profit equation
Lego Problem
• Select reasonable
value
8
7
> 40:
6
• Tables: 40/16 = 2.5
• Chairs: 40/10 = 4
Chairs
5
4
3
2
1
0
0
1
2
3
Tables
© Copyright 2004, Alan Marshall
4
5
17
Graphing Lego Example
> Add profit equation
Lego Problem
• Select reasonable
value
8
7
> 40:
6
• Tables: 40/16 = 2.5
• Chairs: 40/10 = 4
Chairs
5
4
3
2
1
0
0
1
2
3
Tables
© Copyright 2004, Alan Marshall
4
5
18
Graphing Lego Example
> Move profit
equation outwards,
as far as feasible
> Solution: T = 2, C
=2
> Profit:
16(2)+10(2)=52
Lego Problem
8
7
6
Chairs
5
4
3
2
1
0
0
1
2
3
Tables
© Copyright 2004, Alan Marshall
4
5
19
Characteristics Of LPs
>Objective function and constraints are
linear functions
>Constraint types are <, = , or >
>Variables can assume any fractional
value
>Decision variables are non-negative
>Maximize or Minimize single objective
© Copyright 2004, Alan Marshall
20
Characteristics Of LPs
>Objective function and constraints are
linear functions
• Lego: All were linear trade-offs
>Constraint types are <, = , or >
>Variables can assume any fractional
value
>Decision variables are non-negative
>Maximize or Minimize single objective
© Copyright 2004, Alan Marshall
21
Characteristics Of LPs
>Objective function and constraints are
linear functions
>Constraint types are <, = , or >
• Lego: All Constraints implied maximums
(<)
>Variables can assume any fractional
value
>Decision variables are non-negative
>Maximize or Minimize single objective
© Copyright 2004, Alan Marshall
22
Characteristics Of LPs
>Objective function and constraints are
linear functions
>Constraint types are <, = , or >
>Variables can assume any fractional
value
• Lego: Fractional values can be viewed as
work-in-process at the end of the day
>Decision variables are non-negative
>Maximize or Minimize single objective
© Copyright 2004, Alan Marshall
23
Characteristics Of LPs
>Objective function and constraints are
linear functions
>Constraint types are <, = , or >
>Variables can assume any fractional
value
>Decision variables are non-negative
• Lego: Cannot produce negative amounts
>Maximize or Minimize single objective
© Copyright 2004, Alan Marshall
24
Characteristics Of LPs
>Objective function and constraints are
linear functions
>Constraint types are <, = , or >
>Variables can assume any fractional
value
>Decision variables are non-negative
>Maximize or Minimize single objective
• Lego: Maximizing Profit
© Copyright 2004, Alan Marshall
25
Key Definitions
>Feasible solution: one that satisfies all
constraints
• can have many feasible solutions
>Feasible region: set of all feasible
solutions
>Optimal solution: any feasible solution
that optimizes the objective function
• can have ties
© Copyright 2004, Alan Marshall
26
Standard LP Form
>All constraints expressed as equalities
• use slack (<) or surplus (>) variables
>All variables are nonnegative
>All variables appear on the left side of
the constraint functions
>All constants appear on the right side
of the constraint functions
>Formulate Lego problem in standard
form
© Copyright 2004, Alan Marshall
27
Lego - Standard Form
>Maximize profit: Z = 16T + 10C
>Subject to
• For large blocks: 2T + 1C + S1 = 6
• For small blocks: 2T + 2C + S2 = 8
• Non-negativities:, T, C, S1, S2 > 0
>Useful, because of the concept of
slack and surplus
• While we will not formulate this way in
Excel, we will still use these concepts
© Copyright 2004, Alan Marshall
28
Possible LP Outcomes
>Unique optimal solution
>Alternate optimal solutions
>Unbounded problem
>Infeasible problem
© Copyright 2004, Alan Marshall
29
Example: Unique Optimal Soln
>Solve graphically for the optimal
solution:
Max:
z = 6x1 + 2x2
s.t.
4x1 + 3x2 > 12
x1 + x2 < 8
x1, x2 > 0
© Copyright 2004, Alan Marshall
30
Example: Unique Optimal
>There is only one point in the feasible
set that maximizes the objective
function (x1 = 8, x2 = 0)
x2
8
Max 6x1 + 2x2
4x1 + 3x2 > 12
4
x1 + x2 < 8
3
© Copyright 2004, Alan Marshall
8
x1
31
Example: Alternate Solutions
>Solve graphically for the optimal
solution:
Max
z = 6x1 + 3x2
s.t.
4x1 + 3x2 > 12
2x1 + x2 < 8
x1, x2 > 0
© Copyright 2004, Alan Marshall
32
Example: Alternate Solutions
>There are infinite points satisfying
both constraints - objective function
falls on a constraint line
2x1 + x2 < 8
x2 8
4x1 + 3x2 > 12
Max 6x1 + 3x2 4
3
© Copyright 2004, Alan Marshall
4
x1
33
Example: Infeasible Problem
>Solve graphically for the optimal
solution:
Max
z = 2x1 + 6x2
s.t.
4x1 + 3x2 < 12
2x1 + x2 > 8
x1, x2 > 0
© Copyright 2004, Alan Marshall
34
Example: Infeasible Problem
>No points satisfy both constraints
• no feasible region, no optimal solution
x2
2x1 + x2 > 8
8
4x1 + 3x2 < 12
4
3
© Copyright 2004, Alan Marshall
4
x1
35
Example: Unbounded Problem
>Solve graphically for the optimal
solution:
Max
z = 3x1 + 4x2
s.t.
x1 +
3x1 +
x2 > 5
x2 > 8
x1, x2 > 0
© Copyright 2004, Alan Marshall
36
Example: Unbounded Problem
>objective function can be moved
outward without limit; z can be
increased infinitely
x2
8
3x1 + x2 > 8
5
x1 + x2 > 5
Max 3x1 + 4x2
2.67
© Copyright 2004, Alan Marshall
5
x1
37
RECAP
© Copyright 2004, Alan Marshall
38
Characteristics Of LPs
>Objective function and constraints are
linear functions
>Constraint types are <, = , or >
>Variables can assume any fractional
value
>Decision variables are non-negative
>Maximize or Minimize single objective
© Copyright 2004, Alan Marshall
39
Formulation
>Define decision variables: x1, x2, …
>Objective Function (max, min)
>s.t., with constraints listed
• Variables on left side
• Constants on right side
• All variables nonnegative
>NB: “Standard Form” requires
constraints stated as equalities
• add slack/surplus variables
© Copyright 2004, Alan Marshall
40
Possible LP Outcomes
>Unique optimal solution
>Alternate optimal solutions
>Unbounded problem
>Infeasible problem
© Copyright 2004, Alan Marshall
41
Break
15 Minutes
© Copyright 2004, Alan Marshall
42
LP Models: Key Questions
>What am I trying to decide?
>What is the objective?
• Is it to be minimized or maximized?
>What are the constraints?
• Are they limitations or requirements?
• Are they explicit or implicit?
© Copyright 2004, Alan Marshall
43
Example
A chemical company makes and sells a product
in 40-lb. and 80-lb. bags on a common
production line. To meet anticipated orders,
next week’s production should be at least
16,000 lbs. Profit contributions are $2 per 40lb. bag, and $4 per 80-lb. bag. The packaging
line operates 1500 minutes/week. 40-lb. bags
require 1.2 min. of packaging time; 80-lb.
bags require 3 min. The company has 6000
square feet of packaging material available.
Each 40-lb. bag uses 6 square feet, and each
80-lb. bag uses 10 square feet. How many
bags of each type should be produced?
© Copyright 2004, Alan Marshall
44
Model Development
>What do we need to decide?
What are our decision variables?
© Copyright 2004, Alan Marshall
45
Model Development
>What do we need to decide?
x1 = number of 40-lb. bags to produce
x2 = number of 80-lb. bags to produce
© Copyright 2004, Alan Marshall
46
Model Development
>What is the objective?
© Copyright 2004, Alan Marshall
47
Model Development
>What is the objective?
Maximize total profit
© Copyright 2004, Alan Marshall
48
Model Development
>What is the objective?
Maximize total profit
z = 2x1 + 4x2
© Copyright 2004, Alan Marshall
49
Check Your Units!
> Always be sure that your units are
consistent with the problem
> Our decision variable is measured in “Bags”
> Our profit/objective function is in $
Obj.Fn.Coe f.  Dec.Var.Un it  Obj.Fn.Uni t
$
 bags  $
bag
© Copyright 2004, Alan Marshall
50
Model Development
>What are the constraints?
© Copyright 2004, Alan Marshall
51
Model Development
>What are the constraints?
Aggregate production:
Packaging time:
Packaging materials:
Nonnegativity:
© Copyright 2004, Alan Marshall
52
Model Development
>What are the constraints?
Prod: 40x1 + 80x2 > 16,000
Time:
Mat:
NN:
© Copyright 2004, Alan Marshall
53
Model Development
>What are the constraints?
Prod: 40x1 + 80x2 > 16,000
Time: 1.2x1 + 3x2 < 1,500
Mat:
NN:
© Copyright 2004, Alan Marshall
54
Model Development
>What are the constraints?
Prod: 40x1 + 80x2 > 16,000
Time: 1.2x1 + 3x2 < 1,500
Mat: 6x1 + 10x2 < 6,000
NN:
© Copyright 2004, Alan Marshall
55
Model Development
>What are the constraints?
Prod: 40x1 + 80x2 > 16,000
Time: 1.2x1 + 3x2 < 1,500
Mat: 6x1 + 10x2 < 6,000
NN:
x1, x2 > 0
© Copyright 2004, Alan Marshall
56
Check The Units!
>What are the constraints?
Prod: 40x1 + 80x2 > 16,000
Time: 1.2x1 + 3x2 < 1,500
Mat: 6x1 + 10x2 < 6,000
NN:
lbs/bag x bags
min/bag x bags
ft2/bag x bags
x1, x2 > 0
© Copyright 2004, Alan Marshall
57
Complete Model
x1 = no. of 40-lb. bags to produce
x2 = no. of 80-lb. bags to produce
Maximize z = 2x1 + 4x2
subject to
40x1 + 80x2 > 16,000
1.2x1 + 3x2 < 1,500
6x1 + 10x2 < 6,000
x1, x2 > 0
© Copyright 2004, Alan Marshall
58
Excel
>Model Input
• Basic model
• Solver: identify objective function &
constraints
>Results
• Answer Report
• Sensitivity Report
• Limits Report
© Copyright 2004, Alan Marshall
59
Spreadsheet Model
A
1
2
3
4
5
6
7
8
9
B
C
40-lb 80-lb
DecVar
0
E
F
G
0
ObFnCoef
2
4
MinProd'n
MachTime
PackMat
40
1.2
6
80
3
10
© Copyright 2004, Alan Marshall
D
0
0 >=
0 <=
0 <=
Constraint Constraint
Amount
Slack
16000
1500
6000
16000
1500
6000
60
Cell Formulas
A
1
2
3
4
5
6
7
8
9
DecVar
B
C
40-lb 80-lb
0
ObFnCoef
2
MinProd'n
MachTime
PackMat
40
1.2
6
D
E
F
G
Constraint
Amount
Constraint
Slack
0
4 =SUMPRODUCT(B5:C5,$B$3:$C$3)
80 =SUMPRODUCT(B7:C7,$B$3:$C$3) >=
3 =SUMPRODUCT(B8:C8,$B$3:$C$3) <=
10 =SUMPRODUCT(B9:C9,$B$3:$C$3) <=
© Copyright 2004, Alan Marshall
16000 =ROUND(F7-D7,2)
1500 =ROUND(F8-D8,2)
6000 =ROUND(F9-D9,2)
61
Using Solver
© Copyright 2004, Alan Marshall
62
Adding Constraints
© Copyright 2004, Alan Marshall
63
Solver Options
Note Assumptions
ticked - essential!
© Copyright 2004, Alan Marshall
64
Answer Report
Microsoft Excel 8.0e Answer Report
Target Cell (Max)
Cell
Name
$D$5 ObFnCoef
Original Value Final Value
0
2200
Adjustable Cells
Cell
Name
Original Value Final Value
$B$3 DecVar 40-lb
0
500
$C$3 DecVar 80-lb
0
300
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Cell Value
Formula
Status
Slack
1500 $D$8<=$F$8 Binding
0
6000 $D$9<=$F$9 Binding
0
44000 $D$7>=$F$7 Not Binding 28000
65
Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Adjustable Cells
Final
Cell
Name
Value
$B$3 DecVar 40-lb
500
$C$3 DecVar 80-lb
300
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase
Decrease
0
2
0.4
0.4
0
4
1 0.666666667
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Final
Shadow
Constraint Allowable
Value
Price
R.H. Side Increase
1500 0.666666667
1500
300
6000
0.2
6000
1500
44000
0
16000
28000
Allowable
Decrease
300
1000
1E+30
66
Optimal Solution
>Three parts:
• decision variables
• values of decision variables
• value of objective function
>Decision variables:
• basic (non-zero value),
• non-basic (zero)
• Basic variables are “in the solution”; nonbasic are not
© Copyright 2004, Alan Marshall
67
Impact of Possible Changes
>Change existing constraint
• changes slope; may change size of
feasible region
>Add new constraint
• may decrease feasible region (if binding)
>Remove constraint
• may increase feasible region (if binding)
>Change objective
• may change optimal solution
© Copyright 2004, Alan Marshall
68
Sensitivity Analysis
>The next section will deal with the
sensitivity analysis that can be done
simply based on the reports generated,
without rerunning the solution.
>While this can be useful, mastery of this
material is not important for the course,
or programme as you can always simply
run the model again with the changes
>However, we will look at this briefly
© Copyright 2004, Alan Marshall
69
Sensitivity Analysis
>Used to determine how optimal
solution is affected by changes, within
specified ranges: objective function or
RHS coefficients (only 1 at a time)
>Important to managers who must
operate in a dynamic environment
with imprecise estimates of
coefficients
>Sensitivity analysis allows us to ask
certain what-if questions
© Copyright 2004, Alan Marshall
70
Objective Function Coefficients
>If an objective function coefficient
changes, slope of objective function
line changes. At some threshold,
another corner point may become
optimal.
>Question: How much can objective
coefficient change without changing
optimal corner point?
© Copyright 2004, Alan Marshall
71
Geometric Illustration
x2
current optimal
solution
new optimal solution
x1
© Copyright 2004, Alan Marshall
72
Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Adjustable Cells
Final
Cell
Name
Value
$B$3 DecVar 40-lb
500
$C$3 DecVar 80-lb
300
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase
Decrease
0
2
0.4
0.4
0
4
1 0.666666667
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Final
Shadow
Constraint Allowable
Value
Price
R.H. Side Increase
1500 0.666666667
1500
300
6000
0.2
6000
1500
44000
0
16000
28000
Allowable
Decrease
300
1000
1E+30
73
Reduced Costs
(Objective Function Coefficients)
>Reduced cost for decision variable not
in solution (current value is 0) is
amount variable's objective function
coefficient would have to improve
(increase for max, decrease for min)
before variable could enter solution
>Reduced cost for decision variable in
solution is 0
© Copyright 2004, Alan Marshall
74
Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Adjustable Cells
Final
Cell
Name
Value
$B$3 DecVar 40-lb
500
$C$3 DecVar 80-lb
300
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase
Decrease
0
2
0.4
0.4
0
4
1 0.666666667
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Final
Shadow
Constraint Allowable
Value
Price
R.H. Side Increase
1500 0.666666667
1500
300
6000
0.2
6000
1500
44000
0
16000
28000
Allowable
Decrease
300
1000
1E+30
75
Objective Function Ranges
>Interval within which original solution
remains optimal (same decision
variables in solution) while keeping all
other data constant
>Within this range, associated reduced
cost is valid
>Value of the objective function might
change in this range of optimality
© Copyright 2004, Alan Marshall
76
Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Adjustable Cells
Final
Cell
Name
Value
$B$3 DecVar 40-lb
500
$C$3 DecVar 80-lb
300
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase
Decrease
0
2
0.4
0.4
0
4
1 0.666666667
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Final
Shadow
Constraint Allowable
Value
Price
R.H. Side Increase
1500 0.666666667
1500
300
6000
0.2
6000
1500
44000
0
16000
28000
Allowable
Decrease
300
1000
1E+30
77
RHS Coefficient Changes
>When a right-hand-side value
changes, the constraint moves
parallel to itself
>Question: How is the solution
affected, if at all?
>Two cases:
• constraint is binding
• constraint is nonbinding
© Copyright 2004, Alan Marshall
78
Geometric Illustration
x2
Binding constraints have zero slack
Nonbinding constraints have positive slack
Binding constraints
optimal solution
Nonbinding constraint
© Copyright 2004, Alan Marshall
x1
79
Binding Constraints
Microsoft Excel 8.0e Answer Report
Target Cell (Max)
Cell
Name
$D$5 ObFnCoef
Original Value Final Value
0
2200
Adjustable Cells
Cell
Name
Original Value Final Value
$B$3 DecVar 40-lb
0
500
$C$3 DecVar 80-lb
0
300
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Cell Value
Formula
Status
Slack
1500 $D$8<=$F$8 Binding
0
6000 $D$9<=$F$9 Binding
0
44000 $D$7>=$F$7 Not Binding 28000
80
Tightening & Relaxing Constraints
>Tightening a constraint means to
make it more restrictive; i.e.
decreasing the RHS of a less than
constraint, or increasing the RHS of a
greater constraint. This compresses
the feasible region.
>Relaxing a constraint means to make
it less restrictive; i.e., expand the
feasible region.
© Copyright 2004, Alan Marshall
81
Effect of Tightening a Constraint
x2
New optimal solution (z decreases)
Original optimal solution
x1
© Copyright 2004, Alan Marshall
82
Effect of Relaxing a Constraint
x2
optimal solution
x1
© Copyright 2004, Alan Marshall
83
Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Adjustable Cells
Final
Cell
Name
Value
$B$3 DecVar 40-lb
500
$C$3 DecVar 80-lb
300
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase
Decrease
0
2
0.4
0.4
0
4
1 0.666666667
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Final
Shadow
Constraint Allowable
Value
Price
R.H. Side Increase
1500 0.666666667
1500
300
6000
0.2
6000
1500
44000
0
16000
28000
Allowable
Decrease
300
1000
1E+30
84
Dual Prices (RHS Coefficients)
>Amount objective function will improve
per unit increase in constraint RHS value
>Reflects value of an additional unit of
resource (if resource cost is sunk);
reflects extra value over normal cost of
resource (when resource cost is
relevant)
>Always 0 for nonbinding constraint
(positive slack or surplus at optimal
solution)
© Copyright 2004, Alan Marshall
85
Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Adjustable Cells
Final
Cell
Name
Value
$B$3 DecVar 40-lb
500
$C$3 DecVar 80-lb
300
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase
Decrease
0
2
0.4
0.4
0
4
1 0.666666667
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Final
Shadow
Constraint Allowable
Value
Price
R.H. Side Increase
1500 0.666666667
1500
300
6000
0.2
6000
1500
44000
0
16000
28000
Allowable
Decrease
300
1000
1E+30
86
Sensitivity Report
Microsoft Excel 8.0e Sensitivity Report
Adjustable Cells
Final
Cell
Name
Value
$B$3 DecVar 40-lb
500
$C$3 DecVar 80-lb
300
Reduced
Cost
Objective Allowable Allowable
Coefficient Increase
Decrease
0
2
0.4
0.4
0
4
1 0.666666667
Constraints
Cell
Name
$D$8 MachTime
$D$9 PackMat
$D$7 MinProd'n
© Copyright 2004, Alan Marshall
Final
Shadow
Constraint Allowable
Value
Price
R.H. Side Increase
1500 0.666666667
1500
300
6000
0.2
6000
1500
44000
0
16000
28000
Allowable
Decrease
300
1000
1E+30
87
RHS Ranges
>As long as the constraint RHS
coefficient stays within this range, the
associated dual price is valid
>For changes outside this range, must
resolve
© Copyright 2004, Alan Marshall
88
Shadow vs Dual Prices
>Shadow Price: Amount objective
function will change per unit increase
in RHS value of constraint
>For maximization problems, dual
prices and shadow prices are the
same
>For minimization problems, shadow
prices are the negative of dual prices
© Copyright 2004, Alan Marshall
89
Next Class
>We will look at the two handout
exercises
• To be posted on the website, with
solution files
>Decision Theory
>In Lecture 3, we will do additional
problems in both Linear Programming
and Decision Theory
© Copyright 2004, Alan Marshall
90