DECISION MODELING WITH
MICROSOFT EXCEL
Chapter 5
LINEAR OPTIMIZATION:
APPLICATIONS
Part 1
Copyright 2001
Prentice Hall
Introduction
Several specific models (which can be used as
templates for real-life problems) will be examined in
this chapter. These models include:
TRANSPORTATION MODEL
Management must determine how to send
products from various sources to various
destinations in order to satisfy requirements at
the lowest possible cost.
ASSIGNMENT MODEL
Allows management to investigate allocating
fixed-sized resources to determine the optimal
assignment of salespeople to districts, jobs to
machines, tasks to computers …
MEDIA SELECTION MODEL
This model is concerned with designing an
effective advertising campaign.
DYNAMIC (MULTIPERIOD) MODEL
These are models in which coordinated
decision making must occur over more than
one time period.
FINANCIAL AND PRODUCTION PLANNING
These business models illustrate the joint
optimization of both production and financial
resources.
NETWORK MODELS
These models involve the movement or
assignment of physical entities (e.g., money).
The Transportation Model
In this example, the AutoPower Company makes a
variety of battery and motorized uninterruptible
electric power supplies (UPS’s).
AutoPower has 4 final assembly plants in Europe
and the diesel motors used by the UPS’s are
produced in the US, shipped to 3 harbors and then
sent to the assembly plants.
Production plans for the third quarter (July – Sept.)
have been set. The requirements (demand at the
destination) and the available number of motors at
harbors (supply at origins) are shown on the next
slide:
Assembly Plant
(1) Leipzig
(2) Nancy
(3) Liege
(4) Tilburg
No. of Motors Required
400
900
200
500
2000
Supply
Harbor
(A) Amsterdam
(B) Antwerp
(C) Le Havre
No. of Motors Available
500
700
800
2000
Balanced
Demand
Graphical presentation of Supply and Demand:
500
Amsterdam (A)
500
Tilburg (4)
400
Leipzig (1)
700
Antwerp (B)
Liege (3)
200
800
Le Havre (C)
900
Nancy (2)
The Transportation Model
AutoPower must decide how many motors to send
from each harbor (supply) to each plant (demand).
The cost ($, on a per motor basis) of shipping is
given below.
FROM ORIGIN
(A) Amsterdam
TO DESTINATION
Leipzig Nancy
Liege
(1)
(2)
(3)
Tilburg
(4)
120
130
41
59.50
(B) Antwerp
61
40
100
110
(C) Le Havre
102.50
90
122
42
The goal is to minimize total transportation cost.
Since the costs in the previous table are on a per
unit basis, we can calculate total cost based on the
following matrix (where xij represents the number of
units that will be transported from Origin i to
Destination j):
TO DESTINATION
FROM ORIGIN
1
2
3
4
A
120xA1
B
61xB1
C
130xA2
41xA3 59.50xA4
40xB2 100xB3
102.50xC1 90xC2
122xC3
110xB4
42xC4
Total Transportation Cost =
120xA1 + 130xA2 + 41xA3 + … + 122xC3 + 42xC4
The model has two general types of constraints.
1. The number of items shipped from a harbor
cannot exceed the number of items available.
A constraint is required for each origin that
describes the total number of units that can be
shipped.
For Amsterdam: xA1 + xA2 + xA3 + xA4 < 500
For Antwerp:
xB1 + xB2 + xB3 + xB4 < 700
For Le Havre:
xC1 + xC2 + xC3 + xC4 < 800
Note: We could have used an “=“ instead of
“<“ since supply and demand are balanced for
this model. However, the supply inequality
constraints will be binding at optimality giving
the same effect.
2. Demand at each plant must be satisfied.
A constraint is required for each destination
that describes the total number of units
demanded.
For Leipzig: xA1 + xB1 + xC1 > 400
For Nancy:
xA2 + xB2 + xC2 > 900
For Liege:
For Tilburg:
xA3 + xB3 + xC3 > 200
xA4 + xB4 + xC4 > 500
Note: We could have used an “=“ instead of
“>“ since supply and demand are balanced for
this model. However, the demand inequality
constraints will be binding at optimality giving
the same effect.
Here is the spreadsheet model using Excel
and solved with Solver:
=SUM(C9:C11)
= C4*C9
=SUM(C16:C18)
=SUM (C9:F9)
=SUM (C16:F16)
Here is the Sensitivity Report from Solver for the
Transportation Model:
Variations on the Transportation Model
Solving Max Transportation Models
Suppose we now want to maximize the value of the
objective function instead of minimizing it.
In this case, we would use the same model, but now
the objective function coefficients define the
contribution margins (i.e., unit returns) instead of
unit costs.
In the Solver dialog, you would check the Max radio
button before solving the problem.
Additionally, your interpretation of Solver’s
Sensitivity Report would reflect the maximization of
the objective function.
Variations on the Transportation Model
When Supply and Demand Differ
When supply and demand are not equal, then the
problem is unbalanced. There are two situations:
When supply is greater than demand:
In this case, when all demand is satisfied, the
remaining supply that was not allocated at
each origin would appear as slack in the
supply constraint for that origin.
Using inequalities in the constraints (as in the
previous example) would not cause any
problems in Solver.
Variations on the Transportation Model
When demand is greater than supply:
In this case, the LP model has no feasible
solution. However, there are two approaches
to solving this problem:
1. Rewrite the supply constraints to be
equalities and rewrite the demand
constraints to be < .
Unfulfilled demand will appear as slack on
each of the demand constraints when
Solver optimizes the model.
Variations on the Transportation Model
2. Revise the model to append a placeholder
origin, called a dummy origin, with supply
equal to the difference between total
demand and total supply.
The purpose of the dummy origin is to
make the problem balanced (total supply =
total demand) so that Solver can solve it.
The cost of supplying any destination from
this origin is zero.
Once solved, any supply allocated from
this origin to a destination is interpreted as
unfilled demand.
Variations on the Transportation Model
Eliminating Unacceptable Routes
Certain routes in a transportation model may be
unacceptable due to regional restrictions, delivery
time, etc.
In this case, you can assign an arbitrarily large unit
cost number (identified as M) to that route.
Choose M such that it will be larger than any other
unit cost number in the model.
This will force Solver to eliminate the use of that
route since the cost of using it would be much larger
than that of any other feasible alternative.
Variations on the Transportation Model
Integer Valued Solutions
Generally, LP models do not produce integer
solutions.
The exception to this is the Transportation model. In
general:
If all of the supplies and demands in a
transportation model have integer values,
the optimal values of the decision variables
will also have integer values.
Variations on the Transportation Model
Using Alternative Optima to Achieve
Multiple Objectives
Zeros in the Allowable Increase/Decrease columns
for objective coefficients in the Sensitivity Report
indicate that there are alternative optimal solutions.
Using the AutoPower example, examine the effects
of such occurrences.
Suppose that due to a potential trucker’s strike, you
need to find a cheaper transportation schedule that
also minimizes the cost of shipping motors out of
Le Havre harbor. You would need to shift costs
away from Le Havre to reduce AutoPower’s risk.
In this case, the presence of alternative optima
would help avoid some of the risk without increasing
total costs.
From the previous solution, we find that there are an
infinite number of alternative optima that produce a
minimal cost of $121,450.
So, the original objective
can then be recast as an
additional total cost
constraint, thereby
allowing Solver to be
given a new OV to
minimize.
Here is the modified spreadsheet model.
Note the additional constraint $G$19 < $H$19.
Note that the new solution provides feasible
alternatives (no more costly than the original
solution), while minimizing Le Havre’s total costs (a
shift of $18,000 to other routes).
The Assignment Model
In general, the Assignment model is the problem of
determining the optimal assignment of n
“indivisible” agents or objects to n tasks.
For example, you might want to assign
Salespeople to sales territories
Service representatives to service calls
Consultants to clients
Lawyers to cases
Computers to networks
Commercial artists to advertising copy
The important constraint is that each person or
machine be assigned to one and only one task.
The Assignment Model
AutoPower Europe’s Auditing Problem
We will use the AutoPower example to illustrate
Assignment problems.
AutoPower’s European headquarters is in Brussels.
This year, each of the four corporate vice-presidents
will visit and audit one of the assembly plants in
June. The plants are located in:
Leipzig, Germany
Nancy, France
Liege, Belgium
Tilburg, the Netherlands
The issues to consider in assigning the different
vice-presidents to the plants are:
1. Matching the vice-presidents’ areas of
expertise with the importance of specific
problem areas in a plant.
2. The time the management audit will require
and the other demands on each vicepresident during the two-week interval.
3. Matching the language ability of a vicepresident with the plant’s dominant language.
Keeping these issues in mind, first estimate the
(opportunity) cost to AutoPower of sending each
vice-president to each plant.
The following table lists the assignment costs in
$000s for every vice-president/plant combination.
PLANT
Nancy
Liege
(2)
(3)
V.P.
Leipzig
(1)
Tilburg
(4)
Finance (F)
24
10
21
11
Marketing (M)
14
22
10
15
Operations (O)
15
17
20
19
Personnel (P)
11
19
14
13
To determine total cost, make the assignment and
then add up the costs associated with the
assignment.
For example, consider the following assignment:
PLANT
Nancy
Liege
(2)
(3)
V.P.
Leipzig
(1)
Tilburg
(4)
Finance (F)
24
10
21
11
Marketing (M)
14
22
10
15
Operations (O)
15
17
20
19
Personnel (P)
11
19
14
13
Total cost = 24 + 22 + 20 + 13 = 79
The question is, is this the least cost assignment?
The Assignment Model
Solving by Complete Enumeration
Complete enumeration is the calculation of the total
cost of each feasible assignment pattern in order to
pick the assignment with the lowest total cost.
This is not a problem when there are only a few rows
and columns (e.g., vice-presidents and plants).
However, complete enumeration can quickly become
burdensome as the model grows large.
For example, determine the number of alternatives
in the AutoPower (4x4) model. Consider assigning
the vice-presidents in the order F, M, O, P.
1. F can be assigned to any of the 4 plants.
2. Once F is assigned, M can be assigned to any
of the remaining 3 plants.
3. Now O can be assigned to any of the
remaining 2 plants.
4. P must be assigned to the only remaining
plant.
There are 4 x 3 x 2 x 1 = 24 possible solutions.
In general, if there are n rows and n columns, then
there would be n(n-1)(n-2)(n-3)…(2)(1) = n!
(n factorial) solutions. As n increases, n! increases
rapidly. Therefore, this may not be the best method.
The Assignment Model
The LP Formulation and Solution
For this model, let
xij = number of V.P’s of type i assigned to plant j
where i = F, M, O, P
j = 1, 2, 3, 4
Notice that this model is balanced since the total
number of V.P.’s is equal to the total number of
plants.
Remember, only one V.P. (supply) is needed at each
plant (demand).
Here is the spreadsheet model using Excel
and solved with Solver:
=SUM(C10:C13)
=SUM (C10:F10)
= C4*C10
=SUM (C18:F18)
=SUM(C18:C21)
As a result, the optimal assignment is:
PLANT
Nancy
Liege
(2)
(3)
V.P.
Leipzig
(1)
Finance (F)
24
10
21
11
Marketing (M)
14
22
10
15
Operations (O)
15
17
20
19
Personnel (P)
11
19
14
13
Total Cost ($000’s) = 10 + 10 + 15 + 13 = 48
Tilburg
(4)
The Assignment Model
Relation to the Transportation Model
The Assignment model is similar to the
Transportation model with the exception that supply
cannot be distributed to more than one destination.
In the Assignment model, all supplies and demands
are one, and hence integers. Thus, Solver will not
produce any fractional allocations.
As a result, in the Solver solution, each decision
variable cell will either contain a 0 (no assignment)
or a 1 (assignment made).
In general, the assignment model can be formulated
as a transportation model in which the supply at
each origin and the demand at each destination = 1.
The Assignment Model
Unequal Supply and Demand:
The Auditing Problem Reconsidered
Case 1: Supply Exceeds Demand
In this example, suppose the company President
decides to audit the plant in Tilburg. Now there are 4
V.P.’s to assign to 3 plants.
Here is the cost (in $000s) matrix for this scenario:
V.P.
F
M
O
P
No. of V.P.s
Required
1
24
14
15
11
PLANT
2
3
10
21
22
10
17
20
19
14
1
1
1
NUMBER OF V.P.s
AVAILABLE
1
1
1
1
4
3
To formulate this model, simply drop the constraint
that required a V.P. at plant 4 and Solve:
Note that one of the
V.P.s has not been
assigned to a plant.
The Assignment Model
Unequal Supply and Demand:
The Auditing Problem Reconsidered
Case 2: Demand Exceeds Supply
In this example, assume that the V.P. of Personnel is
unable to participate in the European audit. Now the
cost matrix is as follows:
V.P.
1
PLANT
2
3
F
M
O
24
14
15
10
22
17
No. of V.P.s
Required
21
10
20
4
11
15
19
NUMBER OF V.P.s
AVAILABLE
1
1
1
3
1
1
1
1
4
Demand > Supply: Adding a Dummy V.P.
In this form, the model is infeasible.
To fix this, you can
1. Modify the inequalities in the constraints
(similar to the Transportation example)
2. Add a dummy V.P. as a placeholder to the
cost matrix (shown below).
V.P.
1
PLANT
2
3
F
M
O
Dummy
24
14
15
0
10
22
17
0
No. of V.P.s
Required
21
10
20
0
4
11
15
19
0
NUMBER OF V.P.s
AVAILABLE
1
1
1
1
4
1
1
Zero cost to assign the dummy
1
1
4
Dummy supply;
now supply = demand
In the solution, the dummy V.P. would be assigned to
a plant. In reality, this plant would not be audited.
The Assignment Model
Maximization Models
In this Assignment model, the response from each
assignment is a profit rather than a cost.
For example, AutoPower must now assign four new
salespeople to three territories in order to maximize
profit.
The effect of assigning any salesperson to a
territory is measured by the anticipated marginal
increase in profit contribution due to the
assignment.
Here is the profit matrix for this model.
SALESPERSON
A
B
C
D
No. of
Salespeople
Required
TERRITORY
1
2
3
40
18
12
25
30
28
16
24
NUMBER OF
SALESPEOPLE
AVAILABLE
20
22
20
27
1
1
1
1
4
1
1
1
3
This value represents the profit contribution if A is
assigned to Territory 3.
Here is the spreadsheet model using Excel
and solved with Solver:
=SUM(C10:C13)
=SUM (C10:E10)
= C4*C10=SUM (C18:E18)
=SUM(C18:C21)
The Assignment Model
Situations with Unacceptable Assignments
Certain assignments in the model may be
unacceptable for various reasons.
In this case, you can assign an arbitrarily large unit
cost (or small unit profit) number to that assignment.
This will force Solver to eliminate the use of that
assignment since, for example, the cost of making
that assignment would be much larger than that of
any other feasible alternative.
The Media Selection Model
Advertising agencies use Media Selection models to
develop effective advertising campaigns.
The basic question that they try to answer is:
How many “insertions” (ads) should the firm
purchase in each of several possible media
(e.g., radio, TV, newspapers, magazines, and Internet
Web pages)?
Constraints on the decision maker are typically:
advertising budget
the number of ads in each media
other “rules of thumb” from management
The Media Selection Model
The law of diminishing returns may also influence
the Media Selection decision. In other words, the
effectiveness of an ad decreases as the number of
exposures in a medium increases during a specified
period of time.
The objective function of this model is unusual.
Conceptually, the model should find the advertising
campaign that maximizes demand and satisfies the
budget and other constraints.
However, the approach most often used is to
measure the response to an ad in a medium in terms
of exposure units.
The Media Selection Model
An exposure unit is a subjective measure based on:
The quality of the ad
The desirability of the potential market
In other words, it is an arbitrary measure of the
“goodness” of an ad.
An exposure unit can be thought of as a kind of
economic utility.
So the goal is to maximize the total exposure units,
taking into account other properties of the model.
The Media Selection Model
Example: Promoting a New Product
The RollOn company has decided to start a new
product line of motorcycle-like machines with three
oversized tires.
An advertising campaign with a budget of $72,000 is
planned for the introductory month. RollOn decides
to use daytime radio, evening TV, and daily
newspaper ads in its advertising campaign.
The cost per ad in each media are given below:
ADVERTISING
MEDIUM
Daytime Radio
Evening TV
Daily Newspaper
NUMBER OF PURCHASING
UNITS REACHED PER AD
30,000
60,000
45,000
COST PER
AD ($)
1700
2800
1200
The Media Selection Model
Total Exposures
Example: Promoting a New Product
RollOn arbitrarily selects a scale from 0 to 100 for
each ad offering.
It is assumed
Total Exposures vs. Num ber of Radio Ads
that each of the
1200
first 10 radio ads
Slope = 40
1000
has a value of 60
800
exposure units,
600
and each radio
Slope = 60
400
ad after the first
200
10 is rated as
0
having 40
0
5
10
15
20
25
exposures.
Num ber of Ads
The previous graph shows that radio adds suffer
from diminishing returns (as evidenced by the
change in slope from 60 to 40).
RollOn subjectively determines that the first radio
ads are more effective than later ones. In addition,
they feel that the same situation will occur with TV
and newspaper ads.
The exposures per ad for each medium are given
below:
ADVERTISING
MEDIUM
Daytime Radio
Evening TV
Daily Newspaper
FIRST 10 ADS
ALL FOLLOWING
ADS
60
80
70
40
55
35
Here is a plot of the total exposures as a function of
the number of ads in each medium.
Total Exposures vs. Num ber of Ads
1400
TV
Total Exposures
1200
55
1000
35
800
600
70
400
Newspaper
Radio
40
80
60
200
0
0
10
20
Num ber of Ads
30
RollOn wants to ensure that the ad campaign will
satisfy the following important criteria:
1. No more than 25 ads per medium
2. A total of 1,800,000 purchasing units must
be reached across all media
3. At least ¼ of the ads must appear on TV
(blending requirement)
Now, to model this Media Selection model as an LP
model, let
x1 = no. of daytime radio ads up to the first 10
y1 = no. of daytime radio ads after the first 10
x2 = no. of evening TV ads up to the first 10
y2 = no. of evening TV ads after the first 10
x3 = no. of newspaper ads up to the first 10
y3 = no. of newspaper ads after the first 10
The objective function is:
Max 60x1 + 40y1 + 80x2 + 55y2 + 70x3 + 35y3
To determine the constraints, remember:
x1 + y1 = total radio ads
No. exposures
x2 + y2 = total TV ads
x3 + y3 = total newspaper ads
Also remember that the total advertising
expenditure cannot exceed $72,000 and the cost
of each radio ad is $1700, each TV ad is $2800
and each newspaper ad is $1200. Therefore, the
total expenditure constraint is:
1700x1 + 1700y1 + 2800x2 + 2800y2 + 1200x3 +
1200y3 < 72,000
The constraints are:
Cost per ad
Total advertising expenditure less than $72,000:
1700x1 + 1700y1 + 2800x2 + 2800y2 + 1200x3 +
1200y3 < 72,000
No more than 25 ads in a single medium:
x1 + y1 < 25
x2 + y2 < 25
x3 + y3 < 25
The entire campaign must reach at least 1,800,000
purchasing units:
30,000x1 + 30,000y1 + 60,000x2 + 60,000y2 +
45,000x3 + 45,000y3 > 1,800,000
No. purchasing units per ad
Blending Constraint (at least ¼ of the ads must
appear on event TV) :
x2 + y2
x1 + y1 + x2 + y2 + x3 + y3
>¼
Using this constraint in Excel will produce a
Solver “Conditions for Assume Linear Model are
not Satisfied” error message. You can make this
constraint linear by multiplying out the
denominator:
x2 + y2 > .25(x1 + y1 + x2 + y2 + x3 + y3 )
Here is the Excel spreadsheet model after Solving:
= M3*F5
= C3*C9
Here is=C5*I5
the Solver setup:
= C3*I9
End of Part 1
Please continue to Part 2