Linear Optimization Models
Chapter 11
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Introduction
• Optimization problems:
• Can be used to support and improve managerial decision making
• Maximize or minimize some function, called the objective function, and
have a set of restrictions known as constraints
• Can be linear or nonlinear
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2
Introduction
• Typical applications:
• A manufacturer wants to develop a production schedule and an inventory policy
that will satisfy demand in future periods and at the same time minimize the total
production and inventory costs
• A financial analyst would like to establish an investment portfolio from a variety of
stock and bond investment alternatives that maximizes the return on investment
• A marketing manager wants to determine how best to allocate a fixed advertising
budget among alternative advertising media such as web, radio, television,
newspaper, and magazine that maximizes advertising effectiveness
• A company had warehouses in a number of locations. Given specific customer
demands, the company would like to determine how much each warehouse
should ship to each customer so that total transportation costs are minimized
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3
Introduction
• Linear optimization models are also known as linear programs
• Linear programming:
• A problem-solving approach developed to help managers make better
decisions
• Numerous applications in today’s competitive business environment
• For instance, GE Capital uses linear programming to help determine optimal
lease structuring
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4
A Simple Maximization Problem
Problem Formulation
Mathematical Model for the Par, Inc. Problem
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A Simple Maximization Problem
• Illustration:
• Par, Inc. - A small manufacturer of golf equipment and supplies
• Management has decided to move into the market for medium- and
high-priced golf bags
• Par’s distributor to buy all the produced bags by the end of third
month
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6
A Simple Maximization Problem
• Operations involved in manufacturing a golf bag:
•
•
•
•
Cutting and dyeing the material
Sewing
Finishing (inserting umbrella holder, club separators, etc.)
Inspection and packaging
• Table 11.1: Production Requirements Per Golf Bag
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7
A Simple Maximization Problem
• Estimated total time available for the next three months to perform
different operations:
Department
Number of hours
Cutting and Dyeing
630
Sewing
600
Finishing
708
Inspection and Packaging
135
• Required profit contribution:
• Standard bag: $10/unit
• Deluxe bag: $9/unit
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8
A Simple Maximization Problem
• Develop a mathematical model of the Par, Inc. problem to determine
the number of standard bags and the number of deluxe bags to
produce to maximize total profit contribution
• Problem Formulation
• Problem formulation or modeling: Process of translating the verbal
statement of a problem into a mathematical statement (or model)
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9
A Simple Maximization Problem
• General guidelines for problem formulation:
•
•
•
•
•
•
Understand the problem thoroughly
Describe the objective
Describe each constraint
Define the decision variables
Write the objective in terms of the decision variables
Write the constraints in terms of the decision variables
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A Simple Maximization Problem
• Describe each constraint
Constraint
Constraint
1
Number of hours of cutting and dyeing time used must be
less than or equal to the number of hours of cutting and
dyeing time available.
2
Number of hours of sewing time used must be less than or
equal to the number of hours of sewing time available.
3
Number of hours of finishing time used must be less than or
equal to the number of hours of finishing time available.
4
Number of hours of inspection and packaging time used
must be less than or equal to the number of hours of
inspection and packaging time available.
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11
A Simple Maximization Problem
• Define the decision variables
• S = number of standard bags
• D = number of deluxe bags
• Write the objective in terms of the decision variables
• If Par makes $10 for every standard and $9 for every deluxe bag,
Total profit contribution = 10S + 9D = Objective function
• Objective: Max 10S + 9D
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12
A Simple Maximization Problem
• Write the constraints in terms of the decision variables
Hours of cutting and
Hours of cutting and
• Constraint 1:
≤
dyeing time used
dyeing time available
7
S + 1D ≤ 630
10
Hours of sewing
Hours of sewing
• Constraint 2:
≤
time used
time available
1 5
S + D ≤ 600
2 6
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A Simple Maximization Problem
Hours of finishing
Hours of finishing
• Constraint 3:
≤
time used
time available
2
1S + D ≤ 708
3
Hours of inspection and
Hours of inspection and
• Constraint 4:
≤
packaging time used
packaging time available
1
1
S + D ≤ 135
10 4
• Nonnegativity constraints—based on the fact that the number of standard
or deluxe bags produced cannot be negative
S ≥ 0 and D ≥ 0 or S, D ≥ 0
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A Simple Maximization Problem
Mathematical Model for the Par, Inc. Problem
Mathematical model: a set of mathematical relationships
Max 10S + 9D
7
subject to(s.t)
S + 1D ≤ 630 Cutting and dyeing
10
1 5
S + D ≤ 600 Sewing
2 6
2
1S + D ≤ 708 Finishing
3
1
1
S + D ≤ 135 Inspection and packaging
10 4
S, D ≥ 0
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A Simple Maximization Problem
Mathematical Model for the Par, Inc. Problem (contd.)
• This is a linear programming model (or linear program) because the
objective function and all constraint functions are linear functions of
the decision variables
• Linear function: Mathematical function in which each variable
appears in a separate term and is raised to the first power
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Solving the Par, Inc. Problem
The Geometry of the Par, Inc. Problem
Solving Linear Programs with Excel Solver
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Solving the Par, Inc. Problem
• To find the optimal solution to the problem modeled as a linear
program:
• The optimal solution must have the highest objective function value
• The optimal solution must be a feasible solution—a setting of the
decision variables that satisfies all of the constraints of the problem
• Search over the feasible region—a set of all possible solutions
• Find the solution that gives the best objective function value
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Solving the Par, Inc. Problem
The Geometry of the Par, Inc. Problem
• When only two decision variables, the functions of variables are
linear
• If constraints are inequalities, the constraint cuts the space in two
• The line and the area on one side of the line is the space the satisfies that
constraint
• These subregions are called half spaces
• The intersection of the half spaces make up the feasible region
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Figure 11.1: Feasible Region for the Par, Inc. Problem
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20
Figure 11.2:
The Optimal Solution to the Par, Inc. Problem
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Solving the Par, Inc. Problem
• Based on the geometry of Figure 11.2, to solve a linear optimization
problem we only have to search the extreme points of the feasible
region to find the optimal solution
• Extreme points are found where constraints intersect on the
boundary of the feasible region
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Solving the Par, Inc. Problem
Solving Linear Programs with Excel Solver
• The first step is to construct the relevant what-if model
• A what-if model for optimization allows the user to try different values of
the decision variables and see:
• Whether that trial solution is feasible
• The value of the objective function for that trial solution
• Convey to Excel Solver the structure of the linear optimization model
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Figure 11.3: What-If Spreadsheet Model for Par, Inc.
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24
Figure 11.4: Solver Dialog Box and Solution to the Par,
Inc. Problem
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25
Solving the Par, Inc. Problem
• The optimal solution:
• To make 540 Standard bags and 252 Deluxe bags for a profit of $7,668
• Using all the cutting and dyeing time as well as all finishing time, from
cells B19:B22 compared to C19:C22
• The results are consistent with the results obtained in Figures 11.1
and 11.2
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26
Figure 11.5: The Solver Answer Report for the Par, Inc.
Problem
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27
Solving the Par, Inc. Problem
• A binding constraint is one that holds as an equality at the optimal
solution
• The slack value for each less-than-or-equal-to constraint indicates
the difference between the left-hand and right-hand values for a
constraint
• By adding a nonnegative slack variable, we can make the constraint
equality
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A Simple Minimization Problem
Problem Formulation
Solution for the M&D Chemicals Problem
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A Simple Minimization Problem
• Illustration:
• Production requirements for M&D Chemicals:
• The combined production for products A and B must total at least 350
gallons
• Separately a major customer’s order for 125 gallons of product A must
also be satisfied
• Processing time:
• Product A: 2 hours/gallon
• Product B: 1 hour/gallon
• For the coming month, 600 hours of processing time are available
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30
A Simple Minimization Problem
• Production cost: Product A: $2/gallon; Product B: $3/gallon
• Objective: Minimizing the total production cost
Problem Formulation
• To find the minimum-cost production schedule:
• Define the decision variables and the objective function
Let A = number of gallons of product A
B = number of gallons of product B
• Objective function = 2A + 3B
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31
A Simple Minimization Problem
• Linear program for the M&D Chemicals problem:
Min 2A + 3B
s.t.
1A
≥ 125 Demand for product A
1A + 1B ≥ 350 Total production
2A + 1B ≤ 600 Processing time
A, B ≥ 0
• A surplus variable tells how much over the right-hand side the lefthand side of a greater-than-or-equal-to constraint is for a solution
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32
Figure 11.6: Solver Dialog Box and Solution to the M&D
Chemical Problem
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33
Figure 11.7: The Solver Answer Report for the M&D
Chemicals Problem
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34
Special Cases of Linear Program
Outcomes
Alternative Optimal Solutions
Infeasibility
Unbounded
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Special Cases of Linear Program Outcomes
Alternative Optimal Solutions
• Where the optimal objective function contour line coincides with one of
the binding constraint lines on the boundary of the feasible region
• In these situations, more than one solution provides the optimal value
for the objective function
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36
Special Cases of Linear Program Outcomes
• Illustration using the Par, Inc. problem
•
•
•
•
Original objective function: 10S + 9D
Assume the profit for the standard golf bag decreased to $6.30.
Revised objective function: 6.3S + 9D
The optimal solution occurs at two extreme points:
• Extreme point 4 (S = 300, D = 420) and
• Extreme point 3 (S = 540, D = 252)
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37
Figure 11.8: Par, Inc. Problem with an Objective
Function of 6.3S + 9D (Alternative Optimal Solutions)
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38
Special Cases of Linear Program Outcomes
Infeasibility
• Means no solution to the linear programming problem
• No points satisfy all the constraints and the nonnegativity conditions
simultaneously
• Graphically, a feasible region does not exist
• Infeasibility occurs because:
• Management’s expectations are too high
• Too many restrictions have been placed on the problem
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39
Figure 11.9: No Feasible Region for the Par, Inc. Problem with Minimum
Production Requirements of 500 Standard and 360 Deluxe Bags
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40
Special Cases of Linear Program Outcomes
• Interpretation of Infeasibility for the Par, Inc. problem
• Let the management know that the resources available are not sufficient
to make 500 standard bags and 360 deluxe bags
• Provide details to the management on:
• Minimum amounts of resources that must be available
• The amounts currently available
• Additional amounts that would be required to accomplish this level of
production
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41
Table 11.2: Resources Needed to Manufacture 500
Standard Bags and 360 Deluxe Bags
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42
Special Cases of Linear Program Outcomes
• An infeasible problem when solved in Excel Solver:
• Will return a message indicating that no feasible solutions exists—indicating
no solution to the linear programming problem will satisfy all constraints
• Careful inspection is necessary to identify why the problem is infeasible
• One of the approaches is to drop one or more constraints and re-solve the
problem
• If we find an optimal solution for this revised problem, then the constraint(s)
that were omitted, in conjunction with the others, are causing the problem
to be infeasible
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43
Special Cases of Linear Program Outcomes
Unbounded
• The situation in which the value of the solution
• May be made infinitely large—for a maximization linear programming
• May be made infinitely small—for a minimization linear programming
• Without violating any of the constraints
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44
Special Cases of Linear Program Outcomes
• Illustration:
Consider the following linear program with two decision variables,
X and Y:
Max 20X + 10Y
s.t.
1X
≥2
1Y ≤ 5
X, Y ≥ 0
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45
Figure 11.10: Example of an Unbounded Problem
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46
Special Cases of Linear Program Outcomes
• Solving an unbounded problem using Excel Solver:
• Returns a message “Objective Cell values do not converge”
• In linear programming models of real problems:
• The occurrence of an unbounded solution means that the problem
has been improperly formulated
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Sensitivity Analysis
Interpreting Excel Solver Sensitivity Report
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Sensitivity Analysis
• Sensitivity analysis: The study of how the changes in the input
parameters of an optimization model affect the optimal solution
• It helps in answering the questions:
• How will a change in a coefficient of the objective function affect the
optimal solution?
• How will a change in the right-hand-side value for a constraint affect
the optimal solution?
• The shadow price for a constraint is the change in the optimal
objective function value if the right-hand side of that constraint is
increased by one
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Sensitivity Analysis
Interpreting Excel Solver Sensitivity Report
• Consider the M&D chemicals problem:
Min
s.t.
A = number of gallons of product A
B = number of gallons of product B
2A + 3B
1A
≥ 125
1A + 1B ≥ 350
2A + 1B ≤ 600
A, B ≥ 0
Demand for product A
Total production
Processing time
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50
Figure 11.11: Solver Sensitivity Report for the M&D
Chemicals Problem
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51
Sensitivity Analysis
• Classical sensitivity analysis:
• Based on the assumption that only one piece of input data has changed
• It is assumed that all other parameters remain as stated in the original
problem
• When interested in what would happen if two or more pieces of
input data are changed simultaneously:
• The easiest way to examine the effect of simultaneous changes is to make
the changes and rerun the model
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52
General Linear Programming
Notation and More Examples
Investment Portfolio Selection
Transportation Planning
Advertising Campaign Planning
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General Linear Programming Notation and
More Examples
• The general notation for linear programs uses the letter x with a subscript
• In the Par, Inc. problem the decision variables could be denoted as:
• 𝑥1 = number of standard bags
• 𝑥2 = number of deluxe bags
• Advantage: Formulating a mathematical model for a problem that
involves a large number of decision variables is much easier
• Disadvantage: Not being able to easily identify what the decision variables
actually represent in the mathematical model
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54
General Linear Programming Notation and
More Examples
• Par, Inc. model using the general notation:
Max 10𝑥1 + 9𝑥2
s.t.
7
𝑥 + 1 𝑥2 ≤ 630
10 1
1
5
𝑥 + 𝑥 ≤ 600
2 1 6 2
2
1 𝑥1 + 𝑥2 ≤ 708
3
1
1
𝑥 + 𝑥 ≤ 135
10 1 4 2
𝑥1 , 𝑥2 ≥ 0
Cutting and dyeing
Sewing
Finishing
Inspection and packaging
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55
General Linear Programming Notation and
More Examples
Investment Portfolio Selection
• Portfolio selection problems involve situations in which a financial
manager must select specific investments—for example, stocks and
bonds—from a variety of investment alternatives
• Objective: Maximization of expected return or minimization of risk
• Constraints: Restrictions on the type of permissible investments,
state laws, company policy, and so on
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56
General Linear Programming Notation and
More Examples
Illustration
Table 11.3: Investment Opportunities for
Welte Mutual Funds
• Welte Mutual Funds, Inc.,
located in New York City, is
looking for investment
opportunities for $100,000
• The firm’s top financial analyst
identified five investment
opportunities and projected
their annual rates of return
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57
General Linear Programming Notation and
More Examples
• Welte investment guidelines:
• Neither industry (oil or steel) should receive more than $50,000
• Amount invested in government bonds should be at least 25 percent of
the steel industry investments
• The investment in Pacific Oil, the high-return but high-risk investment,
cannot be more than 60 percent of the total oil industry investment
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58
General Linear Programming Notation and
More Examples
• Define the following decision variables:
𝑋1 = dollars invested in Atlantic Oil
𝑋2 = dollars invested in Pacific Oil
𝑋3 = dollars invested in Midwest Steel
𝑋4 = dollars invested in Huber Steel
𝑋5 = dollars invested in government bonds
• Specify the objective: Maximizing return
Max 0.073𝑋1 + 0.103 𝑋2 + 0.064 𝑋3 + 0.075 𝑋4 + 0.045 𝑋5
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59
General Linear Programming Notation and
More Examples
• Define the constraints:
Constraint 1: 𝑋1 + 𝑋2 + 𝑋3 + 𝑋4 + 𝑋5 = 100,000
Constraint 2: 𝑋1 + 𝑋2 ≤ 50,000
𝑋3 + 𝑋4 ≤ 50,000
Constraint 3: 𝑋5 ≥ 0.25(𝑋3 + 𝑋4 )
Constraint 4: 𝑋2 ≤ 0.60(𝑋1 + 𝑋2 )
Nonnegativity constraints: 𝑋1 , 𝑋2 , 𝑋3 , 𝑋4 , 𝑋5 ≥ 0
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General Linear Programming Notation and
More Examples
• Linear programming model for the Welte Mutual Funds investment
problem:
Max 0.073𝑋1 + 0.103 𝑋2 + 0.064 𝑋3 + 0.075 𝑋4 + 0.045 𝑋5
s.t.
𝑋1 + 𝑋2 + 𝑋3 + 𝑋4 + 𝑋5 = 100,000 Available funds
𝑋1 + 𝑋2
≤ 50,000 Oil industry maximum
𝑋3 + 𝑋4
≤ 50,000 Steel industry maximum
𝑋5 ≥ 0.25(𝑋3 + 𝑋4 )
Government bonds minimum
𝑋2 ≤ 0.60(𝑋1 + 𝑋2 )
Pacific Oil restriction
𝑋1 , 𝑋2 , 𝑋3 , 𝑋4 , 𝑋5 ≥ 0
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61
Figure 11.12:
The Solution for the Welte Mutual Funds Problem
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62
General Linear Programming Notation and
More Examples
Transportation Planning
• Transportation problem arises in planning for the distribution of goods
and services from several supply locations to several demand locations
• Quantity of goods available at each supply location (origin) is limited
• Quantity of goods needed at each of several demand locations
(destinations) is known
• Objective: Minimize the cost of shipping goods from the origins to the
destinations
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63
General Linear Programming Notation and
More Examples
• Illustration using Foster Generators problem
• Involves the transportation of a product from three plants to four
distribution centers
• To determine how much of its production should be shipped from
each plant to each distribution center
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64
General Linear Programming Notation and
More Examples
• Production capacities over the next three-month planning period for one type of
generator:
• The three-month forecast of demand for the distribution centers:
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65
Figure 11.13: The Network Representation of the Foster
Generators Transportation Problem
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66
General Linear Programming Notation and
More Examples
• Objective is to determine:
• Routes to be used
• Quantity to be shipped via each route
• Minimum total transportation cost
• Let xij = number of units shipped from origin i to destination j
where i = 1, 2, . . . , m and j = 1, 2, . . . , n
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67
Table 11.4: Transportation Cost Per Unit for the Foster
Generators Transportation Problem ($)
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68
General Linear Programming Notation and
More Examples
• Supply constraints
x11 + x12 + x13 + x14 ≤ 5000 Cleveland supply
x21 + x22 + x23 + x24 ≤ 6000 Bedford supply
x31 + x32 + x33 + x34 ≤ 2500 York supply
• Demand constraints
x11 + x21 + x31 = 6000 Boston demand
x12 + x22 + x32 = 4000 Chicago demand
x13 + x23 + x33 = 2000 St. Louis demand
x14 + x24 + x34 = 1500 Lexington demand
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69
General Linear Programming Notation and
More Examples
• A 12-variable, 7-constraint linear programming formulation of the Foster Generators
transportation problem:
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70
Figure 11.14: Spreadsheet Model and Solution for the
Foster Generator Problem
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71
General Linear Programming Notation and
More Examples
Advertising Campaign Planning:
• Designed to help marketing managers allocate a fixed advertising
budget to various advertising media
• Objective: Maximize reach, frequency, and quality of exposure
• Restrictions: Company policy, contract requirements, and media
availability
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72
General Linear Programming Notation and
More Examples
• Illustration: Relax-and-Enjoy Lake Development Corporation:
• Developing a lakeside community at a privately owned lake
• Primary market includes all middle- and upper-income families within
approximately 100 miles of the development
• Employed the advertising firm of Boone, Phillips, and Jackson (BP&J) to
design the promotional campaign
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73
Table 11.5: Advertising Media Alternatives for the Relax-andEnjoy Lake Development Corporation
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74
General Linear Programming Notation and
More Examples
• Problem Formulation:
• Budget: $30,000
• Restrictions imposed:
• At least 10 television commercials must be used
• At least 50,000 potential customers must be reached
• No more than $18,000 may be spent on television advertisements
• The decision to be made is how many times to use each medium
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75
General Linear Programming Notation and
More Examples
• Define the decision variables:
• DTV = number of times daytime TV is used
• ETV = number of times evening TV is used
• DN = number of times daily newspaper is used
• SN = number of times Sunday newspaper is used
•
R = number of times radio is used
• Objective: Maximizing the total exposure quality units for the
overall media selection plan
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76
General Linear Programming Notation and
More Examples
• Linear programming model for the Relax-and-Enjoy advertising
campaign planning problem:
Max 65DTV + 90ETV + 40DN + 60SN + 20R
DTV
Exposure quality
≤ 15
ETV
≤ 10
DN
≤ 25
SN
Availability of media
≤ 4
R
≤ 30
1500DTV + 3000ETV + 400DN + 1000SN +100R ≤ 30,000
Budget
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77
General Linear Programming Notation and
More Examples
• Linear programming model for the Relax-and-Enjoy advertising
campaign planning problem (contd.):
DTV +
ETV
≥
1500DTV + 3000ETV
≤
1000DTV + 2000ETV + 1500DN + 2500SN + 300R ≥
10 Television
18,000 restrictions
50,000 Customers reached
DTV, ETV, DN, SN, R ≥ 0
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78
Figure 11.15: A Spreadsheet Model and the Solution for the
Relax-and-Enjoy Lake Development Corporation Problem
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79
Figure 11.16: The Excel Sensitivity Report for the Relaxand-Enjoy Lake Development Corporation Problem
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80
Generating an Alternative
Optimal Solution for a Linear
Program
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Generating an Alternative Optimal Solution
for a Linear Program
• Illustration: Consider the Foster Generators transportation
problem
• From Figure 11.14, the optimal solution:
x11 = 1000, x12 = 4000, x13 = 0, x14 = 0
x21 = 2500, x22 = 0, x23 = 2000, x24 = 1500
x31 = 2500, x32 = 0, x33 = 0, x34 = 0
• Optimal cost: $39,500
• For the revised model to be optimal, the solution must give a total
cost of $39,500
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82
Generating an Alternative Optimal Solution
for a Linear Program
• From Figure 11.14:
x13 = x14 = x22 = x32 = x33 = x34 = 0
• If the sum of these variables is maximized and if the optimal
objective function value of the revised problem is positive
• A different feasible solution that is also optimal is found
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83
Generating an Alternative Optimal Solution
for a Linear Program
• Revised model
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84
Table 11.6: An Alternative Optimal Solution to the
Foster Generators Transportation Problem
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85
Generating an Alternative Optimal Solution
for a Linear Program
• In the original solution (Figure 11.14):
• Boston distribution center is sourced from all three plants, whereas each of the other
distribution centers is sourced by one plant
• Hence, the manager in the Boston distribution center has to deal with three different plant
managers, whereas each of the other distribution center managers has only one plant
manager
• The alternative solution (Table 11.6) provides a more balanced solution
• Managers in Boston and Chicago each deal with two plants, and those in St. Louis and
Lexington, which have lower total volumes, deal with only one plant
• Because the alternative solution seems to be more equitable, it might be preferred
• Both the solutions give a total cost of $39,500
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86
Generating an Alternative Optimal Solution
for a Linear Program
• General approach to find an alternative optimal solution to a linear
program:
• Step 1: Solve the linear program
• Step 2: Make a new objective function to be maximized; It is the sum of
those variables that were equal to zero in the solution from Step 1
• Step 3: Keep all the constraints from the original problem; add a constraint
that forces the original objective function to be equal to the optimal
objective function value from Step 1
• Step 4: Solve the problem created in Steps 2 and 3; if the objective function
value is positive, an alternative optimal solution is found
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87