### A.1 Formulating Linear Programs

```Welcome to BA 452
Quantitative Analysis
Getting acquainted
What is Quantitative Analysis?
Quantitative Analysis applies linear programming, game
theory, queuing models, simulation, and decision theory to
help managers make decisions. It emphasizes formulating
and solving complex decision problems, and so differs from
anticipating changes in simplified decision problems in
Managerial Economics (BA 445).
BA 452 Lesson A.1 Formulating Linear Programs
1
Welcome to BA 452
Quantitative Analysis
Getting started
Read and bookmark the online course syllabus:
http://faculty.pepperdine.edu/jburke2/ba452/index.htm
It provides review questions for each lesson, and serves as
a contract specifying our obligations to each other. In
particular, note:
• Linear Algebra (solve 2 equations for 2 variables),
Calculus (take a derivative), and Introduction to
Microeconomics are prerequisites, so review as
needed.
• Before each class meeting, download and read the
PowerPoint lesson, as presented under the “Schedule”
link.
• Have a laptop with Management Scientist installed
BA 452 Lesson A.1 Formulating Linear Programs
2
Readings
Readings
For each lesson, you can use either the 12th edition or the
13th edition of the Anderson, Sweeney, Williams, … text.
For the first lesson in Part I (Lesson I.1), read Chapter 1
BA 452 Lesson A.1 Formulating Linear Programs
3
Overview
Overview
BA 452 Lesson A.1 Formulating Linear Programs
4
Overview
Quantitative Analysis applies linear and nonlinear programming, game theory,
queuing models, simulation, and decision theory to help managers make
profitable decisions.
Linear Programming Problems in managerial applications often maximize profit,
which equals revenue from outputs minus cost of inputs. Profit is a linear
function of output and input decision variables.
Portfolio Selection Problems help financial managers select specific investments
(stocks, bonds, …) to generate returns to either maximize expected return or
minimize risk.
Resource Allocation Problems are Linear Programming Profit Maximization
problems when available input resources are fixed. The opportunity cost of
resources define willingness to pay for inputs.
Resource Allocation Problems with Machines help production managers allocate
specific resources to produce goods to either maximize profit or minimize cost.
Machine use is measured in hours.
BA 452 Lesson A.1 Formulating Linear Programs
5
Quantitative Analysis
Quantitative Analysis
BA 452 Lesson A.1 Formulating Linear Programs
6
Quantitative Analysis
Overview
Quantitative Analysis applies linear and nonlinear
programming, game theory, queuing models, simulation,
and decision theory to help managers make profitable
decisions.
BA 452 Lesson A.1 Formulating Linear Programs
7
Quantitative Analysis
The Harris Corporation



Major electronics company in Melbourne, FL.
Developed a computerized optimization-based
production planning system.
Benefits:
• On-time deliveries increased from 75% to 95%.
• Expanded markets and market share.
• Increased profits by \$115 million annually.
BA 452 Lesson A.1 Formulating Linear Programs
8
Quantitative Analysis
KeyCorp



One of the largest bank holding companies in the US
(\$66.8 billion in assets).
Developed a system to measure branch activities,
customer wait times, teller productivity.
Benefits:
• Customer processing time reduced 53%.
• Customer wait time reduced.
• Cost savings of \$98 million over 5 years.
BA 452 Lesson A.1 Formulating Linear Programs
9
Quantitative Analysis
NYNEX



Major telecommunications provider (16.5 million
customers worldwide).
Developed optimization techniques for network planning.
Benefits:
• Improved quality and reliability of network plans.
• Savings of \$33 million.
BA 452 Lesson A.1 Formulating Linear Programs
10
Quantitative Analysis
The definition of a model:


Models are simplified versions of the things they
represent.
A useful model accurately represents the relevant or
essential characteristics of the object or decision being
studied. (Like a model airplane studied in a wind tunnel.)
BA 452 Lesson A.1 Formulating Linear Programs
11
Quantitative Analysis
Good decisions vs. good outcomes:

A structured, modeling approach to decision making
helps make good decisions, but can’t guarantee good
outcomes because of uncertainty (risk).
• Life insurance is often a good decision, even when it
turns out you do not die that year.
• Other examples of good decisions with bad
outcomes?
• Betting your retirement savings on 17 Black in
Roulette is often a bad decision, even if it turns out 17
Black wins.
• Other examples of bad decisions with good
outcomes?
BA 452 Lesson A.1 Formulating Linear Programs
12
Linear Programming
Linear Programming
BA 452 Lesson A.1 Formulating Linear Programs
13
Linear Programming
Overview
Linear Programming Problems in managerial applications
often maximize profit, which equals revenue from outputs
minus cost of inputs. Profit is a linear function of output
and input decision variables, and linear constraints restrict
permissible decision variables. A key lesson of quantitative
analysis is exposure to the variety of profit-maximization
linear-programming problems.
BA 452 Lesson A.1 Formulating Linear Programs
14
Linear Programming
First, profit-maximization linear-programming problems can
vary by whether outputs are fixed or variable, or whether
inputs are fixed or variable:
 In some problems, outputs are fixed (say, customers
made reservations), so revenue is fixed and the
objective of profit maximization reduces to the objective
of cost minimization.
 In other problems, inputs are fixed (say, airlines make
short-run decisions about using their fixed stock of
planes), so cost is fixed and the objective of profit
maximization reduces to the objective of revenue
maximization.
BA 452 Lesson A.1 Formulating Linear Programs
15
Linear Programming
Second, problems can vary by whether available input
resources are fixed or whether additional inputs may be
bought:
 In some problems, available input resources are fixed
(say, firms make short-run decisions about how much
labor to employ, from 0 up to a fixed maximum), so the
problem is how to best allocate those resources to
produce various outputs.
 In other problems, additional inputs may be bought, so
the problem is to balance the productivity of an input and
its cost.
 In still other problems, inputs can be either made or
bought (say, Sony can either make parts for its
televisions or buy parts).
BA 452 Lesson A.1 Formulating Linear Programs
16
Linear Programming
Third, problems can vary by outputs and inputs are defined:
 In some problems, outputs have different physical
characteristics (say, Toyota produces both cars and
trucks).
 In other problems, outputs occur at different periods in
time (say, Toyota produces cars for sale this year, and
cars for sale next year).
 In other problems, outputs occur at different locations
(say, Toyota offers cars for sale in the US, and cars for
sale in Japan).
Likewise, inputs can have different physical characteristics,
occur at different periods in time, and occur at different
locations
BA 452 Lesson A.1 Formulating Linear Programs
17
Linear Programming
Many linear programming applications are interrelated,
according to the following chart. For example, Assignment
is a type of Transportation Problem, which in turn is a type
of Transshipment Problem, which is a type of Resource
Allocation Problem.
Linear Programming
Profit Maximization
Production
Scheduling
Workforce
Assignment
Resource
Allocation
Make or Buy
Blending
Product
Mix
Revenue
Management
Transshipment
Transportation
Shortest Route
Assignment
BA 452 Lesson A.1 Formulating Linear Programs
18
Linear Programming



Linear programming problems have constraints on
pursuing the objective of maximization or minimization.
A feasible solution satisfies all the constraints.
An optimal solution (or optimum) is a feasible solution
that results in the largest possible objective-function
value when maximizing (or smallest when minimizing).
BA 452 Lesson A.1 Formulating Linear Programs
19
Linear Programming



In a linear-programming problem, the objective function
and the constraints are linear.
Functions are linear when each variable appears in a
separate term raised to the first power and is multiplied
by a constant (which could be 0).
• Thus 5x1 + 7x2 is a linear function, but 5x12 +7x1x2 is
not.
Linear constraints (or, standard linear constraints) are
linear functions that are restricted to be "less than or
equal to", "equal to", or "greater than or equal to" a
constant.
• Thus 2x1 + 3x2 < 19 is a linear constraint, but
2x1 + 3x2 < 19 and 2x1 + 3x1x2 < 19 are not.
BA 452 Lesson A.1 Formulating Linear Programs
20
Linear Programming
The three steps to linear programming:
 Formulate the linear programming problem.
 Solve the problem, using either graphical or computer
analysis.
• In BA 452 lectures, homeworks and exams, you will
solve some simple LP problems graphically, for the
purpose of introducing and better understanding the
concepts.
• In BA 452 lectures, homeworks and exams, you will
solve other complex LP problems by any means
possible, including using program and spreadsheet
templates stored on your laptop.
 Interpret the solution.
BA 452 Lesson A.1 Formulating Linear Programs
21
Linear Programming


Problem formulation (or modeling) is the translation of a
verbal statement of a decision problem into a
mathematical statement.
Here are guidelines for linear programming problem
formulation:
• 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.
BA 452 Lesson A.1 Formulating Linear Programs
22
Resource Allocation
Resource Allocation
BA 452 Lesson A.1 Formulating Linear Programs
23
Resource Allocation
Overview
Resource Allocation Problems are Linear Programming
Profit Maximization problems when available input
resources are fixed. Resource Allocation Problems thus
help production managers allocate various fixed resources
(labor, machine use, storage space, …) to produce various
outputs (cars, trucks, …) to maximize profit or minimize
cost. The opportunity cost of the scarce resources used in
manufacture define the maximum willingness to pay if
additional inputs became available.
BA 452 Lesson A.1 Formulating Linear Programs
24
Resource Allocation
Question: Iron Works, Inc. seeks to maximize profit by
making two products from steel.
 It just received this month's allocation of 19 pounds of
steel.
 It takes 2 pounds of steel to make a unit of product 1,
and 3 pounds of steel to make a unit of product 2.
 The physical plant has the capacity to make at most 6
units of product 1, and at most 8 units of total product
(product 1 plus product 2).
 Product 1 has unit profit 5, and product 2 has unit profit
7.
Formulate the linear program to maximize profit.
BA 452 Lesson A.1 Formulating Linear Programs
25
Resource Allocation
Answer: Here is a mathematical formulation of the
objective.
 Let x1 and x2 denote this month's production level of
product 1 and product 2.
 The total monthly profit =
(profit per unit of product 1) x (monthly production of
product 1)
+ (profit per unit of product 2) x (monthly production of
product 2)
= 5x1 + 7x2
 Maximize total monthly profit: Max 5x1 + 7x2
BA 452 Lesson A.1 Formulating Linear Programs
26
Resource Allocation
Here is a mathematical formulation of constraints.
 The total amount of steel used during monthly production =
(steel used per unit of product 1) x (monthly production of product 1)
+ (steel used per unit of product 2) x (monthly production of product 2)
= 2x1 + 3x2
 That quantity must be less than or equal to the allocated 19 pounds
of steel (the inequality < in the constraint below assumes excess
steel can be freely disposed; if disposal is impossible, then use
equality =) :
2x1 + 3x2 < 19
 The constraint that the physical plant has the capacity to make at
most 6 units of product 1 is formulated
x1 < 6
 The constraint that the physical plant has the capacity to make at
most 8 units of total product (product 1 plus product 2) is
x1 + x2 < 8
BA 452 Lesson A.1 Formulating Linear Programs
27
Resource Allocation
Adding the non-negativity of production completes
the formulation.
Max
s.t.
5x1 + 7x2
x1
< 6
2x1 + 3x2 < 19
x1 + x2 < 8
x1 > 0 and x2 > 0
Objective
function
Standard
constraints
Non-negativity
constraints
“Max” means maximize, and “s.t.” means subject to.
BA 452 Lesson A.1 Formulating Linear Programs
28
Portfolio Selection
Portfolio Selection
BA 452 Lesson A.1 Formulating Linear Programs
29
Portfolio Selection
Overview
Portfolio Selection Problems help financial managers select
specific investments (stocks, bonds, …) to generate returns
to either maximize expected return or minimize risk.
Constraints may restrict permissible investments by state
laws or company policy, and restrict risk.
BA 452 Lesson A.1 Formulating Linear Programs
30
Portfolio Selection
Question: Fidelity Investments manages funds for a variety
of clients. The investment strategy is tailored to each
client’s needs. For a new client, Fidelity has been
authorized to invest up to \$1.2 million in two funds: a stock
fund and a money market fund. Each unit of the stock fund
costs \$50 and returns an expected 10% annually. Each unit
of the money market fund costs \$100 and returns an
expected 4% annually.
BA 452 Lesson A.1 Formulating Linear Programs
31
Portfolio Selection
The client wants to minimize risk subject to the expected
annual income is at least \$60,000. According to Fidelity’s
risk measurement system, each unit invested in the stock
fund has a risk index of 8, and each unit in the money
market fund has index of 3. (A higher index indicates a
riskier investment.) Fidelity’s client also specified that at
least \$300,000 be invested in the money market fund.
Formulate the linear program to minimize the total risk
index of the portfolio.
BA 452 Lesson A.1 Formulating Linear Programs
32
Portfolio Selection
Answer:
 Define decision variables:
• S = number of units purchased in the stock fund
• 50S are the dollars invested in the stock fund
• 5S is the 10% return from the dollars invested in the stock fund
• M = number of units purchased in the money market fund
• 100M are the dollars invested in the money fund
• 4M is the 4% return from the dollars invested in the money fund
 Define objective: Minimize 8S + 3M
 Define constraints:
• 50S + 100M < 1,200,000 (Funds available)
• 5S + 4M > 60,000
(Annual income)
• M > 3,000
(Minimum units in money market)
• S, M > 0
(Non-negativity)
BA 452 Lesson A.1 Formulating Linear Programs
33
Resource Allocation with Machines
Resource Allocation with Machines
BA 452 Lesson A.1 Formulating Linear Programs
34
Resource Allocation with Machines
Overview
Resource Allocation Problems with Machines help
production managers allocate specific resources (including
machine use) to produce goods to either maximize profit or
minimize cost. Machine use is measured in hours, just like
labor use.
BA 452 Lesson A.1 Formulating Linear Programs
35
Resource Allocation with Machines
Question: Engineered Plastic Components, Inc. makes
plastic parts used in automobiles and computers. One of
its major contracts involves the production of plastic printer
cases for a computer company’s portable printers. The
printer cases can be produced on two injection molding
machines. The M-100 machine has a production capacity
of 25 printer cases per hour, and the M-200 machine has a
production capacity of 40 printer cases per hour. Both
machines use the same chemical to produce the printer
cases; the M-100 uses 40 pounds of raw material per hour,
and the M-200 uses 50 pounds per hour.
BA 452 Lesson A.1 Formulating Linear Programs
36
Resource Allocation with Machines
The computer company asked EPC to produce as many of
the cases as possible during the upcoming week; it will pay
\$18 for each case. However, next week is a regularly
scheduled vacation period for most of EPC’s production
employees. During this time, annual maintenance is
performed on all equipment. Because of the downtime for
maintenance, the M-100 is only available for at most 15
hours, and the M-200 for at most 10 hours.
BA 452 Lesson A.1 Formulating Linear Programs
37
Resource Allocation with Machines
The supplier of the chemical used in the production
process informed EPC that a maximum of 1000 pounds of
the chemical material will be available for next week’s
production; the cost for this raw material is \$6 per pound.
In addition to the raw material cost, Jackson Hole estimates
that the hourly cost of operating the M-100 and the M-200
are \$50 and \$75, respectively.
BA 452 Lesson A.1 Formulating Linear Programs
38
Resource Allocation with Machines
However, because of the high setup cost on both
machines, management requires that, if a machine is used
at all, it must be used for at least 5 hours.
Formulate the linear program to maximize profit. To
simplify this problem, you may change the last constraints
to read that each machine must be used for at least 5
hours.
(Do you see the difference between “if a machine is used at
all, it must be used for at least 5 hours” and “each machine
must be used for at least 5 hours”.)
BA 452 Lesson A.1 Formulating Linear Programs
39
Resource Allocation with Machines
Answer:



Define decision variables (assuming positive use of both machines; a
general solution requires binary variables from Part II of the course):
• M1 = number of hours spent on the M-100 machine
• 25 M1 is the production of cases from the M-100 machine
• 18(25) M1 is the revenue from production from the M-100 machine
• 40 M1 is the raw material used by the M-100 machine
• M2 = number of hours spent on the M-200 machine
• 40 M2 is the production of cases from the M-200 machine
• 18(40) M2 is the revenue from production from the M-200 machine
• 50 M2 is the raw material used by the M-200 machine
Total revenue = 18(25) M1 + 18(40) M2 = 450 M1 + 720 M2
Total cost = 6(40) M1 + 6(50) M2 + 50 M1 + 75 M2 = 290 M1 + 375 M2
BA 452 Lesson A.1 Formulating Linear Programs
40
Resource Allocation with Machines


Define objective:
Maximize (profit = revenue-cost) 160 M1 + 345 M2
Define constraints:
• M1
< 15
(M-100 maximum)
•
M2 < 10
(M-200 maximum)
• M1
> 5
(M-100 minimum)
•
M2 > 5
(M-200 minimum)
• 40M1 + 50 M2 < 1000
(Raw Material)
• M1, M2
> 0
BA 452 Lesson A.1 Formulating Linear Programs
41
BA 452
Quantitative Analysis
End of Lesson A.1
BA 452 Lesson A.1 Formulating Linear Programs
42
```