Introduction to Linear Programming

1
Introduction to Linear Programming

• Source
• 1. Operations Research Applications
  Algorithms,4th edition, by Wayne L. Winston
• 2. Jim Orlin PowerPoint presentation

2
What Is a Linear Programming Problem?
Example
Giapettos, Inc., manufactures wooden soldiers
and trains.

• Each soldier built
• Sell for 27 and uses 10 worth of raw
  materials.
• Increase Giapettos variable labor/overhead
  costs by 14.
• Requires 2 hours of finishing labor.
• Requires 1 hour of carpentry labor.
• Each train built
• Sell for 21 and used 9 worth of raw
  materials.
• Increases Giapettos variable labor/overhead
  costs by 10.
• Requires 1 hour of finishing labor.
• Requires 1 hour of carpentry labor.

3
What Is a Linear Programming Problem?

• Each week Giapetto can obtain
• All needed raw material.
• Only 100 finishing hours.
• Only 80 carpentry hours.
• Also
• Demand for the trains is unlimited.
• At most 40 soldiers are bought each week.

Giapetto wants to maximize weekly profit
(revenues expenses). Formulate a mathematical
model of Giapettos situation that can be used
maximize weekly profit.
4
What Is a Linear Programming Problem?

• The Giapetto solution model incorporates the
  characteristics shared by all linear programming
  problems.

x1 number of soldiers produced each week x2
number of trains produced each week
Decision Variables
Objective Function In any linear programming
model, the decision maker wants to maximize
(usually revenue or profit) or minimize (usually
costs) some function of the decision variables.
This function to maximized or minimized is called
the objective function. For the Giapetto
problem, fixed costs are do not depend upon the
the values of x1 or x2.
5
What Is a Linear Programming Problem?

• Giapettos weekly profit can be expressed in
  terms of the decision variables x1 and x2

Weekly profit weekly revenue weekly raw
material costs the weekly variable costs
Weekly revenue 27x1 21x2 Weekly raw material
costs 10x1 9x2 Weekly variable costs 14x1
10x2
Weekly profit (27x1 21x2) (10x1 9x2)
(14x1 10x2 ) 3x1 2x2
6
What Is a Linear Programming Problem?

• Giapettos objective is to chose x1 and x2 to
  maximize 3x1 2x2. We use the variable z to
  denote the objective function value of any LP.
  Giapettos objective function is

Maximize z 3x1 2x2
Maximize will be abbreviated by max and
minimize by min. The coefficient of an
objective function variable is called an
objective function coefficient.
7
What Is a Linear Programming Problem?

• Constraints As x1 and x2 increase, Giapettos
  objective function grows larger. For Giapetto,
  the values of x1 and x2 are limited by the
  following three restrictions (often called
  constraints)

Constraint 1 Each week, no more than 100 hours
of finishing time may be used. Constraint 2
Each week, no more than 80 hours of carpentry
time may be used. Constraint 3 Because of
limited demand, at most 40 soldiers should be
produced.
These three constraints can be expressed
mathematically by the following equations
Constraint 1 2 x1 x2 100 Constraint 2
x1 x2 80 Constraint 3 x1 40
8
What Is a Linear Programming Problem?

• For the Giapetto problem model, combining the
  sign restrictions x1 0 and x2 0 with the
  objective function and constraints yields the
  following optimization model

Max z 3x1 2x2 (objective
function) Subject to (s.t.) 2 x1 x2
100 (finishing constraint) x1 x2
80 (carpentry constraint) x1
40 (constraint on demand for soldiers) x1
0 (sign restriction)
x2 0 (sign restriction)
9
What Is a Linear Programming Problem?

• Concepts of linear function and linear
  inequality
• Linear Function A function f(x1, x2, , xn of
  x1, x2, , xn is a linear function if and only if
  for some set of constants, c1, c2, , cn,
• f(x1, x2, , xn) c1x1 c2x2 cnxn.
• For example, f(x1,x2) 2x1 x2 is a linear
  function of x1 and x2, but f(x1,x2) (x1)2x2 is
  not a linear function of x1 and x2.
• For any linear function f(x1, x2, , xn) and any
  number b, the inequalities inequality f(x1, x2,
  , xn) b and f(x1, x2, , xn) ³ b are linear
  inequalities.

10
What Is a Linear Programming Problem?
A linear programming problem (LP) is an
optimization problem for which we do the
following

1. Attempt to maximize (or minimize) a linear
   function (called the objective function) of the
   decision variables.
2. The values of the decision variables must satisfy
   a set of constraints. Each constraint must be a
   linear equation or inequality.
3. A sign restriction is associated with each
   variable. For each variable xi, the sign
   restriction specifies either that xi must be
   nonnegative (xi 0) or that xi may be
   unrestricted in sign.

11
What Is a Linear Programming Problem?

• Proportionality and Additive Assumptions
• The objective function for an LP must be a
  linear function of the decision variables has two
  implications

1. The contribution of the objective function
from each decision variable is proportional to
the value of the decision variable. For example,
the contribution to the objective function for 4
soldiers is exactly fours times the contribution
of 1 soldier. 2. The contribution to the
objective function for any variable is
independent of the other decision variables. For
example, no matter what the value of x2, the
manufacture of x1 soldiers will always contribute
3x1 dollars to the objective function.
12
What Is a Linear Programming Problem?
Each LP constraint must be a linear inequality or
linear equation has two implications

• The contribution of each variable to the
  left-hand side of each constraint is proportional
  to the value of the variable. For example, it
  takes exactly 3 times as many finishing hours to
  manufacture 3 soldiers as it does 1 soldier.
• 2. The contribution of a variable to the
  left-hand side of each constraint is independent
  of the values of the variable. For example, no
  matter what the value of x1, the manufacture of
  x2 trains uses x2 finishing hours and x2
  carpentry hours

13
What Is a Linear Programming Problem?

• Divisibility Assumption
• The divisibility assumption requires that each
  decision variable be permitted to assume
  fractional values. For example, this assumption
  implies it is acceptable to produce a fractional
  number of trains. The Giapetto LP does not
  satisfy the divisibility assumption since a
  fractional soldier or train cannot be produced.
  The use of integer programming methods necessary
  to address the solution to this problem.
• The Certainty Assumption
• The certainty assumption is that each parameter
  (objective function coefficients, right-hand
  side, and technological coefficients) are known
  with certainty.

14
What Is a Linear Programming Problem?

• Feasible Region and Optimal Solution

The feasible region of an LP is the set of all
points satisfying all the LPs constraints and
sign restrictions.
x1 40 and x2 20 are in the feasible region
since they satisfy all the Giapetto
constraints. On the other hand, x1 15, x2 70
is not in the feasible region because this point
does not satisfy the carpentry constraint 15
70 is gt 80.
Giapetto Constraints 2 x1 x2 100 (finishing
constraint) x1 x2 80 (carpentry
constraint) x1 40 (demand
constraint) x1 0 (sign
restriction) x2 0 (sign
restriction)
15
What Is a Linear Programming Problem?

• For a maximization problem, an optimal solution
  to an LP is a point in the feasible region with
  the largest objective function value. Similarly,
  for a minimization problem, an optimal solution
  is a point in the feasible region with the
  smallest objective function value.

Most LPs have only one optimal solution.
However, some LPs have no optimal solution, and
some LPs have an infinite number of solutions.
The optimal solution to the Giapetto LP is x1
20 and x2 60. This solution yields an
objective function value of
z 3x1 2x2 3(20) 2(60) 180
When we say x1 20 and x2 60 is the optimal
solution, we are saying that no point in the
feasible region has an objective function value
(profit) exceeding 180.
16

• More Linear Programming Models

17
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21
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23
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26
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27
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29
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33
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34
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35
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39
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41
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42
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23
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43
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1, 2, 3, 4)??? ??

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(xA1 xA2 xA3 xA4)
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xB2 0.1 (xB1 xB2 xB3 xB4)
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44
Scheduling Postal Workers

• Each postal worker works for 5 consecutive days,
  followed by 2 days off, repeated weekly.

• Minimize the number of postal workers (for the
  time being, we will permit fractional workers on
  each day.)

45
Formulating as an LP

• Select the decision variables
• Let x1 be the number of workers who start working
  on Monday, and work till Friday
• Let x2 be the number of workers who start on
  Tuesday
• Let x3, x4, , x7 be defined similarly.

46
The linear program
Minimize
z x1 x2 x3 x4 x5 x6 x7
x1 x4 x5 x6 x7 ? 17
subject to
x1 x2 x5 x6 x7 ? 13
x1 x2 x3 x6 x7 ? 15
x1 x2 x3 x4 x7 ? 19
x1 x2 x3 x4 x5 ? 14
x2 x3 x4 x5 x6 ? 16
x3 x4 x5 x6 x7 ? 11
xj ? 0 for j 1 to 7
47
Minimize
z x1 x2 x3 x4 x5 x6 x7
x1 x4 x5 x6 x7 - s1 17
subject to
x1 x2 x5 x6 x7 - s2
13
x1 x2 x3 x6 x7 - s3
15
x1 x2 x3 x4 x7 - s4
19
x1 x2 x3 x4 x5 - s5 14
x2 x3 x4 x5 x6 - s6 16
x3 x4 x5 x6 x7 - s7
11
xj ? 0 , sj ? 0 for j 1 to 7
48
A non-linear objective that often can be made
linear.
Suppose that one wants to minimize the maximum of
the slacks, that is minimize z max (s1, s2,
, s7). This is a non-linear objective. But we
can transform it, so the problem becomes an LP.
49
Minimize z
z ? sj for j 1 to 7.
x1 x4 x5 x6 x7 - s1 17
subject to
x1 x2 x5 x6 x7 - s2
13
x1 x2 x3 x6 x7 - s3
15
x1 x2 x3 x4 x7 - s4
19
x1 x2 x3 x4 x5 - s5 14
x2 x3 x4 x5 x6 - s6 16
x3 x4 x5 x6 x7 - s7
11
xj ? 0 , sj ? 0 for j 1 to 7
The new constraint ensures that z ? max (s1, ,
s7)
The objective ensures that z sj for some j.
50
Non-linear objective that often can be made
linear.
Suppose that the goal is to have dj workers on
day j. Let yj be the number of workers on day
j. Suppose that the objective is minimize Si
yj dj This is a non-linear objective. But
we can transform it, so the problem becomes an LP.
51
Minimize Sj zj
zj ? dj - yj for j 1 to 7.
zj ? yj - dj for j 1 to 7.
x1 x4 x5 x6 x7 y1
subject to
x1 x2 x5 x6 x7 y2
x1 x2 x3 x6 x7 y3
x1 x2 x3 x4 x7 y4
x1 x2 x3 x4 x5 - y5
x2 x3 x4 x5 x6 y6
x3 x4 x5 x6 x7 y7
xj ? 0 , yj ? 0 for j 1 to 7
The new constraints ensure that zj ? yj dj
for each j.
The objective ensures that zj yj dj for
each j.
52
A ratio constraint
Suppose that we need to ensure that at least 30
of the workers have Sunday off.
How do we model this?
(x1 x2 )/x1 x2 x3 x4 x5 x6 x7 ? .3
(x1 x2 ) ? .3 x1 .3 x2 .3 x3 .3 x4 .3
x5 .3 x6 .3 x7
-.7 x1 - .7 x2 .3 x3 .3 x4 .3 x5 .3 x6
.3 x7 lt 0
53
Cost per Ounce and Dietary Requirements for Diet
Problem
54
Example 4-2 Diet Problem
55
Example 4-3 Blending Problem
Formulate the appropriate model for the following
blending problem The sugar content of three
juicesorange, banana, and pineappleis 10, 15,
and 20 percent, respectively. How many quarts of
each must be mixed together to achieve one gallon
(four quarts) that has a sugar content of at
least 17 percent to minimize cost? The cost per
quart is 20 cents for orange juice, 30 cents for
banana juice, and 40 cents for pineapple juice.
Solution Variable definitions O quantity of
orange juice in quarts B quantity of banana
juice in quarts P quantity of pineapple juice
in quarts
56
Example 4-5 Media Selection
The Long Last Appliance Sales Company is in the
business of selling appliances such as microwave
ovens, traditional ovens, refrigerators,
dishwashers, washers, dryers, and the like. The
company has stores in the greater Chicagoland
area and has a monthly advertising budget of
90,000. Among its options are radio advertising,
advertising in the cable TV channels, newspaper
advertising, and direct-mail advertising. A
30-second advertising spot on the local cable
channel costs 1,800, a 30-second radio ad costs
350, a half-page ad in the local newspaper costs
700, and a single mailing of direct-mail
insertion for the entire region costs 1,200 per
mailing. The number of potential buying
customers reached per advertising medium usage is
as follows Radio 7,000 TV 50,000 Newspaper
18,000 Direct mail 34,000 Due to company
restrictions and availability of media, the
maximum number of usages of each medium is
limited to the following Radio 35 TV
2 5 Newspaper 30 Direct mail 18
57
Example 4-5 (contd)
The management of the company has met and decided
that in order to ensure a balanced utilization of
different types of media and to portray a
positive image of the company, at least 10
percent of the advertisements must be on TV. No
more than 40 percent of the advertisements must
be on radio. The cost of advertising allocated to
TV and direct mail cannot exceed 60 percent of
the total advertising budget. What is the optimal
allocation of the budget among the four media?
What is the total maximum audience contact?
58
Example 4-5 (contd)
59
Marketing Research

• Stages of marketing research study development
• Design study.
• Conduct marketing survey.
• Analyze data and obtain results.
• Make recommendations based on the results.

60
Example 4-6 Market Research
Market Facts Inc. is a marketing research firm
that works with client companies to determine
consumer reaction toward various products and
services. A client company requested that Market
Facts investigate the consumer reaction to a
recently developed electronic device. Market
Facts and the client company agreed that a
combination of telephone interviews and
direct-mail questionnaires would be used to
obtain the information from different type of
households. The households are divided into six
categories

1. Households containing a single person under 40
   years old and without children under 18 years of
   age.
2. Households containing married people under 40
   years old and without children under 18 years of
   age.
3. Households containing single parents with
   children under 18 years of age.
4. Households containing married families with
   children under 18 years of age.
5. Households containing single people over 40 years
   old without children under 18 years of age.
6. Households containing married people over 40
   years old without children under 18 years of age.

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Example 4-6 (contd)
Restrictions

1. At least 60 percent of the phone interviews must
   be conducted at households with children.
2. At least 50 percent of the direct-mail
   questionnaires must be mailed to households with
   children.
3. No more than 30 percent of the phone interviews
   and mail-in questionnaires must be conducted at
   households with single people.
4. At least 25 percent of the phone interviews and
   mail-in questionnaires must be conducted at
   households that contain married couples.

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Example 4-6 (contd)
Problem formulation
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Example 4-6 (contd)
Problem solution
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Financial Applications

• Planning Problems for Banks
• Linear programming can be very beneficial in
  banking decisions.
• Financial planning bankers must decide how a
  bank wants to allocate its funds among the
  various types of loans and investment securities.
  
• Portfolio management decisions are based on
  maximizing annual rate of return subject to state
  and federal regulations, and bank policies and
  restrictions.

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Example 4-7 Financial Planning
First American Bank issues five types of loans.
In addition, to diversify its portfolio, and to
minimize risk, the bank invests in risk-free
securities. The loans and the risk-free
securities with their annual rate of return are
given in Table 4-3.
Table 4-3 Rates of Return for Financial Planning
Problem Type of Loan or Security Annual Rate of
Return () Home mortgage (first) 6 Home mortgage
(second) 8 Commercial loan 11 Automobile loan
9 Home improvement loan 10 Risk-free securities
4
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Example 4-7 Financial Planning (contd)
The banks objective is to maximize the annual
rate of return on investments subject to the
following policies, restrictions, and regulations

1. The bank has 90 million in available funds.
2. Risk-free securities must contain at least 10
   percent of the total funds available for
   investments.
3. Home improvement loans cannot exceed 8,000,000.
4. The investment in mortgage loans must be at least
   60 percent of all the funds invested in loans.
5. The investment in first mortgage loans must be at
   least twice as much as the investment in second
   mortgage loans.
6. Home improvement loans cannot exceed 40 percent
   of the funds invested in first mortgage loans.
7. Automobile loans and home improvement loans
   together may not exceed the commercial loans.
8. Commercial loans cannot exceed 50 percent of the
   total funds invested in mortgage loans.

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Example 4-7 Financial Planning (contd)
68
Example 4-7 Financial Planning (contd)
69
Example 4-8 Portfolio Selection
A conservative investor has 100,000 to invest.
The investor has decided to use three vehicles
for generating income municipal bonds, a
certificate of deposit (CD), and a money market
account. After reading a financial newsletter,
the investor has also identified several
additional restrictions on the investments

1. No more than 40 percent of the investment should
   be in bonds.
2. The proportion allocated to the money market
   account should be at least double the amount in
   the CD.

The annual return will be 8 percent for bonds, 9
percent for the CD, and 7 percent for the money
market account. Assume the entire amount will be
invested. Formulate the LP model for this
problem, ignoring any transaction costs and the
potential for different investment lives. Assume
that the investor wants to maximize the total
annual return.
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Example 4-8 Portfolio Selection (contd)
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Production Applications

• Linear programming in production management in
  manufacturing
• Multiperiod production scheduling
• Workforce scheduling
• Make-or-buy decisions.

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Example 4-9 Multiperiod Production Scheduling
Morton and Monson Inc. is a small manufacturer of
parts for the aerospace industry. The production
capacity for the next four months is given as
follows Production Capacity in
Units Month Regular Production Overtime
Production January 3,000 500 February
2,000 400 March 3,000 600 April 3,500 800 The
regular cost of production is 500 per unit and
the cost of overtime production is 150 per unit
in addition to the regular cost of production.
The company can utilize inventories to reduce
fluctuations in production, but carrying one unit
of inventory costs the company 40 per unit per
month. Currently there are no units in inventory.
However, the company wants to maintain a minimum
safety stock of 100 units of inventory during the
months of January, February, and March. The
estimated demand for the next four months is as
follows
Month January February March April Demand
2,800 3,000 3,500 3,000
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Example 4-9 Multiperiod Production Scheduling
(contd)
Continued on next slide.
74
Example 4-9 Multiperiod Production Scheduling
(contd)
75
Example 4-9 Multiperiod Production Scheduling
(contd)
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Radiation Therapy Overview

• High doses of radiation (energy/unit mass) can
  kill cells and/or prevent them from growing and
  dividing
• True for cancer cells and normal cells
• Radiation is attractive because the repair
  mechanisms for cancer cells is less efficient
  than for normal cells

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Conventional Radiotherapy
Relative Intensity of Dose Delivered
78
Conventional Radiotherapy
Relative Intensity of Dose Delivered
79
Conventional Radiotherapy

• In conventional radiotherapy
• 3 to 7 beams of radiation
• radiation oncologist and physicist work together
  to determine a set of beam angles and beam
  intensities
• determined by manual trial-and-error process

80
Goal maximize the dose to the tumor while
minimizing dose to the critical area
With a small number of beams, it is difficult to
achieve these goals.
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Tomotherapy a diagram
82
Radiation Therapy Problem Statement

• For a given tumor and given critical areas
• For a given set of possible beamlet origins and
  angles
• Determine the weight of each beamlet such that
• dosage over the tumor area will be at least a
  target level gL .
• dosage over the critical area will be at most a
  target level gU.

83
Display of radiation levels
84
Linear Programming Model

• First, discretize the space
• Divide up region into a 2D (or 3D) grid of pixels

85
More on the LP

• Create the beamlet data for each of p 1, ...,
  n possible beamlets.
• Dp is the matrix of unit doses delivered by beam
  p.

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Linear Program

• Decision variables w (w1, ..., wp)
• wp intensity weight assigned to beamlet p
  for p 1 to n
• Dij dosage delivered to pixel (i,j)

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An LP model
took 4 minutes to solve.
minimize
In an example reported in the paper, there were
more than 63,000 variables, and more than 94,000
constraints (excluding upper/lower bounds)
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What to do if there is no feasible solution

• Use penalties e.g., Dij ? gL yijand then
  penalize y in the objective.
• Consider non-linear penalties (e.g., quadratic)
• Consider costs that depend on damage rather than
  on radiation
• Develop target doses and penalize deviation from
  the target

89
Optimal Solution for the LP
90
An Optimal Solution to an NLP
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Homework
92
Example 4-10 Workforce Scheduling
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Example 4-11 Make-or-Buy Decisions
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Example 4-12 Agriculture Applications
A farm owner in Des Moines, Iowa, is interested
in determining how to divide the farmland among
four different types of crops. The farmer owns
two farms in separate locations and has decided
to plant the following four types of crops in
these farms corn, wheat, bean, and cotton. The
first farm consists of 1,450 acres of land, while
the second farm consists of 850 acres of land.
Any of the four crops may be planted on either
farm. However, after a survey of the land, based
on the characteristics of the farmlands, Table
4-7 shows the maximum acreage restrictions the
farmer has placed for each crop.
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