Linear Programming
•Concept
•Structure of the problem
•Model Formulation
•Solution
Recap Quiz
1. Operations research supports ……
2. Objective of OR is to get …… solution to the
problem.
3. Phases of OR ……..
4. Basic two types of OR models …..
5. …….. type of models lead to optimisation
6. Model is ………
7. Objective function is…..
8. Constraints are …….
9. OR follows ……. approach towards management.
LPP Concept
• A model consisting of linear relationships
representing a firm’s objective and
resource constraints
• LP is a mathematical modeling technique
used to determine a level of operational
activity in order to achieve an objective,
subject to restrictions called constraints.
Concept
contd…
• Linear programming has nothing to do with
computer programming.
• The use of the word “programming” here
means “choosing a course of action.”
• Linear programming involves choosing a
course of action when the mathematical
model of the problem contains only linear
functions.
Concept
contd…
• A Linear Programming model seeks to
maximize or minimize a linear function,
subject to a set of linear constraints.
• The linear model consists of the following
components:
– A set of decision variables.
– An objective function.
– A set of constraints.
LP Model Formulation
• Problem formulation or modeling is the process
of translating a verbal statement of a problem
into a mathematical statement
• Decision variables
– mathematical symbols representing levels of activity of an
operation
• Objective function
– a linear relationship reflecting the objective of an operation
– most frequent objective of business firms is to maximize profit
– most frequent objective of individual operational units (such as
a production or packaging department) is to minimize cost
• Constraint
– a linear relationship representing a restriction on decision
making
Guidelines for Model Formulation
•
•
•
•
•
Understand the problem thoroughly.
Describe the objective.
Define the decision variables.
Describe each constraint.
Write the objective in terms of the decision
variables.
• Write the constraints in terms of the decision
variables.
Example
• Infoway Company manufactures a printer and
keyboard. The contribution margins of the printer and
keyboard are $30 and $20, respectively. Two types of
skilled labor are required to manufacture these
products: soldering and assembling. A printer requires
2 hours of soldering and I hour of assembling. A
keyboard requires 1 hour of soldering and 1 hour of
assembling. Infoway has 1,000 soldering hours and 800
assembling hours available per week. There are no
constraints on the supply of raw materials. Demand for
keyboards is unlimited, but at most 350 printers are
sold each week. Infoway wishes to maximize its weekly
total contribution margin.
Data summary for LPP
Resource usage per unit of activity
Activity
Amount of resource
available
Resource
1
2
….
n
1
a11
a12
…
a1n
b1
2
a21
a22
…
a2n
b2
…
…
…
…
…
…
m
am1
am2
…
amn
bm
Contribution to Z per
unit of activity
C1
C2
…
Cn
LPP Structure
Maximise/minimise
z = c1x1 + c2x2 + ... + cnxn (Objective function)
subject to:
Constraints
a11x1 + a12x2 + ... + a1nxn (≤, =, ≥) b1
a21x1 + a22x2 + ... + a2nxn (≤, =, ≥) b2
:
am1x1 + am2x2 + ... + amnxn (≤, =, ≥) bm
X1 ≥ 0, …… Xn ≥ 0
xj = decision variables
bi = constraint levels
cj = objective function coefficients
aij = constraint coefficients
(Non – negativity condition)
• EX.: MegaMarketing is planning a concentrated one week
advertising campaign for their new CutsEverything
SuperKnife. The ads have been designed and produced and
now they wish to determine how much money to spend in
each advertising outlet. In reality, they have hundreds of
possible outlets to choose from. We will illustrate their
problem with two outlets: Prime-time TV, and
newsmagazines. For this product, the target segments are
Teenage Boys, Affluent Women (ages 40-49), and Retired
Men. Each minute of primetime TV and page of
newsmagazine advertisement reaches the following
number of people (in millions):
Outlet
Boys
Women
Men
Cost
TV
5
1
3
600
Magazine
2
6
3
500
Target
24
18
24
Example
• The K9 Kondo Company manufactures climate-controlled
doghouses. The company believes that its high-volume customers
are high-income male and female dog owners who want to pamper
their pets. To reach these groups, the marketing manager at K9
Kondo is considering placing one-minute commercials on the
following national TV shows: “New York Dog Show” and “Man's
Best Friend.”
• A one-minute commercial on “New York Dog Show” costs $200,000,
and a one-minute commercial on “Man's Best Friend” costs
$50,000. The marketing manager would like the commercials to be
seen by at least 60 million high-income women and at least 36
million high-income men. Marketing studies show the following:
• Each one-minute commercial on “New York Dog Show” is seen by
six million high-income women and two million high-income men.
• Each one-minute commercial on “Man's Best Friend” is seen by
three million high-income women and three million high-income
men.
Example
• You need to buy some filing cabinets. You know
that Cabinet X costs $10 per unit, requires six
square feet of floor space, and holds eight cubic
feet of files. Cabinet Y costs $20 per unit, requires
eight square feet of floor space, and holds twelve
cubic feet of files. You have been given $140 for
this purchase, though you don't have to spend
that much. The office has room for no more than
72 square feet of cabinets. How many of which
model should you buy, in order to maximize
storage volume?
Assumptions of LP
• In addition to linearity of objective function
and constraints the assumptions are:
• 1. Proportionality:
• The contribution of each activity (xj) to the
value of objective function (z=cjxj) or LHS of
each constraint function (aijxj) is proportional
to the level of activity.
• e.g. Z= 3x + 5y … (For x=1, Z=3 ; x=2, Z=6)
Assumptions of LP
contd…
• Possible violations of proportionality:
– Product development or startup cost ()
– Economies of scale / learning curve effect ()
– Higher marketing costs for higher level of sales ()
• 2. Additivity:
• Every function in LP model is sum of individual
contributions of the respective activities
• e.g. Z= 3x + 5y … (For x=1, y=1; Z = 8)
Assumptions of LP
contd…
• Possible violations of additivity:
– Complementary products improve marketing efficiency
– Competitive products reduce production efficiency
• 3. Divisibility:
• Decision variables in LPP are allowed to have
any fractional values that satisfy all the
constraints
Assumptions of LP
contd…
• 4. Certainty:
• The value assigned to each parameter (aij,cj,bi)
of LP model is assumed to be known constant.
• In reality, LPP is formulated to select future
course of action (uncertainty)
• Sensitivity analysis for optimal solution is
conducted
Example
• Omega mfg. company has discontinued the
production of certain unprofitable product
line. This act created considerable excess
production
capacity.
Management
is
considering devoting this excess capacity to
one or more of three products A, B and C. The
available capacity on machines that might
limit production and number of machine
hours required for each unit of respective
product is given in table in the next slide.
Machine
type
Available
machine hours
per week
Machine hours per unit
Product A
Product B
Product C
Milling
machine
500
9
3
5
Lathe
350
5
4
0
Grinder
150
3
0
2
The sales department indicates that the sales
potential for product C is 20 units per week. The unit
profit would be $50, $20, and $ 25 respectively for
products A,B and C. Formulate a linear programming
model.
•
An advertising company wishes to plan its advertising strategy in three different
media-television, radio and magazines. The purpose of advertising is to reach as
large a number of potential customers as possible. Following data have been
obtained from market survey-
• The company wants to spend not more than Rs. 450000 on advertising.
Following are the further requirements.
• 1. at least 1 million exposures take place among female customers.
• 2. advertising on magazines be limited to Rs 1, 50000
• 3. at least 3 advertising units to be bought on magazine I and 2 units on
magazine II.
• 4. The number of advertising units on television and radio should each
be between 5 and 10.
• Formulate an LPP model for the problem.
• The demand for ice cream during the three months at Cool-Flavors
Parlor is estimated at 500, 600 and 400 (20-gallon cartons)
respectively. Two wholesalers A and B, supply Cool-flavors with its
ice-cream. Although the flavors from the two suppliers are different
they are interchangeable. The maximum number of cartons either
supplier can provide is 400 per month. Also the price the two
suppliers charge change from one month to the next according to
the following schedule:
•
Supplier A
Supplier B
Price per carton in month ($)
June
July August
100
110
120
115
108
125
• To take advantage of price fluctuation, Cool-Flavors Parlor can
purchase more than is needed for a month and store the surplus to
satisfy the demand in a later month. The cost of refrigerating an icecream carton is $5 per month. It is realistic in present situation to
assume that the refrigeration cost is function of the average
number of cartons (bought in the same month) on hand during the
month. Develop a LP model to determine schedule of buying icecream from two suppliers at least cost.
• An assembly line consisting of three consecutive
stations produces two radio models: HFD-1 and
HFD-2. The following table provides the assembly
times for the three workstations.
• The daily maintenance for stations 1, 2, and 3
consumes 10%, 14%, and 12%, respectively, of
the maximum 480 minutes available for each
station each day. Determine the optimal product
mix that will minimize the idle (or unused) times
in the three workstations.
Workstation
1
2
3
Minutes per unit
HFD-1
HFD-2
4
6
5
5
6
4
• You are the investments manager for the Smalltime Mutual Funds
Company, and are trying to determine how to invest a pool of $14
million released by cashing out some of the stock investments. Table
below summarizes the information that you have about a set of five
possible investments. To be as conservative as possible, you
assume that in the event of a loss by the investment, you lose
all of your money. This is a fairly serious assumption, since most
mutual fund investments are likely to lose some but not all of their
value. On the other hand, you also assume that if there is not a
loss, then the investment will grow by the growth rate shown. For
policy reasons, there are limits on how you can invest the money. You
must allocate at least 35% of the total funds available to the balanced
and bond investments. Of all the money put into equity, special equity
and foreign investments, at least half must be in the equity investment.
Finally, the expected lost capital must be less than 10%. Of course,
your overall objective is to maximize the net return on the original pool
of money.
Type of investment Annual growth rate Probability of loss
Equity
0.15
0.18
Special equity
0.21
0.31
Balanced
0.11
0.09
Foreign
0.19
0.19
Bond
0.08
0.03
• SilComputer needs to meet the demand of its largest
corporate and educational customers for notebook
computers over the next four quarters (before its current
model becomes obsolete).
• SilComputer currently has 5,000 notebook computers in
inventory. Expected demand over the next four quarters for
its notebook is 7,000; 16,000; 9,000; and 14,000.
SilComputer has sufficient capacity and material to produce
up to 10,000 computers in each quarter at a cost of $2000
per notebook. By using overtime, up to an additional 2,500
computers can be produced at a cost of $2200 each.
• Computers produced in a quarter can be used either to
meet that quarter's demand, or be held in inventory for use
later. Each computer in inventory is charged $100 to reflect
carrying costs.
• How should SilComputer meet its demand for notebooks at
minimum cost?
• Pune city is one of those cities in India, which are
selected for smart city projects in first phase. The
city of Pune is faced with a severe budget
shortage. Seeking a long-term solution, the city
council votes to improve the tax base by
condemning an inner-city housing area and
replacing it with a modern development.
• The project involves two phases: (1) demolishing
substandard houses to provide land for the new
development, and (2) building the new
development. The following is a summary of the
situation.
Continued on next slide…
• 1. As many as 300 substandard houses can be demolished.
Each house occupies a 0.25-acre lot. The cost of
demolishing a condemned house is $2000.
• 2. Lot sizes for new single-, double-, triple-, and quadruplefamily homes (units) are 0.18, 0.28, 0.4 , and 0.5 acre,
respectively. Streets, open space, and utility easements
account for 15% of available acreage.
• 3. In the new development the triple and quadruple units
account for at least 25% of the total. Single units must be at
least 20% of all units and double units at least 10%.
• 4. The tax levied per unit for single, double, triple, and
quadruple units is $1,000, $1,900, $2,700, and $3,400,
respectively.
• 5. The construction cost per unit for single-, double-, triple-,
and quadruple - family homes is $50,000, $70,000,
$130,000, and $160,000, respectively. Financing through a
local bank can amount to a maximum of $15 million. How
many units of each type should be constructed to maximize
tax collection?
• Progress City is studying the feasibility of introducing a
mass-transit bus system that will alleviate the smog
problem by reducing in-city driving. The study seeks
the minimum number of buses that can handle the
transportation needs.
• After gathering necessary information, the city
engineer noticed that the minimum number of buses
needed fluctuated with the time of the day and that
the required number of buses could be approximated
by constant values over successive 4 hour intervals.
• To carry out the required daily maintenance, each bus
can operate 8 successive hours a day only. Determine
the number of operating buses in each shift that will
meet the minimum demand while minimizing the total
number of buses in operation.
Time slot
Minimum no. of
buses needed
12 AM to 4 AM
4 AM to 8 AM
8 AM to 12 Noon
12 PM to 4 PM
4 PM to 8 PM
8 PM to 12 Midnight
4
8
10
7
12
4
Suppose that buses can run either 8- or 12-hour shifts. If
a bus runs for 12 hours, the driver must be paid for the
extra hours at 150% of the regular hourly pay. Do you
recommend the use of 12-hour shifts?