Transcript ISE 102 Introduction to Linear Programming (LP) Asst. Prof. Dr. Mahmut Ali GÖKÇE

Spring, 2007
ISE 102
Introduction to Linear
Programming (LP)
Asst. Prof. Dr. Mahmut Ali GÖKÇE
Industrial Systems Engineering Dept.
İzmir University of Economics
www.izmirekonomi.edu.tr
Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics
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Spring, 2007
Introduction to Linear Programming
 Many managerial decisions involve trying to
make the most effective use of an organization’s
resources. Resources typically include:
 Machinery/equipment
 Labor
 Money
 Time
 Energy
 Raw materials
 These resources may be used to produce
 Products (machines, furniture, food, or clothing)
 Services (airline schedules, advertising policies, or
investment decisions)
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What is Linear Programming?
 Linear Programming is a mathematical technique
designed to help managers plan and make
necessary decisions to allocate resources
 Linear Programming (LP) is one the most widely
used decision tools of Operations Research &
Management Science (ORMS)
 In a survey of Fortune 500 corporations, 85 % of
those responding said that they had used LP
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Brief History of LP
 LP was developed to solve military logistics
problems during World War II
 In 1947, George Dantzig developed a
solution procedure for solving linear
programming problems (Simplex Method)
 This method turned out to be so efficient for
solving large problems quickly
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History of LP (contd)
 The simultaneous development of the
computer technology established LP as an
important tool in various fields
 Simplex Method is still the most important
solution method for LP problems
 In recent years, a more efficient method for
extremely large problems has been
developed (Karmarkar’s Algorithm)
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LP Problems
 A large number of real problems can be
formulated and solved using LP. A partial list
includes:
Scheduling of personnel
Production planning and inventory control
Assignment problems
Several varieties of blending problems including
ice cream, steel making, crude oil processing
Distribution and logistics problems
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Typical Applications of LP
 Aggregate Planning
Develop a production schedule which
 satisfies specified sales demands in future periods
 satisfies limitations on production capacity
 minimizes total production/inventory costs
 Scheduling Problem
Produce a workforce schedule which
 satisfies minimum staffing requirements
 utilizes reasonable shifts for the workers
 is least costly
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Typical Applications of LP (contd)
 Product Mix (“Blending”) Problem
Develop the product mix which
 satisfies restrictions/requirements for customers
 does not exceed capacity and resource constraints
 results in highest profit
 Logistics
Determine a distribution system which
 meets customer demand
 minimizes transportation costs
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Typical Applications of LP (contd)
 Marketing
Determine the media mix which
meets a fixed budget
maximizes advertising effectiveness
 Financial Planning
Establish an investment portfolio which
meets the total investment amount
meets any maximum/minimum restrictions of
investing in the available alternatives
maximizes ROI
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Typical Applications of LP (contd)
 What do these applications have in
common?
All are concerned with maximizing or
minimizing some quantity, called the
objective of the problem
All have constraints which limit the
degree to which the objective function
can be pursued
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Typical Applications of LP (contd)
Fleet Assignment at Delta Air Lines
 Delta Air Lines flies over 2500 domestic flight legs
every day, using about 450 aircrafts from 10 different
fleets that vary by speed, capacity, amount of noise
generated, etc.
 The fleet assignment problem is to match aircrafts
(e.g. Boeing 747, 757, DC-10, or MD80) to flight legs
so that seats are filled with paying passengers
 Delta is one the first airlines to solve to completion
this fleet assignment problem, one of the largest and
most difficult problems in airline industry
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Fleet Assignment at Delta (contd)
 An airline seat is the most perishable
commodity in the world
 Each time an aircraft takes off with an empty
seat, a revenue opportunity is lost forever
 The flight schedule must be designed to
capture as much business as possible,
maximizing revenues with as little direct
operating cost as possible
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Fleet Assignment at Delta (contd)
 The airline industry combines
the capital-intensive quality of the
manufacturing sector
low profit margin quality of the retail sector
 Airlines are capital, fuel, and labor
intensive
 Survival and success depend on the ability
to operate flights along the schedule as
efficiently as possible
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Fleet Assignment at Delta (contd)
 Both the size of the fleet and the number of
different types of aircrafts have significant
impact on schedule planning
 If the airline assigns too small a plane to a
particular market:
it will lose potential passengers
 If it assigns too large a plane:
it will suffer the expense of the larger
plane transporting empty seats
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Stating the LP Model
 Delta implemented a large scale linear
program to assign fleet types to flight
legs so as to minimize a combination of
operating and passenger “spill” costs,
subject to a variety of operation
constraints
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What are the constraints?
 Some of the complicating factors include:
number of aircrafts available in each fleet
planning for scheduled maintenance (which
city is the best to do the maintenance?)
matching which crews have the skills to fly
which aircrafts
providing sufficient opportunity for crew rest
time
range and speed capability of the aircraft
airport restrictions (noise levels!)
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The result?!
 The typical size of the LP model that Delta
has to optimize daily is:
40,000 constraints
60,000 decision variables
 The use of the LP model was expected to
save Delta $300 million over the 3 years
(1997)
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Formulating LP Models
 An LP model is a model that seeks to
maximize or minimize a linear objective
function subject to a set of constraints
 An LP model consists of three parts:
a well-defined set of decision variables
an overall objective to be maximized or
minimized
a set of constraints
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PetCare Problem
PetCare specializes in high quality care for large
dogs. Part of this care includes the assurance that
each dog receives a minimum recommended
amount of protein and fat on a daily basis. Two
different ingredients, Mix 1 and Mix 2, are
combined to create the proper diet for a dog. Each
kg of Mix 1 provides 300 gr of protein, 200 gr of
fat, and costs $.80, while each kg of Mix 2
provides 200 gr of protein, 400 gr of fat, and costs
$.60. If PetCare has a dog that requires at least
1100 gr of protein and 1000 gr of fat, how many
kgs of each mix should be combined to meet the
nutritional requirements at a minimum cost?
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LP Formulation Steps
 STEP 1: Understand the Problem
 STEP 2: Identify the decision variables
 STEP 3: State the objective function
 STEP 4: State the constraints
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PetCare Problem
PetCare specializes in high quality care for large
dogs. Part of this care includes the assurance that
each dog receives a minimum recommended
amount of protein and fat on a daily basis. Two
different ingredients, Mix 1 and Mix 2, are
combined to create the proper diet for a dog. Each
kg of Mix 1 provides 300 gr of protein, 200 gr of
fat, and costs $.80, while each kg of Mix 2
provides 200 gr of protein, 400 gr of fat, and
costs $.60. If PetCare has a dog that requires at
least 1100 gr of protein and 1000 gr of fat, how
many kgs of each mix should be combined to meet
the nutritional requirements at a minimum cost?
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PetCare: LP Formulation
 STEP 1: Understand the Problem
 STEP 2: Identify the decision variables
x1 : kgs of mix 1 to be used to feed the dog
x2 : kgs of mix 2 to be used to feed the dog
 STEP 3: State the objective function
minimize
0.8 x1 + 0.6 x2
(total cost)
 STEP 4: State the constraints
subject to 300 x1 + 200 x2  1100
200 x1 + 400 x2  1000
x1  0
x2  0
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(protein constraint)
(fat constraint)
(sign restriction)
(sign restriction)
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Furnco Company Problem
Furnco manufactures desks and chairs. Each
desk uses 4 units of wood, and each chair uses 3
units of wood. A desk contributes $40 to profit,
and a chair contributes $25. Marketing restrictions
require that the number of chairs produced must
be at least twice the number of desks produced.
There are 20 units of wood available. Formulate
the Linear Programming model to maximize
Furnco’s profit.
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Furnco Company (contd)
x1 : number of desks produced
x2 : number of chairs produced
maximize 40 x1 + 25 x2
function)
subject to
(objective
4 x1 + 3 x2  20 (wood constraint)
2 x1 - x2  0 (marketing constraint)
x1 , x2  0 (sign restrictions)
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Farmer Jane Problem
Farmer Jane owns 45 acres of land. She is going
to plant each acre with wheat or corn. Each acre
planted with wheat yields $200 profit; each with
corn yields $300 profit. The labor and fertilizer
used for each acre are as follows:
Wheat
Corn
Labor
3 workers 2 workers
Fertilizer 2 tons
4 tons
100 workers and 120 tons of fertilizer are
available. Formulate the Linear Programming
model to maximize the farmer’s profit.
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Farmer Jane (contd)
x1 : acres of land planted with wheat
x2 : acres of land planted with corn
maximize
subject to
200 x1 + 300 x2
x1 +
x2  45
(objective function)
(land constraint)
3 x1 + 2 x2  100 (labor constraint)
2 x1 + 4 x2  120 (fertilizer constraint )
x1 , x2  0
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(sign restrictions)
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Truck-Co Company Problem
Truck-co manufactures two types of trucks: 1 and
2. Each truck must go through the painting shop
and the assembly shop. If the painting shop were
completely devoted to painting type 1 trucks, 800
per day could be painted, whereas if it were
completely devoted to painting type 2 trucks, 700
per day could be painted. Is the assembly shop
were completely devoted to assembling truck 1
engines, 1500 per day could be assembled, and if
it were completely devoted to assembling truck 2
engines, 1200 per day could be assembled. Each
type 1 truck contributes $300 to profit; each type
2 truck contributes $500. Formulate the LP
problem to maximize Truckco’s profit.
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Truckco Company (contd)
x1 : number of type 1 trucks manufactured
x2 : number of type 2 trucks manufactured
maximize
300 x1 + 500 x2
(objective function)
subject to
7 x1 + 8 x2  5600
(painting constraint)
12 x1 + 15 x2  18000 (assembly constraint)
x1 , x2
 0 (sign restrictions)
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McDamat’s Fast Food Restaurant
McDamat's fast food restaurant requires different number of full
time employees on different days of the week. The table below
shows the minimum requirements per day of a typical week:
Day of week
Empl Reqd
Day of week
Empl Reqd
Monday
7
Friday
4
Tuesday
3
Saturday
6
Wednesday
5
Sunday
4
Thursday
9
Union rules state that each full-time employee must work 5
consecutive days and then receive 2 days off. The restaurant wants
to meet its daily requirements using only full time personnel.
Formulate the LP model to minimize the number of full time
employees required.
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McDamat’s Fast Food Restaurant
(contd)
 Defining Decision Variables
xi : number of employees beginning
work on day i
where i = Monday,
…. , Sunday
 Defining the Objective Function
min Z = xmon + xtue + xwed + xthu + xfri +
xsat + xsun
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McDamat’s Fast Food Restaurant (contd)
 Defining the Constraint Set
xmon + xthu + xfri + xsat + xsun  7 (Mon Reqts)
xmon + xtue + xfri + xsat + xsun  3 (Tue Reqts)
xmon + xtue + xwed + xsat + xsun  5(Wed Reqts)
xmon + xtue + xwed + xthu + xsun  9(Thu Reqts)
xmon + xtue + xwed + xthu + xfri  4
(Fri Reqts)
xtue + xwed + xthu + xfri + xsat  6 (Sat Reqts)
xwed + xthu + xfri + xsat + xsun  4 (Sun Reqts)
 Non-Negativity Condition (Sign Restriction)
xmon , …. , xsun  0
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A Multi-Period Production Planning Pr.
Sailco Corporation must determine how many sailboats to produce
during each of the next four quarters. The demand during each of
the next four quarters is as follows:
Quarters
1
2
3
4 .
Demand
40
60
75
25
At the beginning of the first quarter Sailco has an inventory of 10
sailboats.
At the beginning of each quarter Sailco must decide how many
sailboats to produce that quarter. Sailboats produced during a
quarter can be used to meet demand for that quarter.
Capacity
Cost
.
Regular Time
40 (sailboats) $400/sailboat
Overtime
$450/sailboat
Inventory Holding Cost: $20/sailboat
Determine a production schedule to minimize the sum of production
and inventory holding costs during the next four quarters.
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A Multiperiod PP Problem (contd)
 Defining Decision Variables
R1 : regular time production at quarter 1
R2 : regular time production at quarter 2
…
…
Rt : regular time production at quarter t
Ot : overtime production at quarter t
It : inventory at the end of quarter t
 Defining the Objective Function
min 400 R1 + 400 R2 + 400 R3 + 400 R4 + 450 O1 + 450 O2 +
450 O3 + 450 O4 + 20 I1 + 20 I2 + 20 I3 + 20 I4
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A Multiperiod PP Problem (contd)
 Defining the Constraint Set
10 + R1 + O1 - I1 = 40
I1 + R2 + O2 - I2 = 60
I2 + R3 + O3 - I3 = 75
I3 + R4 + O4 - I4 = 25
R1  40
R2  40
R3  40
R4  40
 Non-Negativity Condition (Sign Restriction)
R1, R2, R3, R4, O1, O2, O3, O4, I1, I2, I3, I4  0
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LP Summary
 An LP problem is an optimization problem for
which we do the following:
 We attempt to maximize (or minimize) a linear function
of the decision variables. The function that is to be
maximized (or minimized) is called the objective
function
 The values of the decision variables must satisfy a set
of constraints. Each constraint must be a linear
equation or linear inequality
 A sign restriction is associated with each variable
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Graphical Solution Method
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X2
Chairs
7
Furnco Company
Max
40 x1 + 25 x2
s.t.
4 x1 + 3
x2  20
2 x1 - x2  0
x1 , x2  0
(2)
[166.75] 6.67
6
5
4
(2,4) [180]
3
2
1
0
2
2.5
3.754
Z=100
Z=150
5
6
7
X1
Desks
(1)
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Graphical Solution Method (contd)
X2
Corn
(4)
60
Farner Jane (modified)
max
200 x1 + 300 x2
s.t
x1 + x2  45
3 x1 + 2 x2  100
2 x1 + 4 x2  120
x1
≥ 10
x1 , x 2  0
50
45
40
(20,20)
30
(10,25)
20
(30,15)
10
[2000]
20
30
10
0
(2)
33.3
(3)
(1)
40 45
[6667]
Z=6000
50
60
X1
Wheat
Z=7080
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Special Cases of the Feasible Region
Infeasible
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Redundant Constraint
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More Special Cases of the Feasible
Region
Unbounded Feasible Region
Unbounded Solution
Unbounded Feasible Region
Bounded Optimal Solution
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Special Cases of the Optimal Solution
Multiple Optima
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Unbounded Solution
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