ISE 102

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ISE 102

• Introduction to Linear Programming (LP)
• Asst. Prof. Dr. Mahmut Ali GÖKÇE
• Industrial Systems Engineering Dept.
• Izmir University of Economics

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Introduction to Linear Programming

• Many managerial decisions involve trying to make
  the most effective use of an organizations
  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
  (Karmarkars 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
  (protein constraint)
• 200 x1 400 x2 ? 1000 (fat
  constraint)
• x1 ? 0 (sign restriction)
• x2 ? 0 (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 Furncos
  profit.

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Furnco Company (contd)

• x1 number of desks produced
• x2 number of chairs produced
• maximize 40 x1 25 x2 (objective function)
• subject to 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 farmers profit.

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Farmer Jane (contd)

• x1 acres of land planted with wheat
• x2 acres of land planted with corn
• maximize 200 x1 300 x2 (objective function)
• subject to x1 x2 ? 45 (land constraint)
• 3 x1 2 x2 ? 100 (labor constraint)
• 2 x1 4 x2 ? 120 (fertilizer constraint )
• x1 , x2 ? 0 (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 Truckcos 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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McDamats 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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McDamats 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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McDamats 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

• Furnco Company
• Max 40 x1 25 x2
• s.t. 4 x1 3 x2 ? 20
• 2 x1 - x2 ? 0
• x1 , x2 ? 0

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Graphical Solution Method (contd)

• 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 , x2 ? 0

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Special Cases of the Feasible Region
Infeasible
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
Unbounded Solution