Www.izmirekonomi.edu.tr Asst. Prof. Dr. Mahmut Ali GÖKÇE, Izmir University of Economics Spring,...

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Asst. Asst. Prof. Dr. Prof. Dr. Mahmut Ali Mahmut Ali GÖKÇE, Izmir University of EconomicsGÖKÇE, Izmir University of Economics

SpringSpring, 2007, 2007

ISE 102

Introduction to Linear Introduction to Linear Programming (LP)Programming (LP)

Asst. Prof. Dr. Mahmut Ali GÖKÇEIndustrial Systems Engineering Dept.

İzmir University of Economics

www.izmirekonomi.edu.tr



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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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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)

History of LP (contd)




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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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Aggregate PlanningDevelop a production schedule which satisfies specified sales demands in future periods satisfies limitations on production capacity minimizes total production/inventory costs

Scheduling ProblemProduce a workforce schedule which satisfies minimum staffing requirements utilizes reasonable shifts for the workers is least costly

Typical Applications of LP




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Product Mix (“Blending”) ProblemDevelop the product mix which satisfies restrictions/requirements for customers does not exceed capacity and resource constraints results in highest profit

LogisticsDetermine a distribution system which meets customer demand minimizes transportation costs

Typical Applications of LP (contd)




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Marketing

Determine the media mix which meets a fixed budget maximizes advertising effectiveness

Financial PlanningEstablish 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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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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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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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

Fleet Assignment at Delta (contd)




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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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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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STEP 1: Understand the Problem

STEP 2: Identify the decision variables

STEP 3: State the objective function

STEP 4: State the constraints

LP Formulation Steps




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PetCare ProblemPetCare 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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STEP 1: Understand the Problem STEP 2: Identify the decision variables

x1 : kgs of mix 1 to be used to feed the dogx2 : 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)

PetCare: LP Formulation




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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.

Furnco Company Problem




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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)

Furnco Company (contd)




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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 workersFertilizer 2 tons 4 tons

100 workers and 120 tons of fertilizer are available. Formulate the Linear Programming model to maximize the farmer’s profit.

Farmer Jane Problem




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x1 : acres of land planted with wheat

x2 : acres of land planted with corn


subject to x1 + x2 45 (land constraint)

3 x1 + 2 x2 100 (labor constraint)

2 x1 + 4 x2 120 (fertilizer constraint )


Farmer Jane (contd)




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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.

Truck-Co Company Problem




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x1 : number of type 1 trucks manufactured

x2 : number of type 2 trucks manufactured


subject to 7 x1 + 8 x2 5600 (painting constraint)

12 x1 + 15 x2 18000 (assembly constraint)


Truckco Company (contd)




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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.

McDamat’s Fast Food Restaurant




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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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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

McDamat’s Fast Food Restaurant (contd)




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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/sailboatOvertime $450/sailboat

Inventory Holding Cost: $20/sailboatDetermine 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 1R2 : regular time production at quarter 2

… …

Rt : regular time production at quarter tOt : overtime production at quarter tIt : 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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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

LP Summary




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Furnco CompanyMax 40 x1 + 25 x2

s.t. 4 x1 + 3 x2 20

2 x1 - x2 0 x1 , x2 0

Graphical Solution Method X2

Chairs

X1

Desks

7

6

5

4

3

2

1

2 4 5 6 7

Z=100

2.5 3.75

Z=150

0

(2)

(1)

6.67

(2,4)[180]

[166.75]




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Farner Jane (modified)max 200 x1 + 300 x2

s.t x1 + x2 453 x1 + 2 x2 100

2 x1 + 4 x2 120 x1 ≥ 10 x1 , x2 0

Graphical Solution Method (contd)

X1

Wheat

60

50

40

30

20

10

10 20 40 5030 60

0

X2

Corn(4)

[2000]

Z=6000

33.3

(2)

[6667]

45

45

(1)(3)

(30,15)

(20,20)

(10,25)

Z=7080




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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 RegionUnbounded Solution

Unbounded Feasible RegionBounded Optimal Solution




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Special Cases of the Optimal Solution

Multiple Optima Unbounded Solution

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