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ISE 102. Introduction to Linear Programming (LP) Asst. Prof. Dr. Mahmut Ali GÖKÇE Industrial Systems Engineering Dept. İzmir University of Economics. Many managerial decisions involve trying to make the most effective use of an organization’s resources. Resources typically include:

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### ISE 102

Introduction to Linear Programming (LP)

Asst. Prof. Dr. Mahmut Ali GÖKÇE

Industrial Systems Engineering Dept.

İzmir University of Economics

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)

Introduction to Linear Programming
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
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
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)
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
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

Typical Applications of LP
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
Marketing

Determine the media mix which

meets a fixed budget

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

Typical Applications of LP (contd)

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

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

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

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

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

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
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!)
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)
Formulating LP Models
• An LP model is a model that seeks to maximize or minimize a linear objective functionsubject 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
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 200gr of protein, 400gr of fat, and costs \$.60. If PetCare has a dog that requires at least 1100gr of protein and 1000gr of fat, how many kgs of each mix should be combined to meet the nutritional requirements at a minimum cost?

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
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 300gr of protein, 200gr of fat, and costs \$.80, while each kg of Mix 2 provides 200gr of protein, 400gr of fat, and costs \$.60. If PetCare has a dog that requires at least 1100gr of protein and 1000gr of fat, how many kgs of each mix should be combined to meet the nutritional requirements at a minimum cost?

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 to300x1 + 200x2 1100 (protein constraint)

200x1 + 400x2 1000 (fat constraint)

x1 0 (sign restriction)

x2 0 (sign restriction)

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.

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)

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.

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)

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.

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)

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 weekEmpl ReqdDay of weekEmpl 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 (contd)
• Defining Decision Variables

xi : number of employees beginning work on day iwhere i = Monday, …. , Sunday

• Defining the Objective Function

min Z = xmon + xtue + xwed + xthu + xfri + xsat + xsun

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

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.

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

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

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

X2

Chairs

7

(2)

[166.75]

6.67

6

5

4

(2,4)

[180]

3

2

1

0

2.5

3.75

2

4

7

X1

Desks

5

6

(1)

Z=150

Z=100

Graphical Solution Method

Furnco Company

Max40 x1 + 25 x2

s.t. 4 x1 + 3 x2 20

2 x1 - x2 0

x1 , x2 0

X2

Corn

(4)

60

50

45

40

(20,20)

30

(10,25)

20

(30,15)

10

(3)

(2)

(1)

0

33.3

45

10

20

X1

Wheat

30

40

50

60

[2000]

[6667]

Z=7080

Z=6000

Graphical Solution Method (contd)

Farner Jane (modified)

max 200 x1 + 300 x2

s.tx1 + x2 45

3 x1 + 2 x2 100

2 x1 + 4 x2 120

x1 ≥10

x1 , x2 0

Special Cases of the Feasible Region

Infeasible

Redundant Constraint

More Special Cases of the Feasible Region

Unbounded Feasible Region

Unbounded Solution

Unbounded Feasible Region

Bounded Optimal Solution

Special Cases of the Optimal Solution

Multiple Optima

Unbounded Solution