1. ISE 102 Introduction to Linear Programming (LP)
Asst. Prof. Dr. Mahmut Ali GÖKÇE
Industrial Systems Engineering Dept.
İzmir University of Economics

2. 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)

3. 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

4. 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

5. 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)

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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!)

17. 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)

18. 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

19. 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?

20. 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

21. 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?

22. 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 x x2 (total cost)
STEP 4: State the constraints
subject to 300 x x2  (protein constraint)
200 x x2  1000 (fat constraint)
x1  0 (sign restriction)
x2  0 (sign restriction)

23. 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.

24. Furnco Company (contd)
x1 : number of desks produced
x2 : number of chairs produced
maximize 40 x x2 (objective function)
subject to 4 x1 + 3 x2  20 (wood constraint)
2 x x2  0 (marketing constraint)
x1 , x2  0 (sign restrictions)

25. 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.

26. Farmer Jane (contd) 3 x1 + 2 x2  100 (labor constraint)
x1 : acres of land planted with wheat
x2 : acres of land planted with corn
maximize 200 x x2 (objective function)
subject to x x2  45 (land constraint)
3 x1 + 2 x2  100 (labor constraint)
2 x1 + 4 x2  120 (fertilizer constraint )
x1 , x2  0 (sign restrictions)

27. 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.

28. Truckco Company (contd)
x1 : number of type 1 trucks manufactured
x2 : number of type 2 trucks manufactured
maximize 300 x x2 (objective function)
subject to 7 x x2  (painting constraint)
12 x x2  (assembly constraint)
x1 , x2  0 (sign restrictions)

29. 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 Friday 4
Tuesday Saturday 6
Wednesday Sunday 4
Thursday
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.

30. 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

31. 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

32. 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
Demand
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.

33. 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 R R R R O O O O I I I I4

34. 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

35. 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

36. Graphical Solution MethodX2
Chairs X1
Desks 7 6 5 4 3 2 1
Z=100
2.5
3.75
Z=150
(2)
(1)
6.67
(2,4)
[180]
[166.75]
Furnco Company
Max 40 x x2
s.t. 4 x1 + 3 x2  20
2 x x2  0
x1 , x2  0

37. Graphical Solution Method (contd)X1
Wheat 60 50 40 30 20 10 X2
Corn
(4)
[2000]
Z=6000
33.3
(2)
[6667] 45
(1)
(3)
(30,15)
(20,20)
(10,25)
Z=7080
Farner Jane (modified)
max 200 x x2
s.t x x2  45
3 x1 + 2 x2  100
2 x1 + 4 x2  120
x ≥ 10
x1 , x2  0

38. Special Cases of the Feasible Region
Infeasible
Redundant Constraint

39. More Special Cases of the Feasible Region
Unbounded Feasible Region
Unbounded Solution
Unbounded Feasible Region
Bounded Optimal Solution

40. Special Cases of the Optimal Solution
Multiple Optima
Unbounded Solution