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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

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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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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 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)

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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 x x2 (objective function) subject to 4 x1 + 3 x2 20 (wood constraint) 2 x 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) 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)

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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 x x2 (objective function) subject to 7 x x2 (painting constraint) 12 x x2 (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 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.

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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 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.

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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 R R R R O O O O I I I 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**

X2 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

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**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

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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

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