Presentation on theme: "ISE 102 Introduction to Linear Programming (LP)"— Presentation transcript:
1 ISE 102 Introduction to Linear Programming (LP) Asst. Prof. Dr. Mahmut Ali GÖKÇEIndustrial 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/equipmentLaborMoneyTimeEnergyRaw materialsThese resources may be used to produceProducts (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 resourcesLinear 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 LPLP was developed to solve military logistics problems during World War IIIn 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 fieldsSimplex Method is still the most important solution method for LP problemsIn recent years, a more efficient method for extremely large problems has been developed (Karmarkar’s Algorithm)
6 LP ProblemsA large number of real problems can be formulated and solved using LP. A partial list includes:Scheduling of personnelProduction planning and inventory controlAssignment problemsSeveral varieties of blending problems including ice cream, steel making, crude oil processingDistribution and logistics problems
7 Typical Applications of LP Aggregate PlanningDevelop a production schedule whichsatisfies specified sales demands in future periodssatisfies limitations on production capacityminimizes total production/inventory costsScheduling ProblemProduce a workforce schedule whichsatisfies minimum staffing requirementsutilizes reasonable shifts for the workersis least costly
8 Typical Applications of LP (contd) Product Mix (“Blending”) ProblemDevelop the product mix whichsatisfies restrictions/requirements for customersdoes not exceed capacity and resource constraintsresults in highest profitLogisticsDetermine a distribution system whichmeets customer demandminimizes transportation costs
9 Typical Applications of LP (contd) MarketingDetermine the media mix whichmeets a fixed budgetmaximizes advertising effectivenessFinancial PlanningEstablish an investment portfolio whichmeets the total investment amountmeets any maximum/minimum restrictions of investing in the available alternativesmaximizes 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 problemAll 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 LinesDelta 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 passengersDelta 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 worldEach time an aircraft takes off with an empty seat, a revenue opportunity is lost foreverThe 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 combinesthe capital-intensive quality of the manufacturing sectorlow profit margin quality of the retail sectorAirlines are capital, fuel, and labor intensiveSurvival 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 planningIf the airline assigns too small a plane to a particular market:it will lose potential passengersIf it assigns too large a plane:it will suffer the expense of the larger plane transporting empty seats
15 Stating the LP ModelDelta 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 fleetplanning for scheduled maintenance (which city is the best to do the maintenance?)matching which crews have the skills to fly which aircraftsproviding sufficient opportunity for crew rest timerange and speed capability of the aircraftairport restrictions (noise levels!)
17 The result?!The typical size of the LP model that Delta has to optimize daily is:40,000 constraints60,000 decision variablesThe use of the LP model was expected to save Delta $300 million over the 3 years (1997)
18 Formulating LP ModelsAn LP model is a model that seeks to maximize or minimize a linear objective function subject to a set of constraintsAn LP model consists of three parts:a well-defined set of decision variablesan overall objective to be maximized or minimizeda set of constraints
19 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?
20 LP Formulation Steps STEP 1: Understand the Problem STEP 2: Identify the decision variablesSTEP 3: State the objective functionSTEP 4: State the constraints
21 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?
22 PetCare: LP Formulation STEP 1: Understand the ProblemSTEP 2: Identify the decision variablesx1 : kgs of mix 1 to be used to feed the dogx2 : kgs of mix 2 to be used to feed the dogSTEP 3: State the objective functionminimize 0.8 x x2 (total cost)STEP 4: State the constraintssubject 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 producedx2 : number of chairs producedmaximize 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 ProblemFarmer 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 CornLabor 3 workers 2 workersFertilizer 2 tons 4 tons100 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 wheatx2 : acres of land planted with cornmaximize 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 manufacturedx2 : number of type 2 trucks manufacturedmaximize 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 ReqdMonday Friday 4Tuesday Saturday 6Wednesday Sunday 4ThursdayUnion 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 Variablesxi : number of employees beginning work on day i where i = Monday, …. , SundayDefining the Objective Functionmin Z = xmon + xtue + xwed + xthu + xfri + xsat + xsun
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:QuartersDemandAt 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 CostRegular Time 40 (sailboats) $400/sailboatOvertime $450/sailboatInventory Holding Cost: $20/sailboatDetermine 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 VariablesR1 : 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 tDefining the Objective Functionmin 400 R R R R O O O O I I I I4
35 LP SummaryAn 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 functionThe values of the decision variables must satisfy a set of constraints. Each constraint must be a linear equation or linear inequalityA sign restriction is associated with each variable