LINEAR OPTIMIZATION

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LINEAR OPTIMIZATION

  1. 1. 1 LINEAR OPTIMIZATIONLINEAR OPTIMIZATION BYBY Er. ASHISH BANSODEEr. ASHISH BANSODE M.E. (WATER RESOURCES ENGINEERING)M.E. (WATER RESOURCES ENGINEERING)
  2. 2. 2 Introduction To Linear ProgrammingIntroduction To Linear Programming Today many of the resources needed as inputs toToday many of the resources needed as inputs to operations are in limited supply.operations are in limited supply. Operations managers must understand the impact ofOperations managers must understand the impact of this situation on meeting their objectives.this situation on meeting their objectives. Linear programming (LP) is one way that operationsLinear programming (LP) is one way that operations managers can determine how best to allocate theirmanagers can determine how best to allocate their scarce resourcesscarce resources..
  3. 3. 3 Linear Programming (LP) in OMLinear Programming (LP) in OM There are five common types of decisions inThere are five common types of decisions in which LP may play a rolewhich LP may play a role Product mixProduct mix Production planProduction plan Ingredient mixIngredient mix TransportationTransportation AssignmentAssignment
  4. 4. 4 LP Problems in OM: Product MixLP Problems in OM: Product Mix ObjectiveObjective To select the mix of products or services that resultsTo select the mix of products or services that results in maximum profits for the planning periodin maximum profits for the planning period Decision VariablesDecision Variables How much to produce and market of each product orHow much to produce and market of each product or service for the planning periodservice for the planning period ConstraintsConstraints Maximum amount of each product or serviceMaximum amount of each product or service demanded; Minimum amount of product or servicedemanded; Minimum amount of product or service policy will allow; Maximum amount of resourcespolicy will allow; Maximum amount of resources availableavailable
  5. 5. 5 LP Problems in OM: Production PlanLP Problems in OM: Production Plan ObjectiveObjective To select the mix of products or services that resultsTo select the mix of products or services that results in maximum profits for the planning periodin maximum profits for the planning period Decision VariablesDecision Variables How much to produce on straight-time labor andHow much to produce on straight-time labor and overtime labor during each month of the yearovertime labor during each month of the year ConstraintsConstraints Amount of products demanded in each month;Amount of products demanded in each month; Maximum labor and machine capacity available inMaximum labor and machine capacity available in each month; Maximum inventory space available ineach month; Maximum inventory space available in each montheach month
  6. 6. 6 Recognizing LP ProblemsRecognizing LP Problems Characteristics of LP Problems in OMCharacteristics of LP Problems in OM A well-defined single objective must be stated.A well-defined single objective must be stated. There must be alternative courses of action.There must be alternative courses of action. The total achievement of the objective must beThe total achievement of the objective must be constrained by scarce resources or other restraints.constrained by scarce resources or other restraints. The objective and each of the constraints must beThe objective and each of the constraints must be expressed as linear mathematical functions.expressed as linear mathematical functions.
  7. 7. 7 Steps in Formulating LP ProblemsSteps in Formulating LP Problems 1.1. Define the objective. (min or max)Define the objective. (min or max) 2.2. Define the decision variables. (positive, binary)Define the decision variables. (positive, binary) 3.3. Write the mathematical function for the objective.Write the mathematical function for the objective. 4.4. Write a 1- or 2-word description of each constraint.Write a 1- or 2-word description of each constraint. 5.5. Write the right-hand side (RHS) of each constraint.Write the right-hand side (RHS) of each constraint. 6.6. WriteWrite <<, =, or, =, or >> for each constraint.for each constraint. 7.7. Write the decision variables on LHS of each constraint.Write the decision variables on LHS of each constraint. 8.8. Write the coefficient for each decision variable in eachWrite the coefficient for each decision variable in each constraint.constraint.
  8. 8. 8 Cycle Trends is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from aluminum and steel alloys. The anticipated unit profits are $10 for the Deluxe and $15 for the Professional. The number of pounds of each alloy needed per frame is summarized on the next slide. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly. How many Deluxe and Professional frames should Cycle Trends produce each week? Example: LP FormulationExample: LP Formulation
  9. 9. 9 Aluminum AlloyAluminum Alloy Steel AlloySteel Alloy DeluxeDeluxe 2 32 3 ProfessionalProfessional 44 22 Example: LP FormulationExample: LP Formulation Pounds of each alloy needed per frame
  10. 10. 10 Example: LP FormulationExample: LP Formulation Define the objective Maximize total weekly profit Define the decision variables x1 = number of Deluxe frames produced weekly x2 = number of Professional frames produced weekly Write the mathematical objective function Max Z = 10x1 + 15x2
  11. 11. 11 Example: LP FormulationExample: LP Formulation Write a one- or two-word description of each constraint Aluminum available Steel available Write the right-hand side of each constraint 100 80 Write <, =, > for each constraint < 100 < 80
  12. 12. 12 Example: LP FormulationExample: LP Formulation Write all the decision variables on the left-hand sideWrite all the decision variables on the left-hand side of each constraintof each constraint x1 x2 < 100 x1 x2 < 80 Write the coefficient for each decision in each constraint + 2x1 + 4x2 < 100 + 3x1 + 2x2 < 80
  13. 13. 13 Example: LP FormulationExample: LP Formulation LP in Final FormLP in Final Form Max Z = 10x1 + 15x2 Subject To 2x1 + 4x2 < 100 ( aluminum constraint) 3x1 + 2x2 < 80 ( steel constraint) x1 , x2 >> 0 (non-negativity constraints)
  14. 14. 14 Montana Wood Products manufacturers two- high quality products, tables and chairs. Its profit is $15 per chair and $21 per table. Weekly production is constrained by available labor and wood. Each chair requires 4 labor hours and 8 board feet of wood while each table requires 3 labor hours and 12 board feet of wood. Available wood is 2400 board feet and available labor is 920 hours. Management also requires at least 40 tables and at least 4 chairs be produced for every table produced. To maximize profits, how many chairs and tables should be produced? Example: LP FormulationExample: LP Formulation
  15. 15. 15 Example: LP FormulationExample: LP Formulation Define the objective Maximize total weekly profit Define the decision variables x1 = number of chairs produced weekly x2 = number of tables produced weekly Write the mathematical objective function Max Z = 15x1 + 21x2
  16. 16. 16 Example: LP FormulationExample: LP Formulation Write a one- or two-word description of each constraint Labor hours available Board feet available At least 40 tables At least 4 chairs for every table Write the right-hand side of each constraint 920 2400 40 4 to 1 ratio Write <, =, > for each constraint < 920 < 2400 >> 4040 4 to 11
  17. 17. 17 Example: LP FormulationExample: LP Formulation Write all the decision variables on the left-hand side of eachWrite all the decision variables on the left-hand side of each constraintconstraint x1 x2 < 920 x1 x2 < 2400 x2 >> 4040 4 to 1 ratio  x1 / x2 ≥ 4/1 Write the coefficient for each decision in each constraint + 4x1 + 3x2 < 920 + 8x1 + 12x2 < 2400 x2 >> 4040 x1≥4 x2
  18. 18. 18 Example: LP FormulationExample: LP Formulation LP in Final FormLP in Final Form Max Z = 15x1 + 21x2 Subject To 4x1 + 3x2 < 920 ( labor constraint) 8x1 + 12x2 < 2400 ( wood constraint) x2 - 40 >> 0 (make at least 40 tables)0 (make at least 40 tables) x1 - 4 x2 >> 0 (at least 4 chairs for every table)0 (at least 4 chairs for every table) x1 , x2 >> 0 (non-negativity constraints)
  19. 19. 19 The Sureset Concrete Company produces concrete. Two ingredients in concrete are sand (costs $6 per ton) and gravel (costs $8 per ton). Sand and gravel together must make up exactly 75% of the weight of the concrete. Also, no more than 40% of the concrete can be sand and at least 30% of the concrete be gravel. Each day 2000 tons of concrete are produced. To minimize costs, how many tons of gravel and sand should be purchased each day? Example: LP FormulationExample: LP Formulation
  20. 20. 20 Example: LP FormulationExample: LP Formulation Define the objective Minimize daily costs Define the decision variables x1 = tons of sand purchased x2 = tons of gravel purchased Write the mathematical objective function Min Z = 6x1 + 8x2
  21. 21. 21 Example: LP FormulationExample: LP Formulation Write a one- or two-word description of each constraint 75% must be sand and gravel No more than 40% must be sand At least 30% must be gravel Write the right-hand side of each constraint .75(2000) .40(2000) .30(2000) Write <, =, > for each constraint = 1500 < 800800 >> 600600
  22. 22. 22 Example: LP FormulationExample: LP Formulation Write all the decision variables on the left-hand sideWrite all the decision variables on the left-hand side of each constraintof each constraint x1 x2 = 1500 x1< 800800 x2 >> 600600 Write the coefficient for each decision in each constraint + x1 + x2 = 1500 + x1 < 800 x2 >> 600600
  23. 23. 23 Example: LP FormulationExample: LP Formulation LP in Final FormLP in Final Form Min Z = 6x1 + 8x2 Subject To x1 + x2 = 1500 ( mix constraint) x1 < 800 ( mix constraint) x2 >> 600 ( mix constraint )600 ( mix constraint ) x1 , x2 >> 0 (non-negativity constraints)
  24. 24. 24 LP Problems in GeneralLP Problems in General Units of each term in a constraint must be the same asUnits of each term in a constraint must be the same as the RHSthe RHS Units of each term in the objective function must beUnits of each term in the objective function must be the same as Zthe same as Z Units between constraints do not have to be the sameUnits between constraints do not have to be the same LP problem can have a mixture of constraint typesLP problem can have a mixture of constraint types
  25. 25. 25 LP ProblemLP Problem Galaxy Ind. produces two water guns, the Space RayGalaxy Ind. produces two water guns, the Space Ray and the Zapper. Galaxy earns a profit of $3 for everyand the Zapper. Galaxy earns a profit of $3 for every Space Ray and $2 for every Zapper. Space Rays andSpace Ray and $2 for every Zapper. Space Rays and Zappers require 2 and 4 production minutes per unit,Zappers require 2 and 4 production minutes per unit, respectively. Also, Space Rays and Zappers require .5respectively. Also, Space Rays and Zappers require .5 and .3 pounds of plastic, respectively. Givenand .3 pounds of plastic, respectively. Given constraints of 40 production hours, 1200 pounds ofconstraints of 40 production hours, 1200 pounds of plastic, Space Ray production can’t exceed Zapperplastic, Space Ray production can’t exceed Zapper production by more than 450 units; formulate theproduction by more than 450 units; formulate the problem such that Galaxy maximizes profit.problem such that Galaxy maximizes profit.
  26. 26. 26 R = # of Space Rays to produceR = # of Space Rays to produce Z = # of Zappers to produceZ = # of Zappers to produce Max Z = 3.00R + 2.00ZMax Z = 3.00R + 2.00Z STST 2R + 4Z2R + 4Z ≤ 2400≤ 2400 can’t exceed available hours (40*60)can’t exceed available hours (40*60) .5R + .3Z ≤ 1200.5R + .3Z ≤ 1200 can’t exceed available plasticcan’t exceed available plastic R - S ≤ 450R - S ≤ 450 Space Rays can’t exceed Zappers by moreSpace Rays can’t exceed Zappers by more than 450than 450 R, S ≥ 0R, S ≥ 0 non-negativity constraintnon-negativity constraint LP ModelLP Model
  27. 27. 27 The White Horse Apple Products Company purchases applesThe White Horse Apple Products Company purchases apples from local growers and makes applesauce and apple juice. Itfrom local growers and makes applesauce and apple juice. It costs $0.60 to produce a jar of applesauce and $0.85 tocosts $0.60 to produce a jar of applesauce and $0.85 to produce a bottle of apple juice. The company has a policy thatproduce a bottle of apple juice. The company has a policy that at least 30% but not more than 60% of its output must beat least 30% but not more than 60% of its output must be applesauce.applesauce. The company wants to meet but not exceed demand forThe company wants to meet but not exceed demand for each product. The marketing manager estimates that theeach product. The marketing manager estimates that the maximum demand for applesauce is 5,000 jars, plus anmaximum demand for applesauce is 5,000 jars, plus an additional 3 jars for each $1 spent on advertising. Maximumadditional 3 jars for each $1 spent on advertising. Maximum demand for apple juice is estimated to be 4,000 bottles, plus andemand for apple juice is estimated to be 4,000 bottles, plus an additional 5 bottles for every $1 spent to promote apple juice.additional 5 bottles for every $1 spent to promote apple juice. The company has $16,000 to spend on producing andThe company has $16,000 to spend on producing and advertising applesauce and apple juice. Applesauce sells foradvertising applesauce and apple juice. Applesauce sells for $1.45 per jar; apple juice sells for $1.75 per bottle. The$1.45 per jar; apple juice sells for $1.75 per bottle. The company wants to know how many units of each to producecompany wants to know how many units of each to produce and how much advertising to spend on each in order toand how much advertising to spend on each in order to maximize profit.maximize profit. LP ProblemLP Problem
  28. 28. 28 S = # jars apple Sauce to makeS = # jars apple Sauce to make J = # bottles apple Juice to makeJ = # bottles apple Juice to make SA = $ for apple Sauce AdvertisingSA = $ for apple Sauce Advertising JA = $ for apple Juice AdvertisingJA = $ for apple Juice Advertising Max Z = 1.45S + 1.75J - .6S - .85J – SA – JAMax Z = 1.45S + 1.75J - .6S - .85J – SA – JA STST SS ≥ .3(S + J)≥ .3(S + J) at least 30% apple sauceat least 30% apple sauce S ≤ .6(S + J)S ≤ .6(S + J) no more than 60% apple sauceno more than 60% apple sauce S ≤ 5000 + 3SAS ≤ 5000 + 3SA don’t exceed demand for apple saucedon’t exceed demand for apple sauce J ≤ 4000 + 5JAJ ≤ 4000 + 5JA don’t exceed demand for apple juicedon’t exceed demand for apple juice .6S + .85J + SA + JA ≤ 16000.6S + .85J + SA + JA ≤ 16000 budgetbudget LP ModelLP Model
  29. 29. 29 A ship has two cargo holds, one fore and one aft. The fore cargo hold hasA ship has two cargo holds, one fore and one aft. The fore cargo hold has a weight capacity of 70,000 pounds and a volume capacity of 30,000 cubica weight capacity of 70,000 pounds and a volume capacity of 30,000 cubic feet. The aft hold has a weight capacity of 90,000 pounds and a volumefeet. The aft hold has a weight capacity of 90,000 pounds and a volume capacity of 40,000 cubic feet. The shipowner has contracted to carry loadscapacity of 40,000 cubic feet. The shipowner has contracted to carry loads of packaged beef and grain. The total weight of the available beef isof packaged beef and grain. The total weight of the available beef is 85,000 pounds; the total weight of the available grain is 100,000 pounds.85,000 pounds; the total weight of the available grain is 100,000 pounds. The volume per mass of the beef is 0.2 cubic foot per pound, and theThe volume per mass of the beef is 0.2 cubic foot per pound, and the volume per mass of the grain is 0.4 cubic foot per pound. The profit forvolume per mass of the grain is 0.4 cubic foot per pound. The profit for shipping beef is $0.35 per pound, and the profit for shipping grain is $0.12shipping beef is $0.35 per pound, and the profit for shipping grain is $0.12 per pound. The shipowner is free to accept all or part of the availableper pound. The shipowner is free to accept all or part of the available cargo; he wants to know how much meat and grain to accept in order tocargo; he wants to know how much meat and grain to accept in order to maximize profit.maximize profit. LP ProblemLP Problem
  30. 30. 30 BF = # lbs beef to load in fore cargo holdBF = # lbs beef to load in fore cargo hold BA = # lbs beef to load in aft cargo holdBA = # lbs beef to load in aft cargo hold GF = # lbs grain to load in fore cargo holdGF = # lbs grain to load in fore cargo hold GA = # lbs grain to load in aft cargo holdGA = # lbs grain to load in aft cargo hold Max Z = .35 BF + .35BA + .12GF + .12 GAMax Z = .35 BF + .35BA + .12GF + .12 GA STST BF + GFBF + GF ≤ 70000≤ 70000 fore weight capacity – lbsfore weight capacity – lbs BA + GA ≤ 90000BA + GA ≤ 90000 aft weight capacity – lbsaft weight capacity – lbs .2BF + .4GF ≤ 30000.2BF + .4GF ≤ 30000 for volume capacity – cubic feetfor volume capacity – cubic feet .2BA + .4GA ≤ 40000.2BA + .4GA ≤ 40000 for volume capacity – cubic feetfor volume capacity – cubic feet BF + BA ≤ 85000BF + BA ≤ 85000 max beef availablemax beef available GF + GA ≤ 100000GF + GA ≤ 100000 max grain availablemax grain available LP ModelLP Model
  31. 31. 31 In the summer, the City of Sunset Beach staffs lifeguardIn the summer, the City of Sunset Beach staffs lifeguard stations seven days a week. Regulations require that citystations seven days a week. Regulations require that city employees (including lifeguards) work five days a week andemployees (including lifeguards) work five days a week and be given two consecutive days off. Insurance requirementsbe given two consecutive days off. Insurance requirements mandate that Sunset Beach provide at least one lifeguard permandate that Sunset Beach provide at least one lifeguard per 8000 average daily attendance on any given day. The average8000 average daily attendance on any given day. The average daily attendance figures by day are as follows: Sunday –daily attendance figures by day are as follows: Sunday – 58,000, Monday – 42,000, Tuesday – 35,000, Wednesday –58,000, Monday – 42,000, Tuesday – 35,000, Wednesday – 25,000, Thursday – 44,000, Friday – 51,000 and Saturday –25,000, Thursday – 44,000, Friday – 51,000 and Saturday – 68,000. Given a tight budget constraint, the city would like to68,000. Given a tight budget constraint, the city would like to determine a schedule that will employ as few lifeguards asdetermine a schedule that will employ as few lifeguards as possible.possible. LP ProblemLP Problem
  32. 32. 32 X1 = number of lifeguards scheduled to begin on SundayX1 = number of lifeguards scheduled to begin on Sunday X2 =X2 = ““ ““ ““ ““ “ Monday“ Monday X3 =X3 = ““ ““ ““ ““ “ Tuesday“ Tuesday X4 =X4 = ““ ““ ““ ““ “ Wednesday“ Wednesday X5 =X5 = ““ ““ ““ ““ “ Thursday“ Thursday X6 =X6 = ““ ““ ““ ““ “ Friday“ Friday X7 =X7 = ““ ““ ““ ““ “ Saturday“ Saturday LP ModelLP Model
  33. 33. 33 Min X1 + X2 + X3 + X4 + X5 + X6 + X7Min X1 + X2 + X3 + X4 + X5 + X6 + X7 STST X1 + X4 + X5 + X6 +X7 ≥ 8 (Sunday)X1 + X4 + X5 + X6 +X7 ≥ 8 (Sunday) X1 + X2 + X5 + X6 +X7 ≥ 6 (Monday)X1 + X2 + X5 + X6 +X7 ≥ 6 (Monday) X1 + X2 + X3 + X6 +X7 ≥ 5 (Tuesday)X1 + X2 + X3 + X6 +X7 ≥ 5 (Tuesday) X1 + X2 + X3 + X4 +X7 ≥ 4 (Wednesday)X1 + X2 + X3 + X4 +X7 ≥ 4 (Wednesday) X1 + X2 + X3 + X4 +X5 ≥ 6 (Thursday)X1 + X2 + X3 + X4 +X5 ≥ 6 (Thursday) X2 + X3 + X4 + X5 +X6 ≥ 7 (Friday)X2 + X3 + X4 + X5 +X6 ≥ 7 (Friday) X3 + X4 + X5 + X6 +X7 ≥ 9 (Saturday)X3 + X4 + X5 + X6 +X7 ≥ 9 (Saturday) All variables ≥ 0 and integerAll variables ≥ 0 and integer LP ModelLP Model

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