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ISQA 511 Dr. Mellie Pullman EVALUATING CONSTRAINED RESOURCES W/ LINEAR PROGRAMMING Tinker Toys We need to allocate scarce resources among several alternatives resources= ? alternatives=? Need to get into teams Your job is to produce Tinkertoys with three products (Turnstiles, Robots, & Front Wheel Assemblies) Parts Required and Availability Number of Robot Turnstile Front Wheel Parts left over parts available Part Units required Units required Units required Blue Rods 12 4 Orange Rods 12 1 1 1 Green Rods 10 1 One-hole Spools 18 1 1 2 Multi-hole Spools 6 1 1 Wood Caps 8 1 1 Wood Washers 6 1 Wood Bearings 6 1 Yellow Rods 10 2 1 Objectives 1) Make as many of the three finished products as possible to maximize the total number of toys produced, how many of each type of toy should be made? 2) Make the number of finished products that make the most revenue. Robots@ $30, Turnstiles @ $10, Front Wheel Assemblies @ $20. Maximize number of toys Number of Robot * units Turnstile * units Front Wheel * units Total Parts left parts Parts over available used Part Units Units Units required required required Blue Rods 12 4 Orange Rods 12 1 1 1 Green Rods 10 1 One-hole 18 1 1 2 Spools Multi-hole 6 1 1 Spools Wood Caps 8 1 1 Wood Washers 6 1 Wood 6 1 Bearings Yellow Rods 10 2 1 TOTAL UNITS Maximize Overall Profit Number of Robot * units Turnstile * units Front Wheel * units Total Parts left parts Parts over available used Part Units Units Units required required required Blue Rods 12 4 Orange Rods 12 1 1 1 Green Rods 10 1 One-hole 18 1 1 2 Spools Multi-hole 6 1 1 Spools Wood Caps 8 1 1 Wood Washers 6 1 Wood 6 1 Bearings Yellow Rods 10 2 1 TOTAL UNITS Profit $30 $10 $20 Determining the Optimal Strategy in a constrained resource world Try multiple attempts with different scenarios OR Use Linear Programming (LP) You will need to install Solver on your laptop In Excel: Click Tools Click Add-ins Click Solver Add-in Where to find it in Excel 2007 1 2 3 What is Linear Programming? A sequence of steps that will lead to an optimal solution. Used to allocate scarce resources (energy, food, land) assign labor (shifts, Reg vs. OT, productivity) determine lowest cost and emission transportation schemes solve blending problems (food, chemicals or portfolios) solve many other types of constrained resources problems Four essential conditions: Explicit Objective: What are we maximizing or minimizing? Usually profit, units, costs, emissions, labor hours, etc. Limiting resources create constraints: workers, equipment, parts, budgets, etc. Linearity (2 is twice as good as 1, if it takes 3 hours to make 1 part then it takes 6 hours to make 2 parts) Homogeneity (each worker has an average productivity) Bank Loan Processing A credit checking company requires different processing times for consumer loans. Housing loans (H) require 1 hour of credit review and 4 hours of appraising. Car loans (C) require 1 hour of credit review and 1 hour of appraising. The credit reviewers have 200 hours available; the appraisers have 400 hours available. Evaluating Housing loans yields $10 profit while evaluating Cars yields $5 profit. How many of each loan type should the company take? Graphical Approach (2 variables) Formulate the problem in mathematical equations Plot all the Equations Determine the area of feasibility Maximizing problem: feasible area is on or below the lines Minimization: feasible area is on or above the lines Plot a few Profit line (Iso-profit) by setting profit equation = different values. Answer point will be one of the corner points (most extreme) Equations Maximize Profit : $10 H + $5 C Constrained Resources 1H + 1C < 200 (credit reviewing hours) 4H + 1C < 400 (appraising hours) H>0; C>0 (non-negative) H= ? C=? Farmer Gail (land and resource limits) Farmer Gail in Pendleton owns 45 acres of land. Gail is going to plant each acre with wheat or corn. Each acre planted with wheat yields $200 profit while corn yields $300. The labor and fertilizer needed for each acre given below. 100 workers and 120 tons of fertilizer are available. Wheat Corn Labor /acre 3 workers 2 workers Fertilizer/acre 2 tons 4 tons Farmer’s Wheat and Corn Problem Variables: Acres planted in wheat = W Acres planted in corn = C Objective Function: : Maximize profit $200 W + $300 C Constraints: Labor: 3 W + 2 C < 100 Fertilizer: 2 W + 4 C < 120 Land: 1W + 1 C < 45 Non-Negativity: P1 & P2 > 0 Solver Set-up on Excel These 2 cells will change to find the solution. They represent W & C (our unknowns) Wheat Corn LSE RSE VARIABLES 0 0 =SUMPRODUCT(C2:D2,C3:D3) Profit 200 300 0 =SUMPRODUCT(C2:D2,C5:D5) Labor 3 2 0 100 Fertilizer 2 4 0 120 Land 1 1 0 45 Note: The inequality signs are NOT typed in, they are an option Answer Report Target Cell (Max) Cell Name Original Value Final Value $D$4 Profit Total 0 10000 Adjustable Cells Cell Name Original Value Final Value $B$3 Decision Wheat 0 20 $C$3 Decision Corn 0 20 What does slack Constraints mean here ? Cell Name Cell Value Formula Status Slack $D$6 Workers Function 100 $D$6<=$F$6 Binding 0 $D$7 Fertilize Function 120 $D$7<=$F$7 Binding 0 $D$8 Land Function 40 $D$8<=$F$8 Not Binding 5 $B$3 Decision Wheat 20 $B$3>=0 Not Binding 20 $C$3 Decision Corn 20 $C$3>=0 Not Binding 20 Sensitivity Report Reduced cost: how much more profitable would W or C have to be to be included in the answer? Profit of Wheat could Adjustable Cells increase by $250 or Final Reduced Objective Allowable Allowable decrease by $50 and Cell Name Value Cost Coefficient Increase Decrease $B$3 Decision Wheat 20 0 200 250 50 we would still use $C$3 Decision Corn 20 0 300 100 166.6666667 plant 20 acres. Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $D$6 Workers Function 100 25 100 20 40 $D$7 Fertilize Function 120 62.5 120 40 53.33333333 $D$8 Land Function 40 0 45 1E+30 5 If we could get another worker, each worker contributes $25 (shadow price) to profit for the range (100+20 =120) to (100 - 40=60) or between 60 and 120 workers. So, how much are we willing to pay for an extra worker? How much are we willing to pay for an extra ton of fertilizer? How much for an extra acre of land ? Transportation Networks Transportation model optimizes shipments between coming from m origins to n destinations. Mexico Warehouse Plant Tennessee Warehouse Plant Toronto Warehouse Plant Warehouse Rent'R Cars is a multi-site rental car company in the city. It is trying out a new "return the car to the location most convenient for you" policy to improve customer service. But this means that the company has to constantly move cars around the city to maintain required levels of vehicle availability. The supply and demand for economy cars, and the total cost of moving these vehicles between sites, are shown below. From\To D E F G Supply A $9 $8 $6 $5 50 B $9 $8 $8 $0 40 C $5 $3 $3 $10 75 Demand 50 60 25 30 165 Set up the equations for to determine the minimal moving costs. Note: Variable AD will be the number of cars moved from A to D. Space Allocation Planes: how much space to allocate to people or cargo (profit maximizing) Retail Space: which products to put on display (profit maximizing) Warehouse Space: how much product to store Stereo Warehouse The retail outlet of Stereo Warehouse is planning a special clearance sale. The showroom has 400 square feet of floor space available for displaying the week’s specials, model X receiver and series Y speakers. Each receiver has a wholesale cost of $100, requires 2 square feet of display space, and will sell for $150. The wholesale cost for a pair of speakers is $50, the pair requires 4 square feet of space and will sell for $70. The budget for stocking stereo items is $8000. The sales potential for the receiver is considered to be no more than 60 units. However, the budget-priced speakers appear to have unlimited appeal. The store manager, desiring to maximize gross profit, must decide how many receivers and speakers to stock. Financial Portfolio Selection Welte Mutual funds has just obtained $100,000 and is now looking for investment opportunities. The firm’s top financial analyst recommends these 5 options. The projected rates of return are shown below: Atlantic Oil 7.3% Pacific Oil 10.3% Midwestern Steel 6.4% Huber Steel 7.5% Government Bonds 4.5% neither oil or steel should receive more than $50,000 of the total investment. Government bonds should be at least 25% of the steel industry. The investment in Pacific Oil is risky thus cannot be more than 60% of the total oil industry investment What is the best investment plan for Welte? Knapsack Problems (Binary) You are running away from home and want to take all your favorite things (Ipod, knife, sweater, etc.) but only have so much room in your knapsack. You assign different values to each item and try to maximize the value of what you fit into the knapsack. You take the item (1) or you don’t (0). Note: This is a constraint called “Binary” under SOLVER. Capital Budgeting: Arm&Hammer A&H specializes in sewage and parking lot construction. It has 6 possible projects that could be done but a limited amount of capital and time for the analyst to do project management. You must decide which projects to do. Note: this is the knapsack problem because you either do a project (1) or don’t (0). Capital Budgeting: Arm&Hammer Project Payoff Budget Analyst time TN Sewer $100000 $50000 3.5 NC Sewer 1 $70000 $20000 3.0 NC Sewer 2 $120000 $40000 2.0 Parking Lot AK $40000 $10000 .50 Parking Lot TN $80000 $30000 2.5 Parking Lot, GA $50000 $20000 1.0 AMOUNT AVAILABLE $90000 6.0

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posted: | 1/9/2012 |

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