Evaluating Constrained Resources w/ Linear Programming ISQA 511 Dr. Mellie Pullman Evaluating Constrained Resources w/ Linear Programming

Overview Game problem Terms Algebraic & Graphical Illustration LP with Excel

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 Tinker Toys Game ISQA 459/559

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

Maximize Overall Profit

Value of constrained resources A toy-trader has offered to sell your group two specific toy parts: Orange rods $5/each Wood caps $10/each Are you interested in either of these parts? How many do you want to buy?

Answers Maximizing number of toys: 11 Toys 2 Robots, 3 Turnstiles, & 6 Front Wheels Maximizing revenue: $220 3 Robots, 3 Turnstiles, & 5 Front Wheels

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

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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=?

Graphical Display C H 4H + C < 400 10 H + 5 C H + C < 200 400 300 200 10 H + 5 C 100 H + C < 200 100 200 300 400 H

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

Wheat & Corn Corn Wheat

Solver Set-up on Excel These 2 cells will change to find the solution. They represent W & C (our unknowns)

Note: The inequality signs are NOT typed in, they are an option

Answer Report What does slack mean here ?

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 increase by $250 or decrease by $50 and we would still use plant 20 acres. 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 ?

Types of Problems Transportation Networks/Models Space Allocation Financial Portfolios

Transportation Networks Transportation model optimizes shipments between coming from m origins to n destinations. Mexico Warehouse Plant Tennessee Warehouse Plant Warehouse Toronto Plant Warehouse

Equations Objective:minimize cost of moving cars Constraints: $9AD +$9BD +$5CD+$8AE+$8BE+$3CE +$6AF+$8BF+$3CF+$5AG+$10CG Constraints: Have to at least meet demand @ D,E,F,G AD+BD+CD>50; AE+BE+CE>60; AF+BF+CF>25; AG+BG+CG>30 Can’t exceed supply from A,B,C AD+AE+AF+FG<50; BD+BE+BF+BG<40; CD+CE+CF+CG<75.

Other Sustainability Issues that might benefit from using this LP network solution?

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.

Space Solution Variables x = # of receivers to stock; y = # of speaker pairs to stock Objective? Maximize profit: (Sale Price -cost)X + (Sale Price -cost)Y Constraints? Floor space: 2X+4Y < 400 Budget: 100X+50Y < 8000 Sales Limit X < 60

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?

Financial Solution Variables: Objective? Constraints? A,P,M,H,and G are the dollars allocated to each investment. Objective? Maximize return: .073A+.103P+.064M+.075H + .045G Constraints? Oil/steel: A+P < 50000; M+H < 50000 Gov Bonds: G > .25 (M+H) or G - .25 M - .25 H > 0 Risky oil: P < .60(A+P) or .40 P-.60A < 0

Socially Responsible Investments? Constraints?

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.

Cork’s Wine Tasting Cork is doing a wine tasting of Oregon Pinot Noirs for a select group. As the wine manager, you must decide which wines to select. But, there are of course some limitations. You have a budget (B) of $1000 and do not want to serve more than 30 bottles and only one of each brand. All the bottles will be pulled from the same year vintage and you have identified 64 (n) bottles each with a Wine Spectator rating rj and price pj (they range between $18 and $140)

Equation Set-up Many Different Possible Objectives Maximize rating: Subject to Constraints: Budget Number Bottles Either in or out

Other Possible Objectives? Cheap tasting Objective? Given you want the rating over some overall average Must have best wines from 3 different parts of Oregon equally represented Add a constraint on picking at least 10 from each