EVALUATING CONSTRAINED RESOURCES W/ LINEAR PROGRAMMING

ISQA 511

Dr. Mellie Pullman

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

Number of parts available

Robot Turnstile Front Wheel Parts left over

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 parts available

Robot * units Turnstile * units Front Wheel * units Total Parts used

Parts left over

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

TOTAL UNITS

Maximize Overall Profit

Number of parts available

Robot * units Turnstile * units Front Wheel * units Total Parts used

Parts left over

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

TOTAL UNITS

Profit $30 $10 $20

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

scenariosOR 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=?

Graphical Display

C

H

200

100

300

400

100 200 300 400

H + C < 200

4H + C < 400

10 H + 5 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

Wheat & Corn

0

10

20

30

40

50

0 50 100

Labor

Fert

Land

Wheat = 20, Corn = 20Profit = 10000

Cor

n

Wheat

Solver Set-up on Excel

Wheat Corn LSE RSEVARIABLES 0 0Profit 200 300 0

Labor 3 2 0 100Fertilizer 2 4 0 120Land 1 1 0 45

=SUMPRODUCT(C2:D2,C3:D3)

=SUMPRODUCT(C2:D2,C5:D5)

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

Target Cell (Max)Cell Name Original Value Final Value

$D$4 Profit Total 0 10000

Adjustable CellsCell Name Original Value Final Value

$B$3 Decision Wheat 0 20$C$3 Decision Corn 0 20

ConstraintsCell 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

What does slack mean here ?

Sensitivity Report

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$B$3 Decision Wheat 20 0 200 250 50$C$3 Decision Corn 20 0 300 100 166.6666667

ConstraintsFinal 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

Profit of Wheat could increase by $250 or decrease by $50 and we would still use plant 20 acres.

Reduced cost: how much more profitable would W or C have to be to be included in the answer?

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 willingto 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 betweencoming from m origins to n destinations.

Plant

Plant

Plant

Warehouse

Warehouse

Warehouse

WarehouseMexico

Tennessee

Toronto

Rent'R Cars is a multi-site rental car company in the city. I t is tryingout a new "return the car to the location most convenient f or you"policy to improve customer service. But this means that the companyhas to constantly move cars around the city to maintain required levelsof vehicle availability. The supply and demand f or economy cars, andthe total cost of moving these vehicles between sites, are shownbelow.

From\ To D E F G SupplyA $9 $8 $6 $5 50B $9 $8 $8 $0 40C $5 $3 $3 $10 75Demand 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 f rom A to D.

Equations

Objective:minimize cost of moving cars $9AD +$9BD +$5CD+$8AE+$8BE+$3CE

+$6AF+$8BF+$3CF+$5AG+$10CG Constraints:

Have to at least meet demand @ D,E,F,GAD+BD+CD>50; AE+BE+CE>60; AF+BF+CF>25; AG+BG+CG>30

Can’t exceed supply from A,B,CAD+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: 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 rjj and price pjj (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

n

j

r1

n

jjjXrr

1

n

jjj BXp

1

64,...,2,1,1,0 njX j

n

jjX

1

30

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