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

2. Overview Game problem Terms Algebraic & Graphical Illustration
LP with Excel

3. 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)

4. Parts Required and Availability
Tinker Toys Game
ISQA 459/559

5. 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.
$30, $10, Front Wheel $20.

6. Maximize number of toys

7. Maximize Overall Profit

8. 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?

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

10. 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

11. Where to find it in Excel 2007

12. 1 2 3

13. 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

14. 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)

15. 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?

16. 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)

17. Equations Maximize Profit : $10 H + $5 C Constrained Resources
1H + 1C < (credit reviewing hours)
4H + 1C < (appraising hours)
H>0; C>0 (non-negative)
H= ?
C=?

18. 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

19. 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 workers and 120 tons of fertilizer are available.
Wheat
Corn
Labor /acre
3 workers
2 workers
Fertilizer/acre
2 tons
4 tons

20. 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: W + 2 C < 100
Fertilizer: 2 W + 4 C < 120
Land: W + 1 C < 45
Non-Negativity: P1 & P2 > 0

21. Wheat & Corn
Corn
Wheat

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

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

24. Answer Report
What does slack mean here ?

25. 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 ( =120) to ( =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 ?

26. Types of Problems Transportation Networks/Models Space Allocation
Financial Portfolios

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

29. 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 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.

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

31. 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

32. 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.

33. 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: X+50Y < 8000
Sales Limit X < 60

34. 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?

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

36. Socially Responsible Investments?
Constraints?

37. 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.

38. 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)

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

40. 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