Linear Programming

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Linear Programming

• Getting Started

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Beginner LP

• Tinker Toy problem
• Terms
• Algebraic Graphical Illustration
• LP with Excel

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Tinker Toys

• We need to allocate scarce resources among
  several alternatives
• resources ?
• alternatives?
• Need to get into teams of 5 or 6 people
• Assemble the Tinkertoys into the three products
  (Turnstiles, Robots, Front Wheel Assemblies)

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Parts Required and Availability
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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 _at_ 30, Turnstiles _at_ 10, Front Wheel
  Assemblies _at_ 20.

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Value of constrained resources

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

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Answers

• Maximizing number of toys 11 Toys2 Robots, 3
  Turnstiles, 6 Front Wheels
• Maximizing revenue 2203 Robots, 3 Turnstiles,
  5 Front Wheels

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What is Linear Programming?

• A sequence of steps that will lead to an optimal
  solution.
• Used to
• allocate scarce resources
• assign workers
• determine transportation schemes
• solve blending problems
• solve many other types of problems

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Five essential conditions

• Explicit Objective What are we maximizing or
  minimizing? Usually profit, units, costs, labor
  hours, etc.
• Limiting resources create constraintsworkers,
  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 (machines make identical parts)
• Divisibility products or resources can be
  divided into fractions.

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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?

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Equations

• Maximize Profit 10 H 5 C
• Constrained Resources
• 1H 1C lt 200 (credit reviewing hours)
• 4H 1C lt 400 (appraising hours)
• Hgt0 Cgt0 (non-negative)
• H ?
• C?

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Graphical Display
C
400
4H C lt 400
300
200
10 H 5 C
100
H C lt 200
100
200
300
400
H
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Farmers Wheat and Corn Home Work 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 lt 100
• Fertilizer 2 W 4 C lt 120
• Land 1W 1 C lt 45
• Non-Negativity P1 P2 gt 0

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Wheat Corn
Corn
Wheat
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Answer Report
What does slack mean here ?
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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 (10020 120) to (100 - 4060) 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 ?
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Advanced Linear Programming Applications
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Types of Problems

• Transportation Networks/Models
• Space Allocation
• Financial Portfolios
• Integer Programming Applications
• labor scheduling (see other handouts)
• knapsack problems (Binary 0 or 1 solutions)

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

• Objectiveminimize cost of moving cars
• 9AD 9BD 5CD8AE8BE3CE6AF8BF3CF
  5AG10CG
• Constraints
• Have to at least meet demand _at_ D,E,F,GADBDCDgt50
  AEBECEgt60 AFBFCFgt25 AGBGCGgt30
• Cant exceed supply from A,B,CADAEAFFGlt50
  BDBEBFBGlt40 CDCECFCGlt75.

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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

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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 weeks 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.

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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 2X4Y lt 400
• Budget 100X50Y lt 8000
• Sales Limit X lt 60

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Financial Portfolio Selection

• Welte Mutual funds has just obtained 100,000
  and is now looking for investment opportunities.
  The firms top financial analyst recommends these
  5 options. The projected rates of return are
  shown belowAtlantic Oil 7.3Pacific
  Oil 10.3Midwestern Steel 6.4Huber
  Steel 7.5Government 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?

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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 AP lt 50000 MH lt 50000
• Gov Bonds G gt .25 (MH) or G - .25 M - .25 H gt 0
  
• Risky oil P lt .60(AP) or .40 P-.60A lt 0

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Knapsack Problems (Binary)

• You are running away from home and want to take
  all your favorite things (Walkman, 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 dont (0).
• Note This is a constraint called Binaryunder
  SOLVER.

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Capital BudgetingArmHammer

• AH 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 dont (0).

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Capital BudgetingArmHammer