Integer and Goal Programming

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• Integer and Goal Programming
• Asli Sencer Erdem

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Integer Program

• Structurally like the LP models where the
  solutions must be integers!
• Ex
• Production order quantities
• Number of items sent from plants to warehouses
• Binary decisions
• produce or do not produce
• Assign staff i to job j or not
• Locate facility i in location j or not

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ExRedwood Furniture Model (revisited)
Optimal solution
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When do we round off the LP?

• For problems having small number of variables, IP
  is solved in a short time.
• Bigger problems require a lot of time.
• If no integer solution has been found, a rounded
  LP may be used.
• Which solution to be used is judgemental.

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Solving Integer Programs with a Spreadsheet

• When using Excel Spreadsheet, it is sufficient to
  add a new constraint and define the decision
  variables as integers.
• You can also use WinQSB or QuickQuant to solve LP
  and IP problems.

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Ex Maui Miser Car Rentals

• Budget 500,000.
• Three types of cars Economy vans, compact cars,
  large cars
• Income/vehicle 15,000, 7,600, 10,600
• Costs/vehicle 25,000, 15,000, 21,000
• At least 25 vehicles
• At least 5 vans, at least 5 large cars, at least
  12 passenger cars
• large cars can not exceed the compacts
• Average daily rental rate should be less than 39
• where Daily rental prices/vehicle 50, 30, 40

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Ex Formulation
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Optimal Solution
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Goal Programming

• Maximize profit
• OR
• Minimize cost

Single Goal
Sometimes we have Multiple Goals. Minimize Cost
AND Maximize Customer Satisfaction Maximize
Return AND Minimize Risk What is the optimal
balance in attaining the conflicting goals?
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Basic Idea

• It is impossible to meet all goals
  simultaneously.
• The focus aims at achieving certain targets for
  each goal.
• The overall objective is to find the solution
  that collectively minimizes deviations from these
  targets.

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ExGoal Programming in Redwood Furniture Co.

• Objectives
• Maximize Profit
• Maximize Revenue to maintain marketshare growth
• Maximize training time to increase work-force
  productivity

Each LP has a different optimal solution. An
idea is to set targets for the goals.
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LP with Goal Targets
Idea 1 We can treat targets as the lower bounds
on the goals
Warning When these constraints are added to the
Redwood Product Mix Formulation there may be no
feasible solution satisfying the minimum
requirements of all goals! What should we do
then? Any comments? Answer Try to lower the
targets
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Goal Deviations

• Idea 2 We can allow some goals to be partially
  met!
• Compared to LP, goal programming is flexible.
  Optimal balance between goals can be achieved in
  several ways.
• Goal Deviation Amount of deviation from the goal
  target is a variable to be minimized.

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Defining Goal Deviations

• Yp The amount by which profit deviates from the
  target level

When Yp is positive, When Yp is negative
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Goal Deviation Constraints
Regular Constrains
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Goal Programming Objective

• It costs 1 to fall a dollar below the profit
  target. However exceeding the profit target is
  desirable.
• Falling below revenue target lowers market share,
  lost earnings is 2 for a dollar below revenue
  target.
• Training time upto 30hrs enhances future
  efficiency. It costs 0.50 per hour obtained.
  Beyond 30hrs. training effect is negligable.

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Ex Optimal Solution
Profit and Training levels are 6 and 6hrs. below
their targets. Revenue target is met.
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Ex Maui Miser Case (revisited)

• Achieve annual rental income of at least 300,000
  from fleet addition
• Spend as close to as possible to 500,000
• Do not exceed average daily rental of 39 for the
  vehicles in the fleet addition.
• Penalty Costs
• 1 for each dollar violation of goals 1 and 2.
• 10,000 for each dollar violation of goal 3.

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Ex Maui Miser Case GP Formulation
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Ex Maui Miser Case GP Optimal Solution