Introduction to LP

1
Chapter 2 An Introduction to Linear Programming
Graphical and Computer Methods
Professor Ahmadi
2
Learning Objectives

• Understand basic assumptions and properties of
  linear programming (LP)
• General LP notation
• LP formulation of the model
• A Maximization Problem
• Graphical solution
• A Minimization Problem
• Special Cases
• Formulating a Spreadsheet Model

3
Introduction

• Management decisions in many organizations
  involve trying to make most effective use of
  resources.
• Include machinery, labor, money, time, warehouse
  space, and raw materials.
• May be used to produce products - such as
  computers, automobiles, or clothing or
• Provide services - such as package delivery,
  health services, or investment decisions.

4
Mathematical Programming

• Mathematical programming is used for resource
  allocation problems.
• Linear programming (LP) is the most common type
  of mathematical programming.
• One assumes that all the relevant input data and
  parameters are known with certainty in models
  (deterministic models).
• Computers play an important role in the
  advancement and use of LP.

5
Development of a LP Model

• LP has been applied extensively to various
  problems areas -
• medical, transportation, operations, petroleum
• financial, marketing, accounting,
• human resources, agriculture, and others
• Development of all LP models can be examined in a
  three step process
• (1) formulation
• (2) solution and
• (3) interpretation.

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Properties of a LP Model

• All problems seek to maximize or minimize some
  quantity, usually profit or cost (called the
  objective function).
• LP models usually include restrictions, or
  constraints, that limit the degree to which one
  can pursue the objective.
• There must be alternative courses of action from
  which one can choose.
• The objective and constraints in LP problems must
  be expressed in terms of linear equations or
  inequalities.

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Three Steps of Developing LP Problem

• Formulation.
• Process of translating problem scenario into a
  simple LP model framework with a set of
  mathematical relationships.
• Solution.
• Mathematical relationships resulting from the
  formulation process are solved to identify an
  optimal solution.
• Interpretation and What-if Analysis.
• Problem solver or analyst works with manager to
• Interpret results and implications of problem
  solution.
• Investigate changes in input parameters and model
  variables and impact on problem solution results.

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Basic Assumptions of a LP Model

• Conditions of certainty exist.
• Proportionality in the objective function and
  constraints (1 unit 3 hours, 3 units 9 hours).
• Additivity (the total of all activities equals
  the sum of the individual activities).
• Divisibility assumption that solutions need not
  necessarily be in whole numbers (integers).

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Linear Equations and Inequalities

• This is a linear equation 2X1 15X2 10
• This equation is not linear 5X4X2 15X3
  100
• LP uses, in many cases, inequalities likeX1
  X2 ? C or X1 X2 ? C

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LP Solutions

• The maximization or minimization of some quantity
  is the objective in all linear programming
  problems.
• A feasible solution satisfies all the problem's
  constraints.
• Changes to the objective function coefficients do
  not affect the feasibility of the problem.
• An optimal solution is a feasible solution that
  results in the largest possible objective
  function value, z, when maximizing or smallest z
  when minimizing.

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LP Solutions

• A feasible region may be unbounded and yet there
  may be optimal solutions. This is common in
  minimization problems and is possible in
  maximization problems.
• The feasible region for a two-variable linear
  programming problem can be a single point, a
  line, a polygon, or an unbounded area.
• Any linear program falls in one of three
  categories
• is infeasible
• has a unique optimal solution or alternate
  optimal solutions
• has an objective function that can be increased
  without bound (Unbounded)

12
LP Solutions

• A graphical solution method can be used to solve
  a linear program with two variables.
• If a linear program possesses an optimal
  solution, then an extreme point will be optimal.
• If a constraint can be removed without affecting
  the shape of the feasible region, the constraint
  is said to be redundant.
• A non-binding constraint is one in which there is
  positive slack or surplus when evaluated at the
  optimal solution.
• A linear program which is over-constrained so
  that no point satisfies all the constraints is
  said to be infeasible.

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Guidelines for Model Formulation

• Understand the problem thoroughly.
• Write a verbal statement of the objective
  function and each constraint.
• Define the decision variables.
• Write the objective function in terms of the
  decision variables.
• Write the constraints in terms of the decision
  variables.

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A Simple Maximization Problem

• Olympic Bike is introducing two new lightweight
  bicycle frames, the Deluxe and the Professional,
  to be made from special aluminum and steel
  alloys. The anticipated unit profits are 10 for
  the Deluxe and 15 for the Professional.
• The number of pounds of each alloy needed per
  frame is summarized below. A supplier delivers
  100 pounds of the aluminum alloy and 80 pounds of
  the steel alloy weekly. How many Deluxe and
  Professional frames should Olympic produce each
  week?
• Aluminum Alloy
  Steel Alloy
• Deluxe 2
  3
• Professional 4
  2

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Max. Example Olympic Bike Co.

• Model Formulation
• Verbal Statement of the Objective Function
• Maximize total weekly profit.
• Verbal Statement of the Constraints
• Total weekly usage of aluminum alloy lt 100
  pounds.
• Total weekly usage of steel alloy lt 80
  pounds.
• Definition of the Decision Variables
• x1 number of Deluxe frames produced weekly.
• x2 number of Professional frames
  produced weekly.

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Max. Example Olympic Bike Co.

• Model Formulation (Continued)
• Max 10x1 15x2 (Total Weekly Profit)
• s.t. 2x1 4x2 lt 100 (Aluminum
  Available)
• 3x1 2x2 lt 80 (Steel
  Available)
• x1, x2 gt 0
  (Non-negativity)

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Max. Example Olympic Bike Co.

• Graphical Solution Procedure

x2
(Steel)
lt
3x1 2x2
80
40
z 412.50
Optimal x1 15, x2 17 1/2,
35
30
(aluminum)
lt
2x1 4x2
100
25
20
MAX 10x1 15x2
15
10
5
x1
5 10 15 20 25 30
35 40 45 50
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Slack and Surplus Variables

• A linear program in which all the variables are
  non-negative and all the constraints are
  equalities is said to be in standard form.
• Standard form is attained by adding slack
  variables to "less than or equal to" constraints,
  and by subtracting surplus variables from
  "greater than or equal to" constraints.
• Slack and surplus variables represent the
  difference between the left and right sides of
  the constraints.
• Slack and surplus variables have objective
  function coefficients equal to 0.

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A Simple Minimization Problem

• Solve graphically for the optimal solution
• Min z 5x1 15x2
• s.t. x1
  x2 gt 500
• 
  x1 lt 400
• 
  x2 gt 200
• x1, x2 gt
  0

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Special Cases

• Alternative Optimal Solutions
• Infeasible Solutions
• Unbounded Problem

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Alternative Optimal Solutions

• In the graphical method, if the objective
  function line is parallel to a boundary
  constraint in the direction of optimization,
  there are alternate optimal solutions, with all
  points on this line segment being optimal.
• Example
• Max z 8x1 6x2
• s.t. 4x1 3x2 lt 12
• 9x1 12 x2 lt 36
• x1, x2 gt 0

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Example Alternative Optimal Solutions

• There are numerous optimal points. The objective
  function (8x1 6x2) is parallel to the first
  constraint.

x2
4x1 3x2 lt 12
4
9x1 12 x2 lt 36
3
Objective function
x1
4
3
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Example Infeasible Problem

• Solve graphically for the optimal solution
• Max z 2x1 6x2
• s.t. 4x1 3x2
  lt 12
• 2x1 x2 gt 8
• x1,
  x2 gt 0

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Example Infeasible Problem

• There are no points that satisfy both
  constraints, hence this problem has no feasible
  region, and no optimal solution.

x2
8
4x1 3x2 lt 12
2x1 x2 gt 8
4
x1
3
4
25
Example Unbounded Problem

• Solve graphically for the optimal solution
• Max z 3x1 4x2
• s.t. x1 x2 gt 5
• 3x1 x2 gt 8
• x1, x2 gt 0

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Example Unbounded Problem

• The feasible region is unbounded and the
  objective function line can be moved parallel to
  itself without bound so that z can be increased
  infinitely.

x2
3x1 x2 gt 8
8
x1 x2 gt 5
5
Max 3x1 4x2
x1
5
2.67
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Using Excel for solving LP problems

• Use Solver in Excel and solve the LP problems
  given previously.
• Tools/Solver
• End of chapter 2