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456556 Introduction to Operations Research

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One of the most important models (due to its usefulness in solving a variety of ... 40 each, require 6 square feet of floor space, and hold 24 cubic feet of files. ... – PowerPoint PPT presentation

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Title: 456556 Introduction to Operations Research


1
456/556 Introduction to Operations Research
  • Chapter 3 Introduction to Linear Programming

2
Linear Programming
  • In order to solve OR problems, we need to turn a
    real-world problem into a mathematical model.
  • One of the most important models (due to its
    usefulness in solving a variety of problems) is
    the linear programming model.
  • This type of model is used to address the general
    problem of allocating limited resources among
    competing activities in a best possible (optimal)
    way.
  • All mathematical functions in linear programming
    models are linear.
  • We now look at some simple examples of linear
    programming models that can be solved graphically.

3
456/556 Introduction to Operations Research
  • 3.1 Prototype Examples

4
Example 1 (File Cabinets)
  • An office manager needs to purchase new filing
    cabinets. She knows that Ace cabinets cost 40
    each, require 6 square feet of floor space, and
    hold 24 cubic feet of files. On the other hand,
    each Excello cabinet costs 80, requires 8 square
    feet of file space, and holds 36 cubic feet. Her
    budget permits her to spend no more than 560 on
    files, while the office has space for no more
    than 72 square feet of cabinets. The manager
    desires the greatest storage capacity within the
    limitations imposed by funds and space. How many
    of each cabinet should she buy?

5
Example 1 (cont.)
  • We can formulate this situation as a linear
    programming problem.
  • Let x1 the number of Ace cabinets to be bought.
  • Let x2 the number of Excello cabinets to be
    bought.
  • Let Z the total storage capacity of cabinets
    purchased.
  • Summarize the given information in a table

6
Example 1 (cont.)
7
Example 1 (cont.)
  • We call x1 and x2 decision variables for this
    model.
  • From the bottom row of the table, we get the
    objective function
  • Z 24 x1 36 x2 (1)
  • The objective function gives the amount of
    storage space in cubic feet for a choice of x1
    and x2.
  • In this case, the objective is to maximize Z.

8
Example 1 (cont.)
  • From rows 1 and 2 of the table, we get
    restrictions on our choices of x1 and x2 due to a
    limit on what we can spend and the size of the
    office.
  • 40 x1 80 x2 560
    (2)
  • 6 x1 8 x2 72
    (3)
  • We also want
  • x1 0
    (4)
  • x2 0
    (5)
  • The last two restrictions on x1 and x2 make sense
    physically.
  • We call equations (2) - (5) constraint equations.

9
Example 1 (cont.)
  • Our model for deciding how to allocate file
    cabinets is as follows
  • Maximize Z 24 x1 36 x2
  • Subject to the restrictions
  • 40 x1 80 x2 560 (cost)
  • 6 x1 8 x2 72 (space)
  • and
  • x1 0 x2 0.

10
Example 2 (Feeding Laboratory Animals)
  • Certain laboratory animals must have at least 30
    grams of protein and at least 20 grams of fat per
    feeding period. These nutrients come from food
    A, which costs 18 cents per unit and supplies 2
    grams of protein and 4 of fat and food B, which
    costs 12 cents per unit and has 6 grams of
    protein and 2 of fat. Food B is bought under a
    long-term contract requiring that at least 2
    units of B be used per serving. How much of each
    food should included in a serving to produce the
    minimum cost per serving?

11
Example 2 (cont.)
  • Again, we can formulate this situation as a
    linear programming problem.
  • Let x1 units of food A per serving.
  • Let x2 units of food B per serving.
  • Let Z the cost per serving of food.
  • As before, summarize the given information in a
    table

12
Example 2 (cont.)
13
Example 2 (cont.)
  • The linear programming model for this case is as
    follows
  • Minimize Z 18 x1 12 x2
  • Subject to the restrictions
  • 2 x1 6 x2 30 (protein)
  • 4 x1 2 x2 20 (fat)
  • and
  • x1 0 x2 2.

14
The Graphical Method
  • For linear programming problems involving two
    decision variables, we can solve by graphing
    linear inequalities!
  • For more than two variables, we will have to
    resort to other methods, such as the Simplex
    Method, which we will see in Chapter 4.

15
The Graphical Method (cont.)
  • The basic idea of this method is to sketch a
    graph of all ordered pairs of decision variables
    (x1, x2) that satisfy all given constraint
    equations.
  • The set of all valid pairs of decision variables
    (x1, x2) is called the feasible region.
  • Once the feasible region is found, we look for
    the pair(s) (x1, x2) that maximize (or minimize)
    the objective function.

16
Example 1 (cont.)
  • Lets apply the graphical method to this linear
    programming problem for filing cabinet choices
  • Maximize Z 24 x1 36 x2
  • Subject to the restrictions
  • 40 x1 80 x2 560 (cost)
  • 6 x1 8 x2 72 (space)
  • and
  • x1 0 x2 0.
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