LINEAR PROGRAMMING: GRAPHICAL METHODS 9. Linear

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LINEAR PROGRAMMING GRAPHICAL METHODS
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9. Linear Programming Graphical Methods

• Linear programming (LP) is a mathematical
  technique designed to help managers in their
  planning and decision making. It is usually used
  in an organization that is trying to make most
  effective use of its resources. Resources
  typically include machinery, manpower, money,
  time, warehouse space, or raw materials.
• A few examples of problems in which LP has been
  successfully applied are
• Development of a production schedule that will
  satisfy future demands for a firms product and
  at the same time minimize total production and
  inventory costs.

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• Establishment of an investment portfolio from a
  variety of stocks or bonds that will maximize a
  companys return on investment.
• Allocation of a limited advertising budget among
  radio, TV, and newspaper spots in order to
  maximize advertising effectiveness.
• Determination of a distribution system that will
  minimize total shipping cost from several
  warehouses to various market locations.
• Selection of the product mix in a factory to make
  best use of machine and man hours available while
  maximizing the firms profit.

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• These are but a few of the applications of linear
  programming that we shall discuss in the next few
  units of this book. All LP problems. As you can
  see above, seek to maximize or minimize some
  quantity (usually profit or cost). We refer to
  this property as the objective of an LP problem.
• The second property that LP problems have in
  common is the presence of restrictions, or
  constraints, that limit the degree to which we
  can pursue our objective. We want, therefore, to
  maximize or minimize a quantity (the objective
  function) subject to limited resources (the
  constraints).

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Formulating Linear Programming Problems

• A very common linear programming problem is
  the product mix problem. Two or more products
  are usually formed using limited resources, such
  as personnel and machines. The profit, which the
  firm seeks to maximize, is base on the profit
  contribution per unit of each product. The
  company would like to determine how many units of
  each product they should produce so as to
  maximize overall profit given their limited
  resources.

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EXAMPLE 9.1

• The Dress-Rite clothing manufacturer, which
  produces mens shirts and pajamas, has two
  primary resources available sewing machine time
  (in the sewing department) and cutting machine
  time (in the cutting department). Over the next
  month, Dress-Rite can schedule up to 280 hours of
  work on sewing machines and up to 450 hours of
  work on cutting machines.
• Each shirt produced requires 1 hour of sewing
  time and 1½ hours of cutting time. To output each
  pair of pajamas, 3/4 hour of sewing time and 2
  hours of cutting time are needed.

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• Our goal here is to develop relationships to
  describe these restrictions. One general
  relationship is that the amount of resource used
  is the amount of resource available.
• In the case of the sewing department, for
  example, the total time used is
• (1 hour/shirt) X (number of shirts produced)
  (3/4 hour/pajama) X (number of pajamas produced)
• To express the LP constraints for this problem
  mathematically, we let
• X1 number of shirts produced
• X2 number of pajamas produced

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• Then
• 1st constraint 1X1 ¾ X2 280 (hours of
  sewing machine time availableour first scarce
  resource)
• 2nd constraint 1 ½ X1 2 X2 450
  (hours of cutting machine time availableour
  second scarce resource)
• Note This means that each pair of pajamas
  takes about 2 hours of the cutting resource).

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EXAMPLE 9.2

• In Example 9.1, Dress-Rite established two
  constraints which will keep the company from
  exceeding the machine time available for
  production. But the company wants to determine
  the product mix (of shirts and pajamas) that will
  maximize its profits. Its accounting department
  analyzes cost and sales figures and states that
  each shirt produced will yield a 4 contribution
  to profit and that each pair of pajamas will
  yield a 3 contribution to profit.

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• This information can be used to create the LP
  objective function for this problem.
• Objective function maximize total contribution
  to profit 4X1 3X2
• where again
• X1 number of shirts produced
• X2 number of pajamas produced

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EXAMPLE 9.3

• Electro Corp. manufactures two electrical
  products, air conditioners and fans. The assembly
  process for each is similar in that both require
  a certain amount of wiring and drilling. Each air
  conditioner takes 4 hours of wiring and 2 hours
  of drilling. Each fan must go through 2 hours of
  wiring and 1 hour of drilling. During the next
  production period, 240 hours of wiring time are
  available and up to 100 hours of drilling time
  may be used. Each air conditioner sold yields a
  profit of 27each fan assembled may be sold for
  a 15 profit.
• Electro would like this production mix situation
  formulated as a linear programming problem.

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• Let
• X1 number of air conditioners to be
  produced
• X2 number of fans to be produced
• Objective function maximize profit 27 X1
  15 X2
• 1st constraint 4X1 2X2 240 (hours of
  wiring time available)
• 2nd constraint 2X1 1X2 100 (hours of
  drilling time available)
• Constraints added to ensure nonnegative
  solutions
• X1 0
• X2 0

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• We see in the previous examples that each linear
  programming problem contains
• an objective function,
• certain constraints (resulting from limited
  resources) which keep the firm from producing
  unlimited quantities and hence making unlimited
  profits, and
• certain interactions between variables namely,
  the more units of one product produced, the less
  the firm can make of other products.

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• It is possible for a problem to have many more
  variables, or products, involved as we shall see
  in forthcoming units. It is also possible to have
  more than two constraints in even a simple LP
  problem. Constraints may involve not only less
  than or equal to () inequalities, but actual
  equalities () and greater than or equal to
  () inequalities as well.

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EXAMPLE 9.4

• Referring back to the Electro Corp. in Example
  9.3, we are informed that the firms management
  wishes to reformulate their LP production mix
  problem. In particular, management has become
  very sensitive to wasted, or unused, time on
  their expensive new drilling machine and wants
  all 100 hours of available time used in
  production. The firm also decides that to ensure
  an adequate supply of air conditioners for a
  contract, at least 25 air conditioners should be
  manufactured. Since it incurred an oversupply of
  fans the previous period, it also insists that no
  more than 70 fans be produced.
• We can use this new information to reformulate
  the constraints of Example 9.3

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• Objective function maximize profit 27X1
  15X2 (no change)
• Wiring-time constraint 4 X1 2 X2 240 (no
  change)
• Drilling-time constraint 2X1 1X2 100 (now
  an equality)
• Air-conditioner minimum constraint X1 25
  new greater than or equal to () constraint
• Fan maximum constraint X2 70 (new fan
  constraint)

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GRAPHICAL REPRESENTATION OF CONSTRAINTS

• In order to determine the optimal solution to a
  linear programming problem, we must first
  identify a set, or region, of feasible solutions.
  When there are only two variables in the problem,
  as in the examples and problems above, this can
  be accomplished by plotting the constraints on a
  two-dimensional graph.
• The variable X1 is usually treated as the
  horizontal axis and the variable X2 as the
  vertical axis. Because of the nonnegativity
  constraints (i.e., X1 0, X2 0), we are always
  working in the first (or northeast) quadrant of
  the graph. The graphical method of dealing with
  LP problems is best demonstrated by way of an
  example.

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EXAMPLE 9.5

• The Flair Furniture Company produces inexpensive
  tables and chairs. it has formulated the
  following LP problem
• Maximize profit 7X1 5X2
• subject to 4X1 3X2 24
  (constraint A)
• 2 X1 1X2 10
  (constraint B)
• X1 0, X2 0
• where X1 stands for the number of tables
  produced and X2 stands for the number of chairs
  produced. We would like to graphically represent
  the constraints of this problem.
• The first step is convert the constraint
  inequalities into equalities (or equations)
• 4X1 3X2 24 (A) 2X1 1X2 10 (B)

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The equation on the left (A) is plotted in Figure
9.1(A) and the one on the right in Figure 9.1(B).
To plot
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• The line in Figure 9.1(A), all we needed to do
  was find the points at which the line intersected
  the X1 and X2 axes. When X1 0 (The location
  where the line touches the X2 axis), it implies
  that 3 X2 24 or that X2 8. Likewise, when X2
  0, we see that 4 X1 24 and that X1 6. Thus,
  the A constraint is bounded by the line running
  from (X1 0, X2 8) to (X1 6, X2 0 The
  shaded area represents all points that satisfy
  the original inequality.
• Constraint B is illustrated in Figure 9.1(B) in a
  similar fashion. When X1 0, then X2 10, and
  when X2 0, then X1 5. The B constraint then
  is bounded by the line between (X1 0, X2 10)
  to (X1 5, X2 0) and the shaded area
  represents the original inequality.

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Figure 9.2 shows both constraints together- the
shaded region is the part that satisfies
both restrictions.
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• The shaded region in Figure 9.2 is called the
  area of feasible solutions or simply the feasible
  region. This region must satisfy all conditions
  specified by the programs constraints, and is
  thus the region where all constraints overlap.
  Any point in the region would be a feasible
  solution to the Flair Furniture Company
  problemany point outside the shaded area would
  represent an infeasible solution. Hence, it would
  be feasible to manufacture 3 tables and 2 chairs
  (X1 3, X22), but it would violate the
  constraints to produce 7 tables and 4 chairs.

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THE CORNER-POINT SOLUTION METHOD

• There are several approaches that can be taken in
  solving for the optimal solution once the
  feasible region has been established graphically.
  The simplest one conceptually is called the
  corner-point method. The mathematical theory
  behind linear programming states that the optimal
  solution to any problem (that is, the values of
  the X1 variables which yield the maximum profit
  or minimum cost) will lie at a corner point, or
  extreme point, of the feasible region. Hence, it
  is only necessary to find the values of the
  variables at each cornerthe maximum profit or
  optimal solution will lie at one of them. This
  concept is illustrated in Example 9.6

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• point a (X1 0, X2 0) profit (7)(0)
  (5)(0) 0
• point b (X1 0, X2 8) profit (7)(0)
  (5)(8) 40
• point d (X1 5, X2 0) profit (7)(5)
  (5)(0) 35

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• We skipped corner point c momentarily because to
  accurately establish its coordinates, it is
  necessary to solve for the intersection of the
  two constraint lines. To do so, we apply the
  method of simultaneous equations to
• 4X1 3X2 24 and 2X1 1X2 10
• Multiply the second equation by -2 and add it to
  the first equation
• 4X1 3X2 24
• -4X1 - 2X2 -20 which is (2)(2X1 3X2
  10)
• 1X2 4
• When X2 4, then 4X1 (3)(4) 24, implying that
• 4X1 12 or X1 3. Now we can evaluate point c
• point C (X1 3, X2 4) profit (7)(3)
  (5)(4) 41

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• Because point c produces the highest profit of
  any corner point, the product mix X1 3 tables
  and X2 4 chairs is the optimal solution to
  Flair Furnitures problem. This solution will
  result in a profit of 41.

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• Note Although the optimal solution to Example
  9.6 and Problems 9.6 and 9.7 fell at a corner
  point which was located at the intersection of
  two constraint equations, this is by no means
  always the case. The optimal solution could just
  as easily have been at a corner bordering on the
  X1 or X2 axis (such as points b or d in Example
  9.6). It all depends upon the values of the
  objective function coefficients and the angle of
  the profit line. This topic is discussed next.

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THE ISO-PROFIT-LINE SOLUTION METHOD

• A second approach to graphically solving linear
  programming problems employs the iso-profit line.
  This technique is often more speedy than the
  corner-point method, for we do not have to
  evaluate the profit at every corner. Instead, we
  draw a series of parallel profit lines. Any point
  along a particular iso-profit line will have the
  same profit or value of the objective function.
  The highest profit line (or one farthest from the
  zero origin) which touches the feasible region
  pinpoints the optimal solution.

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EXAMPLE 9.7

• The objective function for an LP problem is given
  as
• maximize profit 2X1 8X2
• The corresponding feasible region has been
  graphed in Figure 9.4. We wish to determine which
  corner point is optimal using the iso-profit-line
  solution method.

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• We may begin by selecting any profit level, for
  example, P200, and drawing its (dashed)
  iso-profit line. Is there a higher possible
  profit line which touches a corner of the
  feasible region? The answer appears to be yes,
  and by drawing a series of parallel profit lines
  farther and farther away from the origin, we
  eventually reach the tip of the feasible region
  (at corner point b). The profit there will be P
  2(X1 0) 8(X2 50) 2(0) 8(50) 400.
  Thus, the maximum profit line is 400 2X1 8X2

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MINIMIZATION PROBLEMS IN LINEAR PROGRAMMING

• Many linear programming problems involve
  minimizing an objective such as cost, instead of
  maximizing a profit function. Basically the same
  graphical approaches may be applied to solving
  these types of problems.

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EXAMPLE 9.8

• The dean of the Western College of Business must
  plan his schools course offering for the fall
  semester.
• Student demands deem it necessary to offer at
  least 30 undergraduate and 20 graduate courses in
  the term.
• Faculty contracts also dictate that at least
  60 courses be offered in total.
• Each undergraduate course taught costs the
  college an average of 2,500 in faculty wages,
  while each graduate course costs 3,000.
• We may formulate this as a minimization LP
  problem as follows.
• Let
• X1 number of undergraduate business courses
  offered in fall
• X2 number of graduate business courses

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• Minimization problems can be solved graphically
  by first establishing the feasible region, and
  then by using either the corner-point method or
  an iso-cost-line approach (analogous to the
  iso-profit approach in maximization problems) to
  find the values of X1 and X2 which yield the
  minimum cost. The corner-point method is employed
  below to solve Example 9.8.

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EXAMPLE 9.9

• To solve Example 9.8 graphically, we construct
  the problems feasible region (Figure 9.5).
  Minimization problems are often unbounded outward
  (that is, on the right side and on the top), but
  this causes no problem in solving them. As long
  as they are bounded inward (on the left side and
  the bottom), corner points may be established.
  The optimal solution will lie at one of the
  corners.

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• In the case above, there are only two corner
  points, a and b. It is easy to determine that at
  point a, X1 40 and X2 20 and that at point b,
  X1 30 and X2 30. The optimal solution is found
  at the point yielding the lowest total cost.
• total cost at a 2,500 X1 3,000 X2
• (2,500)(40)
  (3,000)(20) 160,000
• total cost at b 2,500 X1 3,000 X2
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  (2,500)(30) (3,000)(30) 165,000
• The lowest cost to the college is at point a
  hence, the dean should schedule 40 undergraduate
  courses and 20 graduate courses.

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• As mentioned, the iso-cost-line approach may also
  be used to solve LP minimization problems. As
  with iso-profit lines (see Example 9.7 and
  Problem 9.8), we need not compute the cost at
  each corner point, but instead draw several
  parallel cost lines, The lowest cost line to
  touch the feasible region provides us with the
  optimal solution corner.