Chapter 8 GLOBAL CRITERION METHOD

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Chapter 8 GLOBAL CRITERION METHOD
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Overview

• An MODM technique
• No priori articulation of preference information
• Poineering works
• M.E. Salukvadze (1971) distance minimization
  concept
• C.L.Hwang (1972) minimization of the total
  mean-square loss
• P.L. Yu (1973) minimizing Group regret
• Hwang and Yu have been credited with originating
  the ideal distance minimization method or also
  known as the Global Criterion Method

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Global Criterion Method

• Original MODM Problem
• Max Zl fl(x) l 1, , K
• S.t. gi(x) 0 i 1, 2, , m
• x 0
• Where hold for ?, ? or
• Individual optimum fl(xl) l 1, , K. x is
  different for each j.

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Payoff Matrix
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Global Criterion Method

• The method of global criterion for the above
  vector maximization problem (VMP) solves the
  problem
• General formulation

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Global Criterion Method Comments

• F is in a ratio form no worry of dimensions
• The best solution chosen for the VMP will
  differ greatly according to the value of p
  chosen. Some suggested p 1, some p 2 for the
  Global Criterion. p 1 implies equal
  importance (weights) for all these deviations,
  while p 2 implies that these deviations are
  weighted proportionately with the largest
  deviation having the largest weight
• p is a DMs judgment

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Example

• A company makes two kinds of products. Product A
  is of high quality and product B is of lower
  quality. The respective profits are 0.4 and 0.3
  per product. Each product A requires twice as
  much time as a product B, and if all products
  were of type B, the company could make 500 per
  day. The supply of material is sufficicent for
  only 400 product units per day (both A and B
  combined). The problem assumes that all the
  products for types A and B the factory can make
  could be sold and that the best customer of the
  company wishes to have as many as possible of
  product type A. The manager realizes that two
  objectives (1) the maximization of profit, and
  (2) the maximum production of product type A,
  should be considered in scheduling the production.

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Example Comments/additional information on the
problem

• The DM is unapproachable/could not be made
  available
• The management gives the problem to the analyst
  and asked him to come up with a reasonable
  solution
• The objectives are of different dimensions
• The global criterion method appears to be an
  appropriate approach

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Example The model formulation

• Maximize Z1 0.4x1 0.3x2
• Maximize Z2 x1
• S.t. x1 x2 400 (material supply)
• 2x1 x2 500 (capacity)
• x1, x2 0
• Where x1 and x2 number of units of product
  types A and B, respectively

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Example

• Step 1 Obtain the ideal solutions
• Maximize Z1 0.4x1 0.3x2
• S.t. x ? X
• This is a LP problem whose ideal solution is x1
  
• (100, 300) and Z1 130 at point B in the figure
• Maximize Z2 x1
• S.t. x ? X
• This is again a LP problem whose ideal solution
  is
• x2 (250, 0) and Z2 250 at point C in the
  figure

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Example

• Step 2 construct a payoff table
• x1 x2
• Z1 130 100
• Z2 100 250
• Step 3 obtain the preferred solution
• Case 1 p 1
• The solution to this LP problem is given by x1
  250,
• x2 0, Z1 100, Z2 250 which is point C
  again.
• Disadvantage when p 1, the preferred solution
  is one of the vertices
• Advantage still a LP problem

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Example

• Case 2 p 2
• This is a NLP problem. When solved using the
• sequential unconstrained minimization technique
• (SUMT), the solution is point D
• Disadvantage for p gt 1 it is difficult for the
  DM to decide what value of p to use especially
  when he has differential preferences on the
  objectives.
• Difficult to solve
• F becomes nonlinear

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Maximum Effectiveness Method

• An MODM technique
• Reference V.V., Khomenyuk, appears to have been
  the originator of the method
• Principle the method maximizes the minimum
  attainment by any objective of its respective
  ideal value by making use of the natural
  normalization
• The lth objectives relative attainment of its
  ideal value is denoted ?l
• Assuming all objectives are being maximized, ?l,
  is defined as

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Example
500 400 300 200 100
x1 x2 400
2x1 x2 500
B
D
C
0 100 200 250 300 400 500
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Maximum Effectiveness Method

• The method is also known as the ?-criteria
  optimization
• The final model is
• In the above model solve for x ? compromise
• solution