Chapter 4, Problem 13RP

Explanation of Solution

Optimal solution:

• Consider the job grading of Indiana University. Job at IU is rated on the following factors:
  • Factor 1: Complexity of duties
  • Factor 2: Education required
  • Factor 3: Mental or visual demands
• To determine grade of each job its point value and desired score is tabulated.
• Given table:

	Factor level
Job	1	2	3	Desired score
1	4	4	4	105
2	3	3	2	93
3	2	2	2	75
4	1	1	2	68

• Let the variables $x_1,x_2,x_3,x_4$, be the point total assigned level 1,2,3,4, respectively. The given goals are:
  • Goal 1: When increasing the level of a factory by one, the points should increase by at least 10.
  • So the constraint equations become
    • $x_2-x_1\ge 10$
    • $x_3-x_2\ge 10$
    • $x_4-x_3\ge 10$
  • It is observed that for those set of constraint equations are:
    • $x_2-x_1+s_1^{+}-s_1^{-}=10$
    • $x_3-x_2+s_2^{+}-s_2^{-}=10$
    • $x_4-x_3+s_3^{+}-s_3^{-}=10$
  • Goal 2: Difference between actual point total for each job and desired score should be minimum.
  • So the constraint equations are:
    • $3x_4+s_4^{+}-s_4^{-}=105$
    • $2x_3+x_2+s_5^{+}-s_5^{-}=93$
    • $3x_2+s_6^{+}-s_6^{-}=75$
    • $x_2+2x_1+s_7^{+}-s_7^{-}=68$
  • Any deviation from the desired score is not desirable. So both $s_i^{+}$ and $s_i^{-}$ should be included. To minimize total deviation, the following should be minimized:
    • $d=s_4^{+}-s_4^{-}+s_5^{+}-s_5^{-}+s_6^{+}-s_6^{-}+s_7^{+}-s_7^{-}$
  • It is observed that these set of constraints do not belong to a feasible region. That is all constraints cannot be met. Hence, a cost value is assigned which is incurred if any of the goal is not met. So, introduce deviational variables as follows.
    • $s_i^{+}$=Amount by which we numerically exceed the ith goal.
    • $s_i^{-}$=Amount by which we are numerically under the ith goal