MBA 650: Quantitative Analysis

1
MBA 650 Quantitative Analysis
2
Postoptimality Analysis3-Questions

• What happens to the optimal solution if
• 1. bi changes?
• When does a scarce resource become abundant?
• When does an abundant resource become scarce?
• 2. cj changes?
• 3. aij changes?

3
Question 1 If bi changes

• i. basis stays the same
• ii. RHS changes
• iii. z changes

4
Question 2 If cj changes

• i. basis stays the same
• ii. RHS stays the same
• iii. z changes

5
Question 3 If aij changes

• Who knows?
• Resolve the problem!

6
Reddy Mikks Changes to bi
7
Information Contained in the Optimal Tableau

• 1. Optimal solution
• 2. Status of resources
• 3. Shadow price (or dual price)
• 4. Sensitivity to changes in bi
• Range of feasibility
• 5. Sensitivity to changes in cj
• Range of optimality

8
3- Step Branch Bound Solution Procedure for
Maximization Problems

• 1. Use simplex method to obtain z (UB)
• If solution is all integer, STOP!
• If not, GO to STEP 2!
• 2. Divide previous UB problem into 2-subproblems
• Make an integer out of the variable with the
  largest fractional part
• 3. Solve subproblems
• best noninteger solution (if any) z (new UB)
• best integer solution (if any) z (new LB)

9
Branch Bound Solution Procedure(continued)

• 3. Solve subproblems
• best noninteger solution (if any) z (new UB)
• best integer solution (if any) z (new LB)
• i. if z (new UB) lt z (previous best LB), STOP!
• Previous best LB is optimal
• ii. if both solutions are integer, STOP!
• Max (new LB, previous LBs) is optimal
• iii. if both solutions are infeasible, STOP!
• previous best LB is optimal
• iv. if z (new UB) gt zmax (new LB, previous
  LBs), RETURN to STEP 2!

10
Differences Between Linear Programming and Goal
Programming
11
An Important Concept Deviations

• 1. dj overachievement
• 2. d-j underachievement
• dj, d-j gt 0
• Goal maintain inventory level at 20 units
• if inv level 22, d 22 20 2
• if inv level 18, d- 20 18 2

12
An Important ConceptWeights of Goals

• 1. Ordinal ranking
• P1 gtgtgt P2 gtgtgt P3
• Min z P1d-1 P2d2 P3d-3
• 2. Cardinal ranking
• Weight of 1st goal 1, 2nd goal 2, 3rd goal
  3, etc
• Min z 1d-1 2d2 3d-3
• 3. Mixed ordinal and cardinal
• Ordinal dominates
• Cardinal distinguishes goals at the same priority
  level
• Min z P1d-1 P2d2 5P3d1 3P3d-3

13
GP Example Watch Factory (1 of 3)

• Products
• X1 No. electric watches
• X2 No. mechanical watches
• Capacities
• Assembly department 1000 hours/month
• Test department 200 hours/month
• Wage rates
• Regular time 10/hour
• Overtime 15/month

14
GP Example Watch Factory (2 of 3)
15
GP Example Watch Factory (3 of 3)

• P1 1. Desired profit 60K/month
• P2 2. O/T in test dept not to exceed 1000
• P3 3. No O/T in assembly dept
• P4 4. Produce at least 500 electric watches and
  2000 mechanical watches (weight each according to
  contribution to profit)
• P5 5. Maximize profit