1. Session 2b

2. Decision Models -- Prof. Juran2 Overview More Sensitivity Analysis –Solver Sensitivity Report More Malcolm Multi-period Models –Distillery Example –Project Funding Example

3. Decision Models -- Prof. Juran3 Solver Sensitivity Report Provides sensitivity information about constraint “right-hand sides” and objective function coefficients Shadow prices Allowable increases and decreases

4. Decision Models -- Prof. Juran4 Malcolm Revisited

5. Decision Models -- Prof. Juran5 Shadow Price The effect on the value of the objective function resulting from a one-unit change in the constraint’s right-hand side May be viewed as an upper bound on the value of one additional unit of a constrained resource

6. Decision Models -- Prof. Juran6 Constraints Sensitivity to changes in constraint right-hand sides Allowable increase and decrease define a range within which the constraint right-hand sides can vary without affecting the shadow price

7. Decision Models -- Prof. Juran7 Example How much would Malcolm pay for more molding capacity? How much more capacity would he buy at that price?

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9. 9 If the limit on molding time is exactly 65.5 hours, then three constraints all intersect at one point. In this situation there is no utility in further increasing molding capacity (all other things held constant).

10. Decision Models -- Prof. Juran10 Adjustable Cells Sensitivity to changes in objective function coefficients Allowable increase and decrease define a range within which the objective function coefficients can vary without affecting the decision variable values

11. Decision Models -- Prof. Juran11 Example How much does the profit per unit on the 6-oz product have to go up before Malcolm would want to increase production of that product?

12. Decision Models -- Prof. Juran12 Increases in the profitability of the 6-oz product have the effect of changing the slope of the isoprofit lines.

13. Decision Models -- Prof. Juran13 If the profit on 6-oz glasses is $540, then the objective function is exactly parallel to the storage constraint. In this situation there are an infinite number of optimal solutions – every point on the line segment between two corner points.

14. Decision Models -- Prof. Juran14 This allowable increase of $40 can be seen in the sensitivity report without re-solving the model. Similarly, if the 6-oz. profit drops by $275 or more, a new corner point will be optimal. This section of the report assesses the robustness of the current optimal solution with respect to changes in the objective function coefficients.

15. Decision Models -- Prof. Juran15 Traverso has 1,000 cases on hand of “Mays & McCovey”. 2,700 cases capacity with regular-time labor, $40 per case. Unlimited capacity with overtime labor, $60 per case. Only 80% production yield is “Mays & McCovey” grade. –(Remaining 20%is sold under the bargain-rate brand “Asterisk 762”. ) Employees drink or accidentally break 10% of inventory. $15 per case cost against ending inventory. Multi-Period Models Example: Traverso Distillery

16. Decision Models -- Prof. Juran16 Managerial Formulation Decision Variables We need to decide on production quantities, both regular and overtime, for three quarters (six decisions). Note that on-hand inventory levels at the end of each quarter are also being decided, but those decisions will be implied by the production decisions.

17. Decision Models -- Prof. Juran17 Managerial Formulation Objective Function We’re trying to minimize the total labor cost of production, including both regular and overtime labor, plus inventory cost.

18. Decision Models -- Prof. Juran18 Managerial Formulation Constraints Upper limit on the number of bottles produced with regular labor in each quarter. No backorders are allowed. Production quantities must be non-negative. Mathematical relationships: Inventory balance equations 80% yield on production 10% Shrinkage

19. Decision Models -- Prof. Juran19 Managerial Formulation Note that there is also an accounting constraint: Ending Inventory for each period is defined to be: Beginning Inventory + Production – Demand This is not a constraint in the usual Solver sense, but useful to link the quarters together in this multi-period model.

20. Decision Models -- Prof. Juran20 Mathematical Formulation Decision Variables X ij = Production of type i in period j. Let i index labor type; 0 is regular and 1 is overtime. Let j index quarters; 1 through 3

21. Decision Models -- Prof. Juran21 Mathematical Formulation Non-Decision Variables

22. Decision Models -- Prof. Juran22 Mathematical Formulation Parameters Define C i to be the production cost of type i Define D j to be demand during quarter j

23. Decision Models -- Prof. Juran23 Mathematical Formulation Objective Function

24. Decision Models -- Prof. Juran24 Mathematical Formulation

25. Decision Models -- Prof. Juran25 Solution Methodology

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28. Decision Models -- Prof. Juran28 Solution Methodology

29. Decision Models -- Prof. Juran29 Optimal Solution

30. Decision Models -- Prof. Juran30 Sensitivity Analysis Investigate changes in the holding cost, and determine if Traverso would ever find it optimal to eliminate all inventory. Prepare some graphs showing how Traverso’s optimal decision depends on the holding cost.

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33. Decision Models -- Prof. Juran33 Never optimal to hold inventory at end of 3 rd quarter 1 st and 2 nd Quarters the optimal level depends on cost

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36. Decision Models -- Prof. Juran36 “Tipping points” are at about $6.287 and $19.444.

37. Decision Models -- Prof. Juran37 Sensitivity Analysis Conclusions: It is never optimal to completely eliminate overtime, but sometimes it is optimal to eliminate inventory. In general, as holding costs increase, Traverso will decide to reduce inventories and therefore produce more cases on overtime. Even if holding costs are reduced to zero, Traverso will need to produce at least 1958 cases on overtime. Demand exceeds the total capacity of regular time production. Critical cost points at $6.287 and $19.444.

38. Decision Models -- Prof. Juran38 Multi-Period Models Example: Project Funding

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45. Decision Models -- Prof. Juran45 Summary More Sensitivity Analysis –Solver Sensitivity Report More Malcolm Multi-period Models –Distillery Example –Project Funding Example