Session 2 b Overview More Sensitivity Analysis Solver

Session 2 b

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

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

Malcolm Revisited Decision Models -- Prof. Juran 4

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 Decision Models -- Prof. Juran 5

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

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

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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). Decision Models -- Prof. Juran 9

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 Decision Models -- Prof. Juran 10

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? Decision Models -- Prof. Juran 11

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

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

• 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. Decision Models -- Prof. Juran 14

Multi-Period Models Example: Traverso Distillery • Traverso has 1, 000 cases on hand of “Mays & Mc. Covey”. • 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 & Mc. Covey” 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. Decision Models -- Prof. Juran 15

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. Decision Models -- Prof. Juran 16

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

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 Decision Models -- Prof. Juran 18

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. Decision Models -- Prof. Juran 19

Mathematical Formulation Decision Variables Xij = 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 Decision Models -- Prof. Juran 20

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

Mathematical Formulation Parameters Define Ci to be the production cost of type i Define Dj to be demand during quarter j Decision Models -- Prof. Juran 22

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

Mathematical Formulation Decision Models -- Prof. Juran 24

Solution Methodology Decision Models -- Prof. Juran 25

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Solution Methodology Decision Models -- Prof. Juran 28

Optimal Solution Decision Models -- Prof. Juran 29

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. Decision Models -- Prof. Juran 30

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

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• “Tipping points” are at about $6. 287 and $19. 444. Decision Models -- Prof. Juran 36

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. Decision Models -- Prof. Juran 37

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

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