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Session 2b. Overview. More Sensitivity Analysis Solver Sensitivity Report More Malcolm Multi-period Models Distillery Example Project Funding Example. Solver Sensitivity Report. Provides sensitivity information about constraint “right-hand sides” and objective function coefficients

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overview
Overview
  • More Sensitivity Analysis
    • Solver Sensitivity Report
      • More Malcolm
  • Multi-period Models
    • Distillery Example
    • Project Funding Example

Decision Models -- Prof. Juran

solver sensitivity report
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

malcolm revisited
Malcolm Revisited

Decision Models -- Prof. Juran

shadow price
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

constraints
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

Decision Models -- Prof. Juran

example
Example

How much would Malcolm pay for more molding capacity?

How much more capacity would he buy at that price?

Decision Models -- Prof. Juran

slide9

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

adjustable cells
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

example1
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

slide12

Increases in the profitability of the 6-oz product have the effect of changing the slope of the isoprofit lines.

Decision Models -- Prof. Juran

slide13

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.

Decision Models -- Prof. Juran

slide14
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

slide15

Multi-Period Models

Example: Traverso Distillery

  • 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.

Decision Models -- Prof. Juran

managerial formulation
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

managerial formulation1
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

managerial formulation2
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

managerial formulation3
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

mathematical formulation
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

mathematical formulation1
Mathematical Formulation

Non-Decision Variables

Define Ij to be ending inventory for quarter j

Decision Models -- Prof. Juran

mathematical formulation2
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

mathematical formulation3
Mathematical Formulation

Objective Function

Minimize

Decision Models -- Prof. Juran

mathematical formulation4
Mathematical Formulation

Constraints

For each quarter,

Decision Models -- Prof. Juran

solution methodology
Solution Methodology

Decision Models -- Prof. Juran

solution methodology1
Solution Methodology

Decision Models -- Prof. Juran

optimal solution
Optimal Solution

Decision Models -- Prof. Juran

sensitivity analysis
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

slide33

Never optimal to hold inventory at end of 3rd quarter

  • 1stand 2nd Quarters the optimal level depends on cost

Decision Models -- Prof. Juran

sensitivity analysis1
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, Traversowill 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

slide38

Multi-Period Models

Example: Project Funding

Decision Models -- Prof. Juran

summary
Summary
  • More Sensitivity Analysis
    • Solver Sensitivity Report
      • More Malcolm
  • Multi-period Models
    • Distillery Example
    • Project Funding Example

Decision Models -- Prof. Juran

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