session 2b

Session 2b

Decision Models -- Prof. Juran

2

Overview• More Sensitivity Analysis

– Solver Sensitivity Report• More Malcolm

• Multi-period Models– Distillery Example– Project Funding Example


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Solver Sensitivity Report• Provides sensitivity information

about constraint “right-hand sides” and objective function coefficients

• Shadow prices• Allowable increases and decreases


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Malcolm Revisited1234567891011121314151617

A B C D E F G HMicrosoft Excel 15.0 Sensitivity ReportWorksheet: [01b-01-malc.xlsx]OptimizedReport Created: 9/3/2014 8:20:58 AM

Variable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$B$2 6-oz 6.428571429 0 500 40 275$C$2 10-oz 4.285714286 0 450 550 33.33333333

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$E$7 Molding Capacity 60 78.57142857 60 5.5 22.5$E$8 Demand for 6-oz 6.428571429 0 8 1E+30 1.571428571$E$9 Storage Space 150 2.857142857 150 90 22


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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


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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


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ExampleHow much would Malcolm pay for more molding capacity?

How much more capacity would he buy at that price?


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Malcolm's Glass Problem

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6-oz Glasses (100 cases)

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Demand ConstraintMolding ConstraintStorage ConstraintProfit = $2000Profit = $4000Profit = $6000Profit = $8000Corner Points


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Molding Capacity = 60 Molding Capacity = 62


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Molding Capacity = 64 Molding Capacity = 66


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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).


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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


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ExampleHow 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?


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


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Demand ConstraintMolding ConstraintProfit = $2000Profit = $4000Profit = $6000Profit = $8000Storage ConstraintCorner Points


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6-oz profit = 300 6-oz profit = 400


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6-oz profit = 500 6-oz profit = 600


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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.


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• 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.


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• 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. 1st Quarter 2nd Quarter 3rd Quarter

Cases to ship: 3000 2000 4000

Multi-Period ModelsExample: Traverso Distillery


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Managerial FormulationDecision VariablesWe 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.


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Managerial FormulationObjective FunctionWe’re trying to minimize the total labor cost of production, including both regular and overtime labor, plus inventory cost.


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Managerial FormulationConstraintsUpper 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


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Managerial FormulationNote that there is also an accounting constraint: Ending Inventory for each period is defined to be:

Beginning Inventory + Production – DemandThis is not a constraint in the usual Solver sense, but useful to link the quarters together in this multi-period model.


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Mathematical Formulation Decision VariablesXij = 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


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Mathematical Formulation Non-Decision Variables

Define Ij to be ending inventory for quarter j


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Mathematical Formulation Parameters

Define Ci to be the production cost of type iDefine Dj to be demand during quarter j


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Mathematical Formulation

Minimize

Objective Function


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Mathematical Formulation Constraints

For each quarter,


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Solution Methodology 123456789

101112131415

A B C D E F G H I J K1st Quarter 2nd Quarter 3rd Quarter

Beginning Inventory 1000 -1799 -3417 Objective (178,049)$

Regular Time 1 1 1 Regular Cost 120$ Overtime Cost 180$

Capacity 2700 2700 2700 Inventory Cost (178,349)$

Overtime 1 1 1

Total Production 2 2 2Usable for Mays 2 2 2 Yield 0.8Demand 3000 2000 4000 Shrinkage 0.1Inventory -1998 -3797 -7416 Regular cost 40.00$ Breakage, etc. -200 -380 -742 Overtime cost 60.00$ Ending Inventory -1799 -3417 -6674 Inventory cost 15.00$

=C15

=D4+D8=D10*$H$11

=D2+D11-D12=D13*$H$12=D2+D11-D12-D14

=SUM(H4:H6)

=SUM(B4:D4)*H13=SUM(B8:D8)*H14=SUM(B15:D15)*H15


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Solution Methodology 123456789

101112131415

A B C D E F G H I J K1st Quarter 2nd Quarter 3rd Quarter

Beginning Inventory 1000 0 144 Objective 445,360$

Regular Time 2500 2700 2700 Regular Cost 316,000$ Overtime Cost 127,200$

Capacity 2700 2700 2700 Inventory Cost 2,160$

Overtime 0 0 2120

Total Production 2500 2700 4820Usable for Mays 2000 2160 3856 Yield 0.8Demand 3000 2000 4000 Shrinkage 0.1Inventory 0 160 0 Regular cost 40.00$ Breakage, etc. 0 16 0 Overtime cost 60.00$ Ending Inventory 0 144 0 Inventory cost 15.00$

=C15

=D4+D8=D10*$H$11

=D2+D11-D12=D13*$H$12=D2+D11-D12-D14

=SUM(H4:H6)

=SUM(B4:D4)*H13=SUM(B8:D8)*H14=SUM(B15:D15)*H15


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Optimal Solution

1st Quarter 2nd Quarter 3rd Quarter

Regular Time 2500 2700 2700 Overtime 0 0 2120 Total Production 2500 2700 4820


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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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A B C D E F G H I J K L M N O P QOneway analysis for Solver model in Model worksheet Sensitivity of Total Cost to Inventory cost

Inventory cost (cell $H$15) values along side, output cell(s) along top Data for chart

1st Quarter2nd Quarter3rd Quarter1st Quarter2nd Quarter3rd Quarter1st Quarter2nd Quarter3rd Quarter Tota

l Cos

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-$ 2700 2700 2700 0 0 1958 144 274 0 441,480$ 4414801.00$ 2700 2700 2700 0 0 1958 144 274 0 441,898$ 441897.62.00$ 2700 2700 2700 0 0 1958 144 274 0 442,315$ 442315.23.00$ 2700 2700 2700 0 0 1958 144 274 0 442,733$ 442732.84.00$ 2700 2700 2700 0 0 1958 144 274 0 443,150$ 443150.45.00$ 2700 2700 2700 0 0 1958 144 274 0 443,568$ 4435686.00$ 2700 2700 2700 0 0 1958 144 274 0 443,986$ 443985.67.00$ 2500 2700 2700 0 0 2120 0 144 0 444,208$ 4442088.00$ 2500 2700 2700 0 0 2120 0 144 0 444,352$ 4443529.00$ 2500 2700 2700 0 0 2120 0 144 0 444,496$ 444496

10.00$ 2500 2700 2700 0 0 2120 0 144 0 444,640$ 44464011.00$ 2500 2700 2700 0 0 2120 0 144 0 444,784$ 44478412.00$ 2500 2700 2700 0 0 2120 0 144 0 444,928$ 44492813.00$ 2500 2700 2700 0 0 2120 0 144 0 445,072$ 44507214.00$ 2500 2700 2700 0 0 2120 0 144 0 445,216$ 44521615.00$ 2500 2700 2700 0 0 2120 0 144 0 445,360$ 44536016.00$ 2500 2700 2700 0 0 2120 0 144 0 445,504$ 44550417.00$ 2500 2700 2700 0 0 2120 0 144 0 445,648$ 44564818.00$ 2500 2700 2700 0 0 2120 0 144 0 445,792$ 44579219.00$ 2500 2700 2700 0 0 2120 0 144 0 445,936$ 44593620.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600021.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600022.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600023.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600024.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600025.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600026.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600027.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600028.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600029.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 44600030.00$ 2500 2500 2700 0 0 2300 0 0 0 446,000$ 446000

When you select an output from the dropdown list in cell $M$4, the chart will adapt to that output.


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• Never optimal to hold inventory at end of 3rd quarter• 1st and 2nd Quarters the optimal level depends on cost


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• “Tipping points” are at about $6.287 and $19.444.


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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.


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Multi-Period Models

It is January 1, 2008. Director of Special Projects Rakesh Parameshwar has a planned $20.5 million project, which will require the following expected cash flows between 2008 and 2012:

Date Cash Requirement

($ millions) 01-Jul-08 7.50 01-Jan-09 4.50 01-Jul-09 1.00 01-Jan-10 1.00 01-Jul-10 1.00 01-Jan-11 1.00 01-Jul-11 1.00 01-Jan-12 3.50

Example: Project Funding


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Rakesh turns to his Director of Financial Planning, Christine Reyling, and asks her to ensure that funding is available for the project. Christine is considering buying a portfolio of bonds, with cash flows from the bonds arranged to coincide with the needs of Rakesh’s project. The following bonds are available, and can be purchased in any quantity:

Maturity 01-Jul-08 01-Jan-09 01-Jul-09 01-Jan-10 01-Jul-10 01-Jan-11 01-Jul-11 01-Jan-12 Coupon 7.00% 7.50% 6.75% 0.00% 10.00% 9.00% 10.25% 10.00%

Price 1.00 1.03 1.02 0.81 1.16 1.15 1.23 1.25 What is the minimum cost portfolio of these bonds that will meet the project’s requirements? Assume that any cash can be reinvested at an annual rate of 4%, and don’t worry about discounting.


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A B C D E F G H I JReinv 0.02

BondsPurchased 8.650 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Maturity 01-Jul-08 01-Jan-09 01-Jul-09 01-Jan-10 01-Jul-10 01-Jan-11 01-Jul-11 01-Jan-121 2 3 4 5 6 7 8

Coupon 7.00% 7.50% 6.75% 0.00% 10.00% 9.00% 10.25% 10.00%Price 1.00 1.03 1.02 0.81 1.16 1.15 1.23 1.25

Periods1 01-Jul-08 1.0350 0.0375 0.0338 0.0000 0.0500 0.0450 0.0513 0.05002 01-Jan-09 0.0000 1.0375 0.0338 0.0000 0.0500 0.0450 0.0513 0.05003 01-Jul-09 0.0000 0.0000 1.0338 0.0000 0.0500 0.0450 0.0513 0.05004 01-Jan-10 0.0000 0.0000 0.0000 1.0000 0.0500 0.0450 0.0513 0.05005 01-Jul-10 0.0000 0.0000 0.0000 0.0000 1.0500 0.0450 0.0513 0.05006 01-Jan-11 0.0000 0.0000 0.0000 0.0000 0.0000 1.0450 0.0513 0.05007 01-Jul-11 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0513 0.05008 01-Jan-12 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0500

=IF($A11>C$6,0,(IF(C$6=$A11,(C$7/2)+1,(C$7/2))))


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123456789101112131415161718192021222324252627


BondsPurchased 8.650 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000


Coupon 7.00% 7.50% 6.75% 0.00% 10.00% 9.00% 10.25% 10.00%Price 1.00 1.03 1.02 0.81 1.16 1.15 1.23 1.25


Cash In from Prev Cash from Bonds Cash to Project Surplus Cash1 01-Jul-08 0.00 1.30 7.50 -6.20 >= 02 01-Jan-09 -6.32 1.27 4.50 -9.55 >= 03 01-Jul-09 -9.75 1.23 1.00 -9.52 >= 04 01-Jan-10 -9.71 1.20 1.00 -9.51 >= 05 01-Jul-10 -9.70 1.20 1.00 -9.50 >= 06 01-Jan-11 -9.69 1.15 1.00 -9.55 >= 07 01-Jul-11 -9.74 1.10 1.00 -9.64 >= 08 01-Jan-12 -9.83 1.05 3.50 -12.28 >= 0

=F24*(1+$E$1)=SUMPRODUCT($C$3:$J$3,C16:J16)

=C27+D27-E27


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123456789101112131415161718192021222324252627


BondsPurchased 18.198 11.446 0.000 0.967 7.117 0.000 0.000 0.000 0.000


Coupon 7.00% 7.50% 6.75% 0.00% 10.00% 9.00% 10.25% 10.00%Price 1.00 1.03 1.02 0.81 1.16 1.15 1.23 1.25


Cash In from Prev Cash from Bonds Cash to Project Surplus Cash1 01-Jul-08 0.00 11.88 7.50 4.38 >= 02 01-Jan-09 4.47 0.03 4.50 0.00 >= 03 01-Jul-09 0.00 1.00 1.00 0.00 >= 04 01-Jan-10 0.00 7.12 1.00 6.12 >= 05 01-Jul-10 6.24 0.00 1.00 5.24 >= 06 01-Jan-11 5.34 0.00 1.00 4.34 >= 07 01-Jul-11 4.43 0.00 1.00 3.43 >= 08 01-Jan-12 3.50 0.00 3.50 0.00 >= 0


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Summary• More Sensitivity Analysis

– Solver Sensitivity Report• More Malcolm

• Multi-period Models– Distillery Example– Project Funding Example

session 2b

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