Consolidated_Sensitivity_Duality_MertonTruckCase.pdf

7/27/2019 Consolidated_Sensitivity_Duality_MertonTruckCase.pdf

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(1) Merton Truck Company

(2) Sensitivity Analysis

(3) Duality


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Case Discussion

Case context

Case facts


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Some specific questions we would like

to address

Best product mix?

Which of the 3 options suggested is/ are

better? (and why?)

Other options?


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Before we formulate

I know we are operating at capacity in some

of our production lines.

Is this true?

If so, which production line(s)?


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Decision Variables?


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What if there were no constraints?

Demand?

What is actually constraining production?


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As I see it, we are losing $1205 for each

Model 101 truck we sell. ??

What is missing in the meeting?


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Objective function?

Max contribution or max profit?

Does it matter?

What if economies of scale?


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Contributions

Unit contribution for Model 101 trucks

= selling price variable costs

= selling price (direct material + direct labor

+ variable overhead)

= $39,000 ($24,000 + $4,000 + $8,000) = $3,000.

Similarly, contribution per unit of Model 102 truck= $38,000 ($20,000 + $4,500 + $8,500) = $5,000.


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Objective Function?


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Constraints?


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max 3000 x + 5000 y

s.t.

Capacity Constraints

Non-negativity Constraints

Integer Constraints


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Solving the LP Relax

Graphical approach

Simplex and other algorithms. (Well not getinto details)

Excel Solver


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Graphical approach

Corner points

Iso-contribution/ Iso-profit


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Start from O(0, 0)

There are unused capacities

Manufacture Model 101 trucks, i.e. increase x.

By how much?


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Move to A (2500, 0). Corresponding contribution = 7.5million. (Note: Same contribution at E(0, 1500) too).

Model 101 assembly capacity is tight (binding).

Other constraints are slack (have excess capacities).

Manufacture model 102 trucks. Increase y by howmuch?


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We move to B (2500, 500). Contribution = 10million.

Now, each unit increase in y requires unitdecrease in x. This tradeoff is worthwhile asthis increases total contribution by $2000.

Move to C (2000, 1000). Contribution = 11million.


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This is optimal. (Note that further one unitincrease in y can be achieved only by

decreasing x by 2).

Which of the constraints are binding and

which slack?

What are the respective slacks?

Shadow-prices? (Well come back to this afterlooking at iso-contribution).


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Iso-contribution line

Production mixes at all points on this line willyield the same contribution.

Can there be many iso-contribution lines?

Slope of iso-contribution lines in this case is

-3/5, negative of ratio of coefficients in objective

function.

If infinitely many solutions, then?


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In this case, iso-contribution and iso-profit

lines are the same.

Iso-cost line in case of minimization problems


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Sensitivity Analysis

Shadow prices and reduced costs

Allowable Increase

Allowable Decrease

Ranging

100% rule


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Shadow price of Engine Assembly

Constraint

Consider the metal stamping constraint. It istight. Merton must reduce one Model 101 truckin order to increase Model 102 truck by one unit.

Engine assembly constraint is tight (binding) too.If we produce one less Model 101 truck andincrease the engine assembly capacity by onehour, we can produce one more Model 102 truck.

Net increase in contribution = $2000, which is theshadow price of the engine assemblyconstraint.


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Merton can increase the engine assembly

capacity by 500, before running out of

capacity in some other department (i.e.

before some other constraint becomes tight).This is the allowable increase.


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Engine Assembly Constraint (Allowable increase,allowable decrease and shadow price):

Capacity Optimal Value (x, y)

4500 12,000,000 (1500, 1500)

4000 11,000,000 (2000, 1000)

3500 10,000,000 (2500, 500)


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Metal Stamping Constraint (Allowable increase,allowable decrease and shadow price):

Capacity Optimal Value (x, y)

6500 11,250,000 (2500, 750)

6000 11,000,000 (2000, 1000)

5000 10,500,000 (1000, 1500)


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Graphical Approach to Sensitivity

Analysis

What if we have an additional machine-hour ofengine capacity?

The line CD will move outward and thecoordinates of C and D will change.

New coordinates of C are (1999, 1001)

Increase in contribution = $2000 (the shadowprice for the engine assembly constraint)


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Till when will this shadow price hold? What is theallowable increase?

Till C overlaps with D.

Allowable increase = 500, as seen earlier.

At this point, 3 constraints are binding. Further increase inengine assembly capacity will make the correspondingconstraint slack (and Model 102 assembly constraint andmetal stamping capacity constraint will be binding).


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Questions listed towards end of the case


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Sourcing Out Engine Assembly

?

Using shadow prices?

Shadow price of engine assembly constraint is$2000.

Allowable increase is 500 machine-hours.

Therefore?


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Model 103?

?

Recall that the first two constraints are tight.

Recall that the shadow prices for the 4

constraints are $2000, $500, 0, 0 respectively.


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Cost to company in lost contribution= sum of resource use for each constraint times the respective

shadow price

= $2000*(4000/5000)*2000 + $500*(6000/4000) + $0*1 + $0*0

= $2350

The additional contribution from the production and sale of oneModel 103 truck is $2000.

Therefore?


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We have just priced out Model 103 trucks

and shown that the reduced cost is $350

Cross-check with corresponding LP

formulation using Solver.

Add constraint z = 1, and observe the change

in contribution.


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Overtime production

?

Reformulate by adding two more decision

variables

What are the decision variables?

x_Overtime, y_Overtime


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Additional constraints?

Engine assembly operations overtime.

Mertons contribution increases by $0.7million.

Therefore?


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Question 5

?

Additional constraint

Hence, ?


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Complementary Slackness

If slack is non-zero, shadow price is zero.

If slack is zero?

If optimal solution calls for nonzero values ofdecision variables, corresponding reduced costsare 0.

When optimal value for a decision variable iszero?


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Slack Shadow

Engine Assembly 0 2000

Metal Stamping 0 500

M101 Assembly 1000 0



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Duality

Dual LP formulation for Merton Truck Company


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Slack Shadow

Engine Assembly 0 2000Metal Stamping 0 500



Primal (P):

Dual (D):

Slack Shadow

Constraint 1 0 2000

Constraint 2 0 1000


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Slack Shadow Dual DV

Engine Assembly 0 2000 2000Metal Stamping 0 500 500

M101 Assembly 1000 0 0

M102 Assembly 1500 0 0

Primal (P):

Dual (D):

Slack Shadow Primal DV

Constraint 1 0 2000 2000

Constraint 2 0 1000 1000


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Observations

Ref Duality_MertonTruck.xlsx for Primal and Dual LP formulations and

solutions.

The shadow price of a primal constraint is the value (at optimal) of thecorresponding dual DV (Decision Variable).

Allowable increase and allowable decrease alongside constraints are theallowable increase and allowable decrease of the capacities of thecorresponding constraints (i.e. point up to which the same shadow priceholds).

Similarly for reduced costs.

Allowable increase and allowable decrease alongside decision variablesare the allowable increase and allowable decrease of the coefficients ofthe corresponding variables up to which the same corner point remainsoptimum.

Consolidated_Sensitivity_Duality_MertonTruckCase.pdf

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