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1. Risk Pooling Problems

Case on Risk Pooling of the book, page 49

Compare the two systems-

a) Without risk pooling

b) With risk pooling with the following parameters: 99% service level

Assume all other parameters are same as given in the case. Provide qualitative and quantitative solutions

Qualitative solution

With random demand, it is very likely that a higher-than-average demand at one retailer will offset by a lower-

than-average demand at another. As the number of retailers served by a warehouse goes up, the likelihood that

the two will offset will also increase, which is inline with the forecasting principle that states aggregate forecastsare more accurate. This argument is in favor of the centralized system (single warehouse).

Product A

week 1 2 3 4 5 6 7 8av.Dd SD CoV S Stock R Q

Ms33 45 37 38 55 30 18 58 39.25 13.18

0.335796

30.7094

69.9594 132

NJ46 35 41 40 26 48 18 55

38.625

11.26873

0.291747

26.25615

64.88115 131

Total79 80 78 78 81 78 36

113

77.875

19.37419

0.248786

45.14187

123.0169 186

Product A: Massachusetts New JerseyTotalAVG =

Total

Weekly

Deman

Product

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d/numb

er of

Weeks

=39.2538.62577.875Expecteddemandduringlead

time =L * AVG=1*39.251*38.6251*77.875

= 39.2538.62577.875SD =

(weekly dd-AVG)2/n-1

=

B

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13.1811.2719.37CoV =

SD/AVG=13.18/39.25 =0.340.290.25SafetyStock =Z *SD

*L= 2.33*13.18*1=30.7126.2745.14ReorderPoint =

Expected ddduringleadtime +safetystock= 39.25

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+30.71=69.9664.88123

OrderQuantity (Q) =(2K*AVG)/hK =$60,h =$0.27=

(2*60*39.25)/.27= 132units131186

Week

1 2 3 4 5 6 7 8 Av.

DD

STD CoV S Stock R Q

Ms0 3 3 0 0 1 3 0 1.25 1.3919

411.1135

533.2432

234.4932

2324

NJ2 4 0 0 3 1 0 0 1.25 1.4790

21.1832

163.4461

164.6961

1624

Total2 7 3 0 3 2 3 0 2.5 2.0615

530.8246

214.8034

187.3034

1834

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Product B: Massachusetts New JerseyTotalAVG = Total Weekly Demand/number of Weeks

=1.25 1.252.5Expected demand during lead time = L * AVG

= 1*1.25 1*1.251*2.5

= 1.25 1.252.5SD = (weekly dd-AVG)2/n-1

= 1.39 1.472.06CoV = SD/AVG

= 1.39/1.25 = 1.11 1.480.82Safety Stock = Z *SD *L

= 2.33 *1.39 *1 =3.24 3.454.80Reorder Point = Expected dd during lead time + safety stock

= 1.25 +3.24 =4.49 4.724Order Quantity (Q) = (2K*AVG)/h K =$60, h = $0.27

= (2*60*1.25)/.27= 24 units 24

34SD measures the absolute variability of customer demands, while the coefficient of variationmeasures variability relative to average demandExample: Product A has a much larger SD (13.18, 11.27 & 19.37) while Product B has asignificant larger CoV (1.11, 1.48 & .82)

The average demandProduct A SD CoV Safety Stock R Q

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M 39.25 13.18 .34 30.7 70 132NJ 38.625 11.27 .29 26.26 65 131

Total 77.875 19.37 .25 45.14 124 186Average dd faced by the centralized distribution system equals the sum of the two

The variability faced by the centralized distribution is much smaller than the sum of the twoboth in terms of SD and CoVAverage inventory for product A at the Massachusetts = SS + Q/2

= 30.7 + 66 =97at the New Jersey = 26.26 + 65.5 =92 = 189

average inventory for Product A in the centralized system = 45.14 + 93 = 138Thus, average inventory for product A is reduced by 37% (189-138)/138 if the centralizedsystem is usedAverage inventory for Product B at the Massachusetts = 3.24 +12 = 15.24Average inventory for Product B at the New Jersey = 3.45 + 12 = 15.45 =31

Average inventory for Product B at the centralized system is = 4.80 + 17 = 22Thus, average inventory for Product B is reduced by 41% (31-22)/22 if the centralized system isused2. Example 7-6, page 243Given:Wholesale price $80Selling price 125Production cost 35Salvage value 20Profit in decentralized system

Each dealer can order 12000 unitsProfit per retailer = $540,000Manufacturers profit = $440000*2 = $880,000

Total supply chain Profit = $1,080,000 +$880,000 =$1,960,000Profit in centeralized systemBoth Retailers can order 26,000 units altogether

B. Supply Contract ProblemsRetail price $28

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Wholesale price 20Salvage value 50% of retail price = 14

The store estimates that demand during the season is normal with a mean of 1000 and a SD of250

a).Quantity

250 500 750 1000 1250 1500 1750

AverageP

2000 4000 6000 8000 6500 5000 3500

Given the information, the optimal order Q for the store is 1000 unitsWhether the optimal quantity is less than or greater than the expected average demand or notrequires to look at the marginal profit and marginal loss figuresMarginal profit = retail price wholesale price

= $28-20 = $8

Marginal loss = wholesale price salvage value= $20 -14 = 6Thus, in general terms if MP MP, then the optimal order quantity is greater than the averagedemand. This is only true if maximizing average profit is in fact the goal of the firm.b). With buy-back contract

The publisher offers $17 price for unsold copies; and the store is required to ship back theunsold copies which results in a $1.00 cost per copyQ 250 500 750 1000 1250 1750 2000Ave. P 2000 4000 6000 8000 7000 6000 5000

Given the information, the optimal order quantity Q for the store is still 1000 units2. Example 4.3 of the text book, page 130

Two companies involved in the supply chain: a retailer and a manufacturerGivenRetail Price Rs. 125Wholesale price = Rs. 80Salvage value = Rs. 20For the manufacturerFixed Production cost is Rs. 100,000

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The variable production cost per unit equals Rs. 35Determine the profit of the supply chain under two scenarios:

a) No coordination (sequential supply chain)8000 10000 12000 14000 16000 18000 price 125

0.11 0.11 0.27 0.23 0.17 0.1 WSprice 80

SV 20

VC 35

Manu FC10000

026000

035000

044000

050500

054500

058500

0

Ex P 28600 3850011880

011615

0 92650 58500

Ret36000

045000

054000

060500

064500

068500

0

Ex P 39600 4950014580

013915

010965

0 68500

The optimal order quantity is 12,000 unitsTotal supply chain profitRetailers profit + manufacturers profit= Rs. 540,000 + 440,000 = Rs. 980,000

b) Buy-back contract with a buy-back price of Rs. 50With buy-back contract

8000 10000 12000 14000 16000 18000 price 125

0.11 0.11 0.27 0.23 0.17 0.1 WS price 80

SV 20

VC 35

FC1000

00

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BB 50

Manu26000

035000

044000

051500

057500

063500

0

28600 38500 118800 118450 97750 63500

Retailer

360000

450000

540000

635000

735000

835000

39600 4950014580

014605

012495

0 83500