5 Uncoordinated Supply Chain (1)

Uncoordinated Supply Chain

ByProf. M. K. Tiwari

Dept of IE&MIIT Kharagpur

Co-ordination In Supply Chain

Coordination in Supply Chain Refer as the coordination of information, materials and financial flow between organizations in supply chain.

Brings many organizations as an united team with well established communication channels and optimized resource allocation.

Why Supply Chain Suffers?

When each member of supply chain tries to maximize their own profit.

When each member or group of supply chain tries to optimize individually instead of coordinating their efforts.

Why Coordination is Important in SCM? Communication and Coordination among members of

a supply chain enhances its effectiveness which lead to the benefit of whole supply chain.

For success in the global marketplace requires whole supply chains to compete against other supply chains.

Kind of coordination involve in SC

Horizontal Coordination Coordination among entities involve at same level

of Supply Chain. Example: Coordination between supplier to supplier or within the firm.

Vertical Coordination Coordination among entities involve at different

levels of Supply Chain. Example: Coordination between supplier to retailer or distributor to retailer

Problems in SCM due to Low Involvement of coordination

1. Location Decision of Franchisees of One Organization

2. Warehouse Decision for Organization

3. Lot sizing problem with deterministic demand

4. Demand Forecasting in Supply Chain

5. Product Pricing and Marginal cost Problem between Suppliers and Retailers

6. Lot sizing problem with stochastic demand in a News-vendor environment

Location Decision of Franchisees of One Organization

A franchise has multiple outlet to serve customers, spread out over a town, a city or country.

Problem for franchise is, where they have to locate their franchisees to get maximum profit in Supply Chain.

In two ways they can select location Two or more Franchisees whose location are coordinated by

Franchisor. Two or more Franchisees that control their own location.

Example: Location Decision of Franchisees

Isaac’s Ice Cream had been selling ice-creams in the city, now Isaac wanted to expand his market to reach summertime tourist by selling his ice-creams through small-carts along the boardwalk on 4 mile beach.

Isaac company decided to open two franchisees on the beach in 4 mile boardwalk.

Now Isaac company has two option to establish these franchisees;

Two Franchisees whose locations are coordinated by Isaac company (Franchisor).

Two Franchisees that control their own locations.

Suppose that a franchisor wishes to open two ice cream parlor along a stretch of road 4-mile long.

Potential customers cluster with mile marker [MM] 0,1,2,3 & 4 and each cluster has n number of customer.

Customer demand is sensitive primarily to distance traveled by customer.

MM 00n

customer

MM 01n

customer

MM 02n

customer

MM 04n

customer

MM 03n

customer

4 Mile Beach with n customers on each clusters

1 Mile 2 Mile 3 Mile 4 Mile

Franchisee 1 Franchisee 1

Case 1: Franchisor choosing location for both FranchiseesCase 2: Two Franchisees that control their own locations

Franchisee 1

Franchisee 2

Two Franchisees whose locations are coordinated by Franchisor

If the franchisor can locate these franchisees anywhere on the 4-mile of the road, the franchisor will try to maximize total demand of supply chain.

Demand for franchise will be maximized when the franchise 1(F1) is located at MM1 and franchise 2(F2) is located at MM3.

Two Franchisees whose locations are coordinated by

Franchisor Total demand depends on distance traveled by customer, hence, Demand D given as;

For Franchise 1 demand D1

1 ( 1) ( 0) ( 1)2

Number of customer on each mile

Constant, 0

Constant, 4

naD na b na b b

D na b d

where d=distance traveled by customer

Two Franchisees whose locations are coordinated by

Franchisor For Franchise 2 demand D2

2 ( 1) ( 0) ( 1)2

Number of customer on each mile

Constant, 0

Constant, 4

naD na b na b b

( 1) ( 0) ( 1) ( 0) ( 1)

D na b na b na b na b na b

D na b

Total demand for Supply Chain;

Two Franchisees that control their own locations

In this case, both franchisee try to maximize their own profit and demand, knowing that the other franchisee exists and reacting accordingly.

In this case best location for each one is MM2 and if both franchisees chooses MM2 then;

Total demand D;

( 2) ( 1) ( 0) ( 1) ( 2)

D na b na b na b na b na b

D na b

Warehouse Decision for Organization

The warehouse is a point in the logistics system where a firm stores or hold raw materials, semi finished goods or finished goods.

The firms can use distributed warehousing or centralized warehousing for storage system.

Example: Warehouse Decision for Organization

Isaac’s Ice Cream has grown and now selling their products over the other state through 200 retail-outlets, which are equally distributed between these two states.

In first state, Isaac company leased warehouse space near each shop.

In second state, Isaac company tried storing goods for all 100 shops in that state at a central location.

Example: Warehouse Decision for Organization

In second state, company pays only for storage space and ordering and receiving costs.

Firm has always carried safety stock to protect against unusual high demand.

In centralized warehousing, two benefits are involved;

Economies of scale in setup costs and holding costs Risk pooling in stochastic demand environment

Economic Order Quantity Costs

Benefits of centralized warehousing in terms of economies-of-scale given by EOQ,

For distributed warehousing;

D Annual Demand

H hoslding Costs

S Setup Costs

N Number of Clients

1 22 / , 2 / ................., 2 /

NR R R

EOQ DS H EOQ DS H EOQ DS H

For N Clients

EOQ N DS H

Benefits of centralized warehousing in terms of economies-of-scale given by EOQ,

For Centralized Warehousing

2( ) /C

For N Clients

EOQ ND S H

D Annual Demand

H hoslding Costs

S Stup Costs

N Number of Clients

In this condition supplier combine the whole demand instead of single client demand.

The saving percent for centralized warehousing with respect to distributed warehousing;

2 / 2 / % 100

( ) 2 / % 100

% 1 100

N DS H NDS HSaving

N DS H

N N DS HSaving

N DS H

N NSaving

NSaving

EOQ of Distributed SC

EOQ of Coordinated SC

Numerical Example: Economic Order Quantity Costs

With regard to EOQ costs, Saving %= [1-(√N)/N]x100

Number of Clients Cost Saving %

2 29.29

3 42.26

4 50.00

5 55.28

6 59.18

7 62.20

8 64.64

9 66.67

10 68.38

Number of Clients Cost Saving %

11 69.85

12 71.13

20 77.64

25 80.00

40 84.42

50 85.86

100 90.00

1000 96.84

2500 98.00

Solving as Saving %=(1-√N/N)*100

=(1- √7/7)*100=(1-0.3779)*100

=62.20%

Risk pooling benefits in Centralized Warehousing: Newsvendor Environment

Suppose that ith firm choosing its optimal order quantity has expected overage and underage costs equal to Kσi

. Where σi is firms i’s standard deviation of demand

And K is constant.

For distributed SC Each client has same overage and underage cost per

unit, but with normal probability demand distribution with mean µ and variance σ2;

For N client overage and underage cost = NKσ

Risk pooling benefits in Centralized Warehousing: Newsvendor EnvironmentFor Centralized SC

If supplier combines the demands of its all clients N, Normal probability demand distribution with mean Nµ and

variance Nσ2,

For N client overage and underage cost ,

% 1 100

Overage Underage Costs K N

Overage Underage Costs N K

NSaving

Risk pooling benefits in Centralized Warehousing: Safety

Stock & Service Level The safety stock equals to zσ, where z represents the

number of standard deviation over the mean to achieve a desired cycle service level,

In distributed warehousing system,

Safety Stock Level for Supply Chain = zσN In centralized warehousing system, Supplier combines the demand for all clients

Safety Stock Level for Supply Chain = zσ√N

Risk pooling benefits in Centralized Warehousing: Safety Stock & Service

Level Saving Cost % for it coordinated SC,

Service Levels can improve in centralized warehousing system by improving z value in centralized warehousing;

% 1 100N

SavingN

old new

new old

z N z N

Standard deviation in

distributed SC

Standard deviation in

coordinated SC

Numerical Example: Cycle Service Level Znew

Number of Clients

70.00%

Zold = 0.5244

80.00%

Zold =0.8416

90.00%

Zold =1.2816

2 77.08% 88.30% 96.50%

3 81.81% 92.75% 98.68%

4 85.29% 95.38% 99.48%

5 87.95% 97.01% 99.79%

6 90.05% 98.04% 99.92%

7 91.73% 98.70% 99.97%

8 93.10% 99.14% 99.99%

9 94.22% 99.42% 99.99%

10 95.14% 99.61% 100.00%

15 97.51% 99.94% 100.00%

25 99.56% 100.00% 100.00%

50 99.99% 100.00% 100.00%

100 100.00% 100.00% 100.00%

Service levels from Z table

2 , at z =0.5244 ;

0.7416

At 0.7416 Service level=77.08%

new old

For clients

Lot sizing problem with deterministic demand

Coordinated Lot Sizes with Deterministic Demand

Some product has an expensive setup cost and a very fast production rate.

And it is cheapest to produce it in lot size instead of producing small number size.

It is optimal for the supplier’s lot size of production (lot size for supplier) to be an integer multiple of the retailer’s lot size.

Total annual supply chain setup cost and holding cost are given as;

1..........(1)

2 2s s r r

n QD D QTC S H S H

Annual demand

Supplier's setup cost

Retailer's setup cost

Supplier's holding cost

Retailer's holding cost

Retailer's order size

Supplier's integer lot-size multiplier

Supplier's lot-size

the greatest integer x x

Supplier's annual setup costs

1Supplier's annual average holding costs

Retailer's annual setup costs

Retailer's annual average holding costs2

Differentiate equation (1) of total supply chain annual setup and holding cost TC with respect to Q;

1..........(1)

( 1)............................(2)

(2) 0;

( 1)0;

s s r r

n QD D QTC S H S H

DS n H DS HTC

Q Q n Q

Putting equation equal to

DS n HDSTC

Q Q n Q

( 1)2 2

s s r r

H DS DS Hn

2( 1) ;

2 2s r r

DS DS Hn n n n

DSH QH

2( 1) ;

2 2s r r

DS nH Hn n n

20...................(3)s

20.......................(3)

211 1 4 1

We have to maximize the lot-size;

From equation

By Formula

b b acx

Supplier’s multiple

Integer for Quantity

When the parties optimize independently, the retailer orders Q* and the supplier orders (n*Q* ),

where,* 2

When they optimize jointly, they go through these steps;1:

4 ( )11 1 0,

2s r s

S H Hn Max

and ( 1)sr s r

SS S H n H H

Therefore;

Numerical Example: Coordinated Lot Sizes with Deterministic

DemandFor example, consider a product with annual demand D=25,000 unit, Ss=$200, Sr=$40.50, Hs=$2.00, and Hr=$2.50;

1 8 25000 2001 1

2 2 900

11 5.07

* * 3 900

2700 Units

* 2 2500 40.5900 Unit

Therefore;

Hence;

Numerical Example: Coordinated Lot Sizes with Deterministic

Demand *

3 1 90025000 25000 900200 2 40.50 2.5

3 900 2 900 2

If they Jointly optimize their lot-size;

1 4 200(2.5 2)1 1 0,

2 40.50 2

11 2.42 1

Numerical Example: Coordinated Lot Sizes with

Deterministic Demand

* 2 25000 240.52193 Unit

and ( 1)

20040.50 and (1 1)2.0 2.5

$240.5 and $2.50

sr s r

SS S H n H H

Deterministic Demand Retailer orders 2193 unit and so does the supplier orders 1x2193=2193 unit and total setup and holding cost = $5483, and its 7.1 % lower than individual optimized order quantity holding and setup cost.

In jointly optimization retailer’s holding and setup cost is increase and so it should be compensate by supplier by giving some quantity discount to retailer.

Deterministic DemandBenefits of lot-sizing;

Ss/Sr Qnew/Qold Cost Saving %

1 1.41 5.72

2 1.73 13.40

3 2.00 20.00

4 2.24 25.46

5 2.45 30.01

10 3.32 44.72

15 4.00 52.94

20 4.58 58.34

50 7.14 72.53

100 10.5 80.29

Cost Saving

1 2 1 (

1 2 1 2

rS SS S S

Demand Forecasting in Supply Chain

Coordinated Demand Forecasting Demand of products varies from downstream to

upstream in supply chain due to bullwhip effect in supply chain.

As demand of products varies in supply chain, So forecasting of demand of product also varies from downstream to upstream.

Due to lack of communication between retailers, distributor, wholesaler and supplier demand forecasting may suffer in supply chain.

Example: Coordinated Demand Forecasting

Wholesaler and Retailer work individually without sharing any information Retailers Wholesaler

Periods Customer's Next Period Onhand Back Order Order Placed In-transit Next Period On-hand Back Orders Order In-Transit Order(B) Forecast(C) Inventory(D) (E) (F) Inventory(G) Forecast (I) (J) Placed(K) Inventory(L)

0 5 0 0 5 (H) 5 0 0 51 5 5 5 0 0 0 0 10 0 0 02 5 5 0 0 5 5 5 5 0 0 03 5 5 0 0 5 5 5 0 0 5 54 5 5 0 0 5 5 5 0 0 5 55 5 5 0 0 5 5 5 0 0 5 56 20 20 0 15 35 5 35 0 30 65 657 20 20 0 30 20 50 20 15 0 5 58 20 20 0 0 20 20 20 0 0 20 209 20 20 0 0 20 20 20 0 0 20 2010 20 20 0 0 20 20 20 0 0 20 2011 50 50 0 30 80 20 80 0 60 140 14012 30 30 0 40 10 70 10 70 0 0 013 30 30 0 0 30 30 30 40 0 0 014 30 30 0 0 30 30 30 10 0 20 2015 30 30 0 0 30 30 30 0 0 30 3016 10 10 20 0 0 0 0 30 0 0 017 10 10 10 0 0 0 0 30 0 0 018 50 50 0 40 90 30 90 0 60 150 15019 10 10 0 20 0 60 0 90 0 0 020 10 10 30 0 0 0 0 90 0 0 0

Total 385 70 175 410 395 150 490

Table 1

Next period Forecast=Current

consumer’s Demand

Wholesaler and Retailer sharing consumer’s demand information Retailers Wholesaler

Periods Customer's Next Period Onhand Back Order Order Placed In-transit Next Period On-hand Back Orders Order In-Transit Order(B) Forecast(C) (D) (E) (F) Inventory(G) Forecast Inventory(I) (J) Placed(K) Inventory(L)

0 5 0 0 (H) 5 0 0 51 5 5 0 0 5 5 5 5 0 0 02 5 5 0 0 5 5 5 0 0 5 53 5 5 0 0 5 5 5 0 0 5 54 5 5 0 0 5 5 5 0 0 5 55 5 5 0 0 5 5 5 0 0 5 56 20 20 0 15 35 5 20 0 30 50 507 20 20 0 30 20 50 20 0 0 20 208 20 20 0 0 20 20 20 0 0 20 209 20 20 0 0 20 20 20 0 0 20 2010 20 20 0 0 20 20 20 0 0 20 2011 50 50 0 30 80 20 50 0 60 110 11012 30 30 0 40 10 70 30 40 0 0 013 30 30 0 0 30 30 30 10 0 20 2014 30 30 0 0 30 30 30 0 0 30 3015 30 30 0 0 30 30 30 0 0 30 3016 10 10 20 0 0 0 10 30 0 0 017 10 10 10 0 0 0 10 30 0 0 018 50 50 0 40 90 30 50 0 60 110 11019 10 10 0 20 0 60 10 50 0 0 020 10 10 30 0 0 0 10 50 0 0 0

Total 385 65 175 410 220 150 455

Table 2

Equations for Table 1; For Retailer,

Next period forecast= Consumer current demand

C5 = B5*On-hand Inventory =

Max[( Previous On-hand Inventory + Previous In Transit Inventory – Previous Back order – Current Consumer demand),0]

D5= MAX(D4 + G4 – E4 – B5, 0)= Max(0+5-0-5,0) = 0

*Back Order =

Max[( Previous Backorder + Current Consumer demand – Previous On-hand Inventory – Previous In Transit Inventory), 0]

E5 = MAX( E4 + B5 – D4 – G4, 0) = Max(0+5-0-5, 0) = 0

*Order Placed by Retailer =

Max[( Next Period forecast – (On-hand Inventory + wholesaler’s Previous Backorder – Retailer’s Previous Backorder)), 0]

F5 = MAX ( C5 – (D5 + J4 – E5), 0) = Max[5-(0+0-0), 0] = 5

*In Transit Inventory for Retailer =

Min[( Order Placed by Retailer + Wholesaler’s Previous Backorder), (Wholesaler’s On-hand Inventory + wholesaler’s In Transit Inventory)]

G5 = MIN (F5 + J4 , I4 + L4)= Min(5+0, 0+5)= 5 Equations for Table 1; For Wholesaler;

*Next Forecast = Order Placed by Retailer

H5 = B5

*Wholesaler’s On-hand Inventory =

Max[( Previous On-hand Inventory + Previous In Transit Inventory – Previous Backorder – Order Placed by Retailer ) , 0]

I5 = MAX ( I4 + L4 – J4 – F5 , 0 ) = Max(0+5-0-5, 0)=0*Wholesaler’s Backorder =

Max[( Wholesaler’s Previous Backorder + Order Placed by Retailer - Previous Wholesaler’s On-hand Inventory – Previous Wholesaler’s In Transit Inventory) , 0]

J5 = MAX ( J4 + F5 – I4 – L4 , 0 ) = Max(0+5-0-5, 0)= 0

*Order Placed by Wholesaler =

Max[( Next Period forecast – (Wholesaler’s Current On-hand Inventory – Current Backorder for Wholesaler) , 0]

K5 = MAX ( H5 – (I5 – J5) , 0 )=Max[5-(0-0), 0]= 5 For Table 2, Everything will remain same except Next period forecast of

wholesaler. Next Period forecast for wholesaler = Current Consumer

demand

H5 = B5

From table 1, wholesaler’s forecast equal to the order received from retailer in current period.

And therefore wholesaler’s on-hand inventory is very high due to low information sharing between them.

From table 2, retailer and wholesaler are sharing the information of customer demand.

Therefore wholesaler’s forecasting is equal to retailer’s forecasting.

When demand information is shared, the wholesaler’s total on-hand inventory held over 20 periods is 42% smaller.

In the uncoordinated case wholesaler overreacting to

the retailer’s catch-up order and assuming that consumer demand will be larger in future.

= (395-230)/395 = 42%

Product Pricing and Marginal cost Problem between Suppliers

and Retailers

Coordinated Pricing

Pricing of products is important factor for demand and demand vary according to pricing.

The Supply Chain loses money when the firms do not coordinate their pricing.

In traditional way, supplier first set the wholesale price and the retailer react accordingly and set his own price according to his marginal cost.

Coordinated Pricing

In pricing, can explain by taking two cases; Case 1: A System with One Retailer and One Supplier Case 2; A System with One Retailer and N-1 Supplier

Suppose P = Retail Price of Product

Q = Quantity Sold

Retailer's Demand Curve;

900 2 .......(4)P Q

Case 1: A System with One Retailer and One Supplier

Let Marginal cost for supplier and retailer equal to $90 and $10 respectively.

Total Revenue for Retailer = PxQ

Marginal Revenue for Retailer is the derivative of total revenue (eq.1) with respect to Q;

2900 2 ................(5)P Q Q Q

Retailer's Marginal Revenue 900 4 .......(6)Q

Taking Retailer and supplier as a one firm. Total Marginal Cost for Supply Chain=$90+$10

Optimal quantity Q* given as;

Total Channel profits = Q(P-C) ……..(7)

= 200[500-($90+$10)]=$80,000Where C = Supply Chain Marginal Costs

900 4 100

200 Unit

900 2 200 $500

Taking Retailer and supplier as two individual part of Supply chain.

From eq.6, wholesaler know that Retailer will set marginal cost according to wholesaler’s price charged.

So, 900-4Q=10 + WWhere W = Wholesale price charged

Therefore demand curve for Supplier;

W = 890 – 4Q ……(8)

Therefore supplier’s total revenue W x Q = 890Q-4Q2

Marginal Costs = 890 – 8Q ……(9)From this equation;

90 = 890 – 8Q (As marginal cost for supplier is $90)

Q* = 100 UnitW* = 890 – 4 x 100 (From Equation 8)

W* = $490Total revenue of Supplier = 100[$490 - $90] = $40,000

(From eq. 7)

Retailer also will sell same quantity as supplier’s.Retail Price P = 900 – 2 x 100

Retail Price P* = $700

(From equation (4))

Total revenue for Retailer = 100[$700-($10+$490)]

= $20,000

Total Channel Profit = $40,000 + $ 20,000

= $ 60,000

Which is 33% lesser than coordinated pricing, Cooperative optimization produces more than independent optimization would produce.

Case 2: A System with One Retailer and N-1 Supplier

Now in this case, One Retailer and N-1 Suppliers are involve.

In this, supply chain consisting of one retailer, and retailer’s supplier and retailer’s supplier’s supplier and so on.

In this case Retailer’s linear demand curve given as;

Where (a, b>0 )

(Retailer faces a deterministic linear demand curve of the form of P1)

..........(10)P a bQ

Now let represent the system profit under coordination pricing and represent the system profit under uncoordinated pricing.

Let Ci be the marginal cost of firm i (i=1,2,3……N) and where i = 1 denotes the retailer, i = 2 denotes the retailer’s supplier and i = 3 denotes the retailer's supplier’s supplier.

denote the price charged by firm i.

is a decision variable and represent the quantity sold to the final customer.

represent the optimal quantity for profit maximization.

For Coordinated Supply Chain If there is coordination among the N firms, all the N

firms are considered as one organization, Thus Marginal revenue;

and Marginal Cost given as;

Retail Price given as;

..............(11)

P Q aQ bQ a bQQ Q

arginal cost= iiM C

1 ( ) / 2 .......(12)iiP a C

With the exception of firm N(the most upstream member of supply chain), Ci doest not include the purchase price.

Let Pi denote the pricing charge by firm i.

The decision variable Q represents the quantity sold to the final customer and Q* represents the optimal(profit-maximizing) quantity.

The retailer faces a deterministic linear demand curve of the form of

P1=a – bQ

equating Marginal revenue 2 with marginalcost ( ),

2TheValueof Q putting in equation(10)

a bQ C we get

a C a CP a bQ a b

For Coordinated System, Value of eq. (12) putting in eq. (10);

So, total Profit given as;C

a Ca bQ

a CbQ a

*1( ) .........(14)c ii

* 1 ............(13)

Q a Cb

Marginal Cost For i Firm

Price Charged by i Firm

Quantity Sold

Optimal Quantity For Profit Maximization

From equation 13 and 14;

Total profit in coordinated Supply Chain;

c i iii

a C P Cb

a Ca C C

a Ca C

1 ..............(15)

Case 2: A System with One Retailer and N-1 Supplier (For Uncoordinated Supply Chain) The tier 1 supplier(i=2) knows that the retailer will chose the

quantity by equating its marginal revenue with its marginal cost. Marginal revenue= P1 = a-2bQ

Marginal cost =C1 +P2 where C1= marginal cost of retailer

a-2bQ = C1+P2

P2=( a-C1)-2bQ

C2= marginal cost of retailer’s supplier

C3= marginal cost of retailer’s supplier’s supplier

Continuing in this fashion up the supply chain, we get1

2 ...............(16)

where m=1,2,3......N and where m is m firm

P a C bQ

22 2 1

3 13 1 2

Marginal revenueof 2

Marginalcost

[ ( )] 4

[ ( )] 2

P P Q a C Q bQQ Q

a C bQ

a C bQ P C

P a C C bQ

Putting m=N+1;

1 .................(17)

P a C bQ

a C bQ

Q a Cb

System contain One Retailer and N-1

Suppliers therefore PN+1=0

The Profit of Firm m equals;

Putting all values;

mU m m mQ P P C

Pm+1= 1

a C bQ

1* 1 * *

m mm m

U i i mi i

Q a C bQ a C bQ C

PmPm+1

* 1 * *

m mm m

U i m ii i

Q a C C bQ a C bQ

From above equations;

i m ii i

a C C a C

* 1 * *

m mm m

U i ii i

Q a C bQ a C bQ

* * 1 *2 2m

m mU Q bQ bQ

* 1 *2 (2 1)m

mU Q bQ

21 *2m

mU b Q

Putting Value of from equation (17);

Profit For firm ‘m’;

b a Cb

2...........(18)

Multiplying by 4 in

numerator & denominator

Similarly total profit for Supply Chain;

Putting values of these profits;

1 2 ..........mU U

2 22 2 3 2

2 24 2 2 1

2 2 ........

.............(19)

N NN N

U i ii i

N m NN N

i ii i

a C a Cb b

2 2 3 2 1

2 2 0 1 2 1

2 1 12 2

12 2 ... 2

12 2 2 2 ... 2

1 2 12

12 2 1

NN N N

2 2 22

System Profit Ratio

1 12 2

4 411 2 2 2 2

4 42 22 2

2 4 2 4

N NN N

i ii i

N NN N

a C a Cb b

sum series will become geometric series and after summing this series by geometric sum;

System Profit Ratio in Coordinated SC vs. Uncoordinated SC is

1(2 2 ) ......(20)

2 22 2 1Profit Ratio

N NC U

Lot sizing problem with stochastic demand in a News-vendor

environment

Coordinated Lot Sizing with Stochastic Demand in

Newsvendor Environment Problem arises when a retailer must make a one-time purchase of a single product to meet uncertain demand.

Problem of deciding the size of a single order that must be placed before observing demand when there are overage and underage costs.

Let O = the overage cost per unit

U = the underage cost per unit

F(Q*)= U/(O+U),

where F(x) = Cumulative distribution function over random demand X.

Ps & Pr be the price charged by Supplier and Retailer.

Cs & Cr be the manufacturing cost for supplier & retailer’s cost per unit

Qc* & Qu

* optimal quantity in coordinated and uncoordinated system respectively.

Newsvendor Environment

For Uncoordinated SC If retailer acts independently, its underage and

overage costs are;

Where V = salvage value of any unsold unit

( ) ...............(21)u r r sU P C P

.............(22)u r sO C P V

( ) .....(23)u u u r r s r

U O U P C P P V

Newsvendor Environment For Coordinated SC And if the firm coordinate in supply chain, the

system’s underage and overage costs are;

...........(25)c r sO C C V

...........(24)c r r sU P C C

( ) .....(26)c c c r r s r

U O U P C C P V

f(x) is the density function of random demand X.

In independent optimization, total profit ;

Newsvendor Environment *( )uQ

* * * *

( ) ( ) ( ) ( ) ( )

( ) ( ) ........(27)

u u u u u u s s u

r r s u r

Q xU Q x O f x x Q U f x x P C Q

P C C Q P V F x x

Expected profit:

f(x) = density function of random demand x

( ) ( )Q

P Q xU Q x O f x dx QUf x dx

0 0(1) 2 3

( ) ( ) ( ) ( )

0 ( ) ( )

( ) ( ) ( )

( ) ( )

( )( )

QIndependent of Q

P Q x U O f x dx OQf x dx QUf x dx

P QNow O f x dx Uf x dx

OF Q Uf x dx Uf x dx Uf x dx

OF Q U f x dx U f x dx

P QOF Q U UF Q f

( ) 1x dx

For maximum Profit:

F(Q*) is the cumulative distribution function over random demand x.

OF Q U UF Q

In case of Un Coordinated

Optimal quantity = Qu*

Underage cost per unit = Uu = Pr-Cr-Ps

Overage cost per unit = Ou = Cr + Ps –V

In case of Coordinated

Optimal quantity = Qu*

Underage cost per unit = Uc = Pr-Cr-Cs

Overage cost per unit = Ou = Cr + Cs –V

*( ) uu

*( ) cc

Un coordinated system:

The total profit

* * * *

0 Supplier profit

Retiler profit

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

u u u u u u s s u

Q xU Q x O f x x Q U f x x P C Q

x U O f x x O Q f x x Q U f x x P C Q

Q U O x f x

( ) ( ) ( )u

u u u u s s u

Q Part

Part Part Part

x Q O f x x Q U f x x P C Q

Now integrating by parts

Part-1

uvdx u vdx vdx dxdx

( ) . ( )

( ) ( )

u u u u

u u u u u u

U O x f x x

xU O x f x x f x x dx

U O xF x F x dx

U O Q F Q F x dx

U O Q F Q U O F x dx

Part 2

Part 3

O Q f x x

O Q F Q

( ) ( ) ( )

( ) ( )

Q U f x x

Q U f x x f x dx f x dx

Q U f x dx f x dx

Q U F Q

Putting all the value, we get

* * * * *

* * * *

( ) ( ) ( )

( ) ( )

putting the va

u u u u u u u u u u

u u u s s u

u u u u u u u u u u s s u

u u u u s s

u s s u u u

Q U O Q F Q U O F x dx O Q F Q

U Q F Q P C Q

U O U O Q F Q U O F x dx U Q P C Q

U O F x dx Q U P C

U P C Q U O F x dx

lueof U and O , weget

u r r s u rQ P C C Q P V F x dx

For coordinated:

The profit

( ) ( ) ( ) ( )

( ) ( ) ( )

c c c c c c

Q xU Q x O f x x Q U f x x

U O xf x x Q O f x x Q U f x x

Now integrating

Part 1

Part 2

( ) ( )

c c c c

c c c c c c

U O xf x x

xU O x f x x f x x dx

U O Q F Q F x dx

U O Q F Q U O F x dx

( ) ( )

c c c c c

Q O f x x

Q O f x x Q O F Q

Part 3

( ) ( )

( ) ( ) ( )

( ) ( )

c c c c

Q U f x x Q U f x x

Q U f x x f x x f x x

Q U f x x f x x

Q U F Q

Thus the profit is

* * * *

c c c c c c c c c

c c c c

c r s r c r

U O Q F Q U O F x dx Q O F Q

Q U F Q

U Q U O F x dx

Putting thevalueof U and O we get

Q P C C Q P V F x dx

Now the profit change due to coordination:

( ) . ( )

r s r c r

r r s u r

r s r c u r

r r c u c

r r sr c u

P C C Q P V F x dx

P C C Q Q P V F x dx

P V F x dx P V Q Q F Q

P C CP V Q Q

( ) ( )

This equation proves that pr

r c u r r s

r r s c u r

P V F x dx Q Q P C C let eqn A

P C C Q Q P V F x dx let eqn B

Here Part Part fromeqn A

ofit in coordinated supply chain is always more

than profit of uncoordinated supply chain.

So the coordination always leads to a improved profit

The order size will be increased if there is coordination between to firms because Cs<Ps.

The supplier’s profit increase with joint optimization, but retailer’s profits decrease.

Therefore some profit of supply chain should be redistributed towards the retailer.

If demand is uniformly distributed between a and b, the expected profit for the uniform distribution with ordering quantity Q;

for a x bf x b a

for other x

Pr ( ) ( )

1( ) ( )

ofit xU Q x O f x dx QUf x dx

UQ U O F x

F x f x dx dxb a

x aF x

In case of Coordinated SC

( )2 2

x aQ QU U O dx

U OQU xdx adx

U O xQU a Q a

U O Q aQU aQ a

U O Q aQ QU aQ let eqn c

For optimal condition

Now putting the value of Q in equation c, we get

b-a( ) =aU+

b a O UU b a

Q aO U

Example: Lot Sizing with Stochastic Demand in

Newsvendor Environment For Numerical example: In this example we can use MS Excel commands to solve this problem.

Commands involved in MS Excel for solving this problem are; NORMSDIST NORMDIST NORMINV

Derivation of formulas used in profit calculations when doing numerical examples.

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) __

Q Q Q Q

xU Q x O f x dx QUf x dx

O U xf x dx QO f x dx QU f x dx f x dx f x dx

O U xf x dx QO f x dx QU f x dx f x dx

O U xf x dx Q O U f x dx QU

_( )let eq D

For coordinated case

Let demand density function normally distributed with mean µ, standard deviation σ

f x dx e

Now putting the value of f(x) and dx in equation D, we get

* *2 2

( ) ( )

1( ) ( )

z zz z

O U xf x dx Q O U f x dx QU

O U xf x dx O U z e dz

O U z e dz

O U e dz z e dz

O U F z

( ) ( )

zlet w

dw zdz

Qe dz f

SothePart

O U F z f z

( ) ( )

( ). ( )

Q O U f x dx

Q O U e dz

Q O U F z

Thus the total profit

( ) ( ) ( ). ( )

,0,1,0

part part part

O U F z f z Q O U F z QU

Q QO U NORMSDIST NORDIST

QQ O U NORMSDIST Q U

Numerical Example

Uncoordinated and coordinated supply chain:

1. Demand function is normally distributed

2. Demand function is uniformly distributed between a to b.

Example 1

Demand is normally distributed.

Mean (µ) = 1000 units

Standard deviation(σ) = 500 unit

Supplier manufacturing cost (Cs)= $20

Retailer supplier cost (Cr) = $20

Retailer price of product (Pr) = $100

Price charged by supplier (Ps) = $50

Product salvage value V = $10

Solution:

Uu = Pr – Cr – Ps = 100-20-50 =$30

Ou = Pr – Cr – Ps = 20 + 50 - 10 = $60

Uc = Pr – Cr – Cs = 100 – 20 – 20 =$60

Oc = Cr + Cs –V = 20+20-10 =30

Using Excel

Qu*= NORMINV(1/3, 1000,500) = 784.63= 785

Using table, in cumulative std. normal , z value corresponding to (FQu*=0.333) is -0.43

* 30 1( ) 0.333

30 60 3u

10000.43

Now in case of uncoordinated

Retailer Profit

* * * *u u

( ) ( )

1using ( )

Finally we will reach to the following formula

= O ( ) ( ) O

u s s u s u

xU Q x O f x dx QU f x dx

U F z f z Q U F z Q U

Q QU NORMSDIST NORMDIST

Note here F = Cumulative distribution function =

QQ O U NORMSDIST Q U

Q QNORMSDIST

( ) StandardNormaldistributionfunction

QNORMDIST

Q* = 785*

785 10000.43

( ) ( ) 0.333598

( ) ( ) 0.363714

F z NORMSDIST z

f z NORMDIST z

πU = (30+60) [1000*0.333598 – 500*0.3]-785*(60+30)*0.333598+783*30

= $13638 ( Retailer profit )

Supplier profit = (Ps - Cs)Q* = (50 - 20)*785 = 23550

Total channel profit = Retailer profit + Supplier profit

= 13638 + 23550 = $37188

For Coordinated

Using table, z value corresponding to [F(Q*)=2/3 is 0.43.

60 30 3Using Excel

2 Q ,1000,500 12153

NORMINV

10000.43

Similarly in case of profit formula for channel,

( ) ( )

For calculation in Excel

( ) ( )

1215 10000.43

T c c s s

c c s c

O U F z f z

Q O U F z Q U

O U NORMSDIST z NORMDIST z

Q O U NORMSDIST z Q U

πT = (60+30)[1000 NORMSDIST(0.43)- σ NORMDIST(0.43)]

- 1215 (60+30) NORSDIST(0.43) + 1215*60

πT = $43579

Now Δ π = πc- πU = 43579 – 37188 = $6391

% Increase in profit due to coordination

= (6391/37188)*100 = 17.18%

If the demand is uniformly distributed between 5000 to 15000 unit.

Case 1: Uncoordinated

a = 5000, b= 15000

Uu = $30 Ou = $60

305000 (15000 5000)

UQ a b a

Q units

So retailer profit:

15000 5000 3030 5000

2 30 60

$200000

b a UU a

Supplier profit =

( Ps – Cs ) * Qu* = (50 - 20) * 8333 = $249990

Total channel profit

= retailer profit + supplier profit = $200000 +$249990 = $449000

Case- II: Coordinated

Uc = $60, Oc = $30

Q* = 5000 + (15000 - 5000) * [60 / (30+60)] = 11667 units

Total profit2

15000 5000 605000 60 $500000

2 60 30

b a UU a

So change in profit

Δ π = πc- πu = $500000 - $449000 = $51000

% increase (due to coordination in Supply Chain)

= ( Δ π / πu)*100 = (51000/449000)*100 = 11.35%

Summary

In supply chain management, communication and coordination can greatly enhance the effectiveness of Supply Chain.

Through coordination we can improve total profit of supply chain management, inventory control, pricing control and demand forecasting.

In SCM, the actions of rational managers of firms independently create natural inefficiencies.

By coordination and communication we can reduce these inefficiencies.

As with any group of entities, when all member effectively integrated their efforts, synergies may emerge and SC profit also increase.

5 Uncoordinated Supply Chain (1)

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