5 Uncoordinated Supply Chain (1)
-
Upload
akash-shejule -
Category
Documents
-
view
31 -
download
3
Transcript of 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
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.
Example: Location Decision of Franchisees
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.
Example: Location Decision of Franchisees
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.
Example: Location Decision of Franchisees
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
n
a a
b b
4
0
( )ii
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
n
a a
b b
( 1) ( 0) ( 1) ( 0) ( 1)
(5 3)
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)
(5 6)
D na b na b na b na b na b
D na b
Warehouse Decision for Organization
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 /
;
2 /
NR R R
D
EOQ DS H EOQ DS H EOQ DS H
For N Clients
EOQ N DS H
Economic Order Quantity Costs
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.
Economic Order Quantity Costs
The saving percent for centralized warehousing with respect to distributed warehousing;
2 / 2 / % 100
2 /
( ) 2 / % 100
2 /
% 100
% 1 100
N DS H NDS HSaving
N DS H
N N DS HSaving
N DS H
N NSaving
N
NSaving
N
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 ,
2 &
&
% 1 100
Overage Underage Costs K N
Overage Underage Costs N K
NSaving
N
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
z N z
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 ;
2 *
0.7416
At 0.7416 Service level=77.08%
old
new old
new
For clients
z z
z
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;
Coordinated Lot Sizes with Deterministic Demand
1..........(1)
2 2s s r r
n QD D QTC S H S H
nQ Q
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
s
r
s
r
D
S
S
H
H
Q
n
nQ
the greatest integer x x
Supplier's annual setup costs
1Supplier's annual average holding costs
2
Retailer's annual setup costs
Retailer's annual average holding costs2
s
s
r
r
DS
nQ
n QH
DS
Q
QH
Coordinated Lot Sizes with Deterministic Demand
Differentiate equation (1) of total supply chain annual setup and holding cost TC with respect to Q;
2 2
2 2
1..........(1)
2 2
( 1)............................(2)
2 2
(2) 0;
( 1)0;
2
s s r r
s s r r
s sr
n QD D QTC S H S H
nQ Q
DS n H DS HTC
Q Q n Q
Putting equation equal to
DS n HDSTC
Q Q n Q
2 2
02
( 1)2 2
r
s s r r
H
H DS DS Hn
Q n Q
Coordinated Lot Sizes with Deterministic Demand
2
2( 1) ;
2 2s r r
rs
r
DS DS Hn n n n
DSH QH
2
2( 1) ;
2 2s r r
s
DS nH Hn n n
H Q
22
2 s
s
DSn n
H Q
22
20...................(3)s
s
DSn n
H Q
2= r
r
DSQ
H
Coordinated Lot Sizes with Deterministic Demand
2
2
2
2
2
(3);
20.......................(3)
;
4;
2
211 1 4 1
2
We have to maximize the lot-size;
811 1
2
s
s
s
s
s
s
From equation
DSn n
H Q
By Formula
b b acx
a
DSn
H Q
DSn
H Q
Supplier’s multiple
Integer for Quantity
Coordinated Lot Sizes with Deterministic Demand
When the parties optimize independently, the retailer orders Q* and the supplier orders (n*Q* ),
where,* 2
Q = r
r
DS
H
2
811 1
2s
s
DSn
H Q
and
Coordinated Lot Sizes with Deterministic Demand
When they optimize jointly, they go through these steps;1:
4 ( )11 1 0,
2s r s
r s
Step
S H Hn Max
S H
**
2 :
and ( 1)sr s r
Step
SS S H n H H
n
* 2S
Q DH
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;
*2
*
*
1 8 25000 2001 1
2 2 900
11 5.07
2
3
n
n
n
* * 3 900
2700 Units
s
Q n Q
Q
* 2 2500 40.5900 Unit
2.5Q
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
$5902
TC
TC
If they Jointly optimize their lot-size;
*
1 4 200(2.5 2)1 1 0,
2 40.50 2
11 2.42 1
2
n Max
n
Numerical Example: Coordinated Lot Sizes with
Deterministic Demand
* 2 25000 240.52193 Unit
2.5Q
**
and ( 1)
20040.50 and (1 1)2.0 2.5
1
$240.5 and $2.50
sr s r
SS S H n H H
n
S H
S H
* 2S
Q DH
Numerical Example: Coordinated Lot Sizes with
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.
Numerical Example: Coordinated Lot Sizes with
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
Cost Saving
1 2 1 (
1 2 1 2
)s
s r r
r
s
s r
r
rS SS S S
S
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
Example: Coordinated Demand Forecasting
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
Example: Coordinated Demand Forecasting
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
Example: Coordinated Demand Forecasting
*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
Example: Coordinated Demand Forecasting
*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
Example: Coordinated Demand Forecasting
*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
Example: Coordinated Demand Forecasting
Example: Coordinated Demand Forecasting
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.
Example: Coordinated Demand Forecasting
= (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
Case 1: A System with One Retailer and One Supplier
Taking Retailer and supplier as a one firm. Total Marginal Cost for Supply Chain=$90+$10
=$100
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
Q
Q
P
Case 1: A System with One Retailer and One Supplier
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)
Case 1: A System with One Retailer and One Supplier
Case 1: A System with One Retailer and One Supplier
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
Case 2: A System with One Retailer and N-1 Supplier
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.
C
U
Case 2: A System with One Retailer and N-1 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.
iP
*Q
Q
Case 2: A System with One Retailer and N-1 Supplier
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;
2 2
..............(11)
P Q aQ bQ a bQQ Q
arginal cost= iiM C
1P
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
1
equating Marginal revenue 2 with marginalcost ( ),
2TheValueof Q putting in equation(10)
2 2
i
i
i i
a bQ C we get
a CQ
b
a C a CP a bQ a b
b
Case 2: A System with One Retailer and N-1 Supplier
For Coordinated System, Value of eq. (12) putting in eq. (10);
So, total Profit given as;C
*
2
2
ii
ii
a Ca bQ
a CbQ a
*1( ) .........(14)c ii
Q P C
* 1 ............(13)
2
N
ii
Q a Cb
th
th
*
Marginal Cost For i Firm
Price Charged by i Firm
Quantity Sold
Optimal Quantity For Profit Maximization
i
i
C
P
Q
Q
Case 2: A System with One Retailer and N-1 Supplier
From equation 13 and 14;
Total profit in coordinated Supply Chain;
1
2
1( )
2
1
2 2
1
2 2
1
4
N
c i iii
Ni
i iii
Nii
ii
N
ii
a C P Cb
a Ca C C
b
a Ca C
b
a Cb
2
1
1 ..............(15)
4
N
c ii
a Cb
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
1
1
2 ...............(16)
where m=1,2,3......N and where m is m firm
mm
m ii
th
P a C bQ
22 2 1
1
2 3 2
1 3 2
3 1 2
3 13 1 2
Marginal revenueof 2
( ) 4
Marginalcost
( ) 4
[ ( )] 4
[ ( )] 2
P P Q a C Q bQQ Q
a C bQ
P P C
a C bQ P C
P a C C bQ
P a C C bQ
Case 2: A System with One Retailer and N-1 Supplier
Putting m=N+1;
11
1
1
*
1
2
0
2 0
1 .................(17)
2
NN
N ii
N
NN
ii
N
iNi
P a C bQ
but P
a C bQ
Q a Cb
System contain One Retailer and N-1
Suppliers therefore PN+1=0
Case 2: A System with One Retailer and N-1 Supplier
The Profit of Firm m equals;
Putting all values;
*1( )
mU m m mQ P P C
Pm+1= 1
2m
mi
i
a C bQ
1* 1 * *
1 1
2 2m
m mm m
U i i mi i
Q a C bQ a C bQ C
PmPm+1
1
* 1 * *
1 1
2 2m
m mm m
U i m ii i
Q a C C bQ a C bQ
Case 2: A System with One Retailer and N-1 Supplier
From above equations;
1
1 1
m m
i m ii i
a C C a C
* 1 * *
1 1
2 2m
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
Case 2: A System with One Retailer and N-1 Supplier
Putting Value of from equation (17);
Profit For firm ‘m’;
2
1
1
12
2m
Nm
U iNi
b a Cb
*Q
21
21
2
2m
m N
U iNi
a Cb
22 1
1
2...........(18)
4m
m N N
U ii
a Cb
Multiplying by 4 in
numerator & denominator
Similarly total profit for Supply Chain;
Putting values of these profits;
1 2 ..........mU U
Case 2: A System with One Retailer and N-1 Supplier
2 22 2 3 2
1 1
2 24 2 2 1
1 1
2 2
4 4
2 2 ........
4 4
.............(19)
N NN N
U i ii i
N m NN N
i ii i
a C a Cb b
a C a Cb b
2
2 2 3 2 1
1
2
2 2 0 1 2 1
1
2 1 12 2
1
2
2 2
1
2
2 2
1
12 2 ... 2
4
12 2 2 2 ... 2
4
1 2 12
4 2 1
12 2 1
4
12
4
NN N N
U ii
NN N
ii
NNN
ii
NN N
ii
NN N
ii
a Cb
a Cb
a Cb
a Cb
a Cb
2 2
2
2 2 2
1
2
12 2
4
N
NN N
ii
a Cb
2 2
2 2 2
1 12
2 2 2
1
2 2 22
2 2 2
2
2
2 2
System Profit Ratio
1 12 2
4 4
12 2
4
4 411 2 2 2 2
4 42 22 2
2 4 2 4
4 2 4
2 2 1
2 1
C U
U
N NN N
i ii i
NN N
ii
N NN N
N N
N N
N N
N
N N
N
a C a Cb b
a Cb
Case 2: A System with One Retailer and N-1 Supplier
sum series will become geometric series and after summing this series by geometric sum;
System Profit Ratio in Coordinated SC vs. Uncoordinated SC is
U
2
2 2 2
1
1(2 2 ) ......(20)
4
NN N
U ii
a Cb
2 22 2 1Profit Ratio
2 1
N NC U
NU
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.
Coordinated Lot Sizing with Stochastic Demand in
Newsvendor Environment
For Uncoordinated SC If retailer acts independently, its underage and
overage costs are;
Where V = salvage value of any unsold unit
Coordinated Lot Sizing with Stochastic Demand in
Newsvendor Environment
( ) ...............(21)u r r sU P C P
.............(22)u r sO C P V
;
( ) .....(23)u u u r r s r
Ratio
U O U P C P P V
Coordinated Lot Sizing with Stochastic Demand in
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
Ratio
U O U P C C P V
f(x) is the density function of random demand X.
In independent optimization, total profit ;
Coordinated Lot Sizing with Stochastic Demand in
Newsvendor Environment *( )uQ
*
*
*
* * * *
0
*
0
( ) ( ) ( ) ( ) ( )
( ) ( ) ........(27)
u
u
u
Q
u u u u u u s s u
Q
Q
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
0
( ) ( )Q
Q
P Q xU Q x O f x dx QUf x dx
0 0(1) 2 3
0
*
0 0
*
0 0
* *
( ) ( ) ( ) ( )
0 ( ) ( )
( ) ( ) ( )
( ) ( )
( )( )
Q Q
QIndependent of Q
Q
Q
Q Q
Q
Q
P Q x U O f x dx OQf x dx QUf x dx
P QNow O f x dx Uf x dx
Q
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
Q
0
( ) 1x dx
For maximum Profit:
F(Q*) is the cumulative distribution function over random demand x.
* *
*
( )0
0 ( )
( )
P Q
Q
OF Q U UF Q
UF Q
O U
In case of Un Coordinated
Optimal quantity = Qu*
Underage cost per unit = Uu = Pr-Cr-Ps
Overage cost per unit = Ou = Cr + Ps –V
So
In case of Coordinated
Optimal quantity = Qu*
Underage cost per unit = Uc = Pr-Cr-Cs
Overage cost per unit = Ou = Cr + Cs –V
*( ) uu
u u
UF Q
O U
*( ) cc
c c
UF Q
O U
Un coordinated system:
The total profit
*
*
* *
*
*
* * * *
0 Supplier profit
Retiler profit
* * *
0 0
*
0
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
u
u
u u
u
u
Q
u u u u u u s s u
Q
Q Q
u u u u u u s s u
Q
Q
u u 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
*
*
* * *
0 4
1 2 3
( ) ( ) ( )u
u
Q
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
.du
uvdx u vdx vdx dxdx
*
* * *
*
*
*
*
0
0 0 0
00
* *
0
* *
0
( )
( ) . ( )
( ) ( )
( ) ( )
( ) ( )
u
u u u
u
u
u
u
Q
u u
Q Q Q
u u
u u
Q
u u u u
Q
u u u u u u
U O x f x x
xU O x f x x f x x dx
x
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
*
*
0
* *
( )uQ
u u
u u u
O Q f x x
O Q F Q
*
* *
*
*
*
*
0 0
*
0 0
* *
( )
( ) ( ) ( )
( ) ( )
1 ( )
u
u u
u
u
u u
Q
Q Q
u u
Q
Q
u u
u u u
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
*
*
*
*
* * * * *
0
* * *
* * * *
0
*
0
*
0
( ) ( ) ( )
1 ( )
( ) ( )
0 ( )
( )
putting the va
u
u
u
u
Q
u u u u u u u u u u
u u u s s u
Q
u u u u u u u u u u s s u
Q
u u u u s s
Q
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
*
u u
* *
0
lueof U and O , weget
( )uQ
u r r s u rQ P C C Q P V F x dx
For coordinated:
The profit
*
*
* *
*
* * *
0
* *
0 0
1 2 3
( ) ( ) ( ) ( )
( ) ( ) ( )
c
c
c c
c
Q
c c c c c c
Q
Q Q
c c c c c c
Q
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
*
* * *
*
*
0
0 0 0
* *
0
* *
0
( )
( ) ( )
( )
( )
c
c c c
c
c
Q
c c
Q Q Q
c c
Q
c c c c
Q
c c c c c c
U O xf x x
xU O x f x x f x x dx
x
U O Q F Q F x dx
U O Q F Q U O F x dx
*
*
*
0
* * *
0
( )
( ) ( )
c
c
Q
c c
Q
c c c c c
Q O f x x
Q O f x x Q O F Q
Part 3
* *
* *
*
*
* *
*
0 0
*
0 0
* *
( ) ( )
( ) ( ) ( )
( ) ( )
1 ( )
c c
c c
c
c
c c c c
Q Q
Q Q
c c
Q
Q
c 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
*
*
*
* * * *
0
* *
*
0
* *
0
( )
1
( )
,
( )
c
c
c
Q
c c c c c c c c c
c c c
Q
c c c c
c c
Q
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:
*
*
*
*
*
*
* *
*
0
*
0
* *
* * *
* *
( )
( )
( )
( ) . ( )
c
u
c
u
c
u
c u
Q
r s r c r
Q
r r s u r
Q
r s r c u r
Q
Q
r r c u c
Q
r r sr c u
r
Q Q
P C C Q P V F x dx
P C C Q P V F x dx
P C C Q Q P V F x dx
Now
P V F x dx P V Q Q F Q
P C CP V Q Q
P
V
*
*
*
*
* *
* *
1
2
( ) ( )
( ) ( )
2 1,
0
This equation proves that pr
c
u
c
u
Q
r c u r r s
Q
Q
r r s c u r
QPart
Part
P V F x dx Q Q P C C let eqn A
Now
P C C Q Q P V F x dx let eqn B
Here Part Part fromeqn A
thus
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.
Coordinated Lot Sizing with Stochastic Demand in
Newsvendor Environment
If demand is uniformly distributed between a and b, the expected profit for the uniform distribution with ordering quantity Q;
Coordinated Lot Sizing with Stochastic Demand in
Newsvendor Environment
1
0
for a x bf x b a
for other x
.
Pr ( ) ( )
( )
1( ) ( )
( )
Q b
a Q
Q
a
x x
a a
ofit xU Q x O f x dx QUf x dx
UQ U O F x
Now
F x f x dx dxb a
x aF x
b a
In case of Coordinated SC
2
2 22
2 2
2
2
( )2 2
Q
a
Q Q
a a
Q
a
x aQ QU U O dx
b a
U OQU xdx adx
b a
U O xQU a Q a
b a
U O Q aQU aQ a
b a
U O Q aQ QU aQ let eqn c
b a
For optimal condition
*
*
*
*
2*
Now putting the value of Q in equation c, we get
b-a( ) =aU+
2 U+O
UF Q
O U
Q a U
b a O UU b a
Q aO U
UQ
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 Q Q
Q
Q 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
So
222
1( )
2
x
f x dx e
let x
z
dz dx
Now putting the value of f(x) and dx in equation D, we get
* 2
* 2
* *2 2
3
1 2
2
2
2 2
*
( ) ( )
1
1( ) ( )
2
1( )
2
1 1
2 2
( )
Q Q
Q z z
z z
z zz z
s
O U xf x dx Q O U f x dx QU
Part
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
* 2
21
2
z z
ze dz
The
* 2
2
2
2
1
2
* *
2
1
2
1
( ) ( )
z z
Q
ws
s s
Now
ze dz
zlet w
dw zdz
Qe dz f
SothePart
O U F z f z
* 2
2
2
( ) ( )
1( )
2
( ). ( )
Q
z z
Part
Q O U f x dx
Q O U e dz
Q O U F z
Thus the total profit
* * *
* *
** *
1 2 3
( ) ( ) ( ). ( )
,0,1,0
.
s s s
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
uu u
UF Q
O U
*
*
*
10000.43
500
785
u
u
u
Qz
Q
Q
Now in case of uncoordinated
Retailer Profit
22
* * * *u u
* *
u
( ) ( )
1using ( )
2
Finally we will reach to the following formula
= O ( ) ( ) O
O
Q
u u u
Q
x
u s s u s u
u
xU Q x O f x dx QU f x dx
f x e
xz
U F z f z Q U F z Q U
Q QU NORMSDIST NORMDIST
*
* *
* *
.
Note here F = Cumulative distribution function =
u u u
QQ O U NORMSDIST Q U
Q QNORMSDIST
2
*
*
* 2
( ) StandardNormaldistributionfunction
1
2
s
z
s
f z
QNORMDIST
f z e
Now
Q* = 785*
*
785 10000.43
500
Qz
* *
* *
( ) ( ) 0.333598
( ) ( ) 0.363714
s
s
So
F z NORMSDIST z
f z NORMDIST z
So
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.
So
*
*
60 2
60 30 3Using Excel
2 Q ,1000,500 12153
cUF Q
O U
NORMINV
*
*
*
10000.43
500
1215
Qz
Q
Q
Similarly in case of profit formula for channel,
* *
* * *
* *
* * *
*
( ) ( )
( )
For calculation in Excel
( ) ( )
( )
1215 10000.43
500
T c c s s
c c s c
T c c
c c 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
Now z
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)
30 60
8333
u
u u
UQ a b a
U O
Q units
So retailer profit:
2
2
2
15000 5000 3030 5000
2 30 60
$200000
uu
u u
b a UU a
U O
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
2
2
Total profit2
15000 5000 605000 60 $500000
2 60 30
cc
c c
b a UU a
O U
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.