Outline

IntroductionIn many industries, the output quantity of a

production process is not deterministic.After the production decision is made, there are

many factors influencing the output quantity.

Example: In agriculture sector: Weather is an important

factor that affects most agriculture related industries, and it is almost impossible to accurately forecast the future weather when the planting decisions are made.

In Semiconductor Industries: The quality of the chips manufactured in very expensive facilities (fabs) is uncertain due to disturbances such as a small amount of dust content in the air, a small timing error in production, etc. Therefore, the same input might yield different output.

A similar phenomenon can be observed in many other industries. Therefore, the same input might result in different output.

So, Random Yield is very common in reality.

Effect of Random Yield on supply chain:

All types of decisions, 1.) pricing decisions2.) ordering policies3.) production decisions etc.

are different in the random yield case from those in deterministic yield case.

Random Yield in one part of the supply chain will influence the whole supply chain.

It will cause overproduction risk or underproduction risk.

Literature Review

ObjectiveTo determine how random yield influences the decisions of independent decision makers in the two level supply chains and how random yield affects the performance of the two level supply chain.

We will discuss following questions: 1.) In different institutional settings, how is the risk

from yield randomness distributed among the parties in the supply chain?

2.) How does random yield affect the performance of the supply chain under these institutional settings?

3.) By studying different contracts, we will try to find

Which contract is better for certain industry? Which contract makes the supplier better off? Which contract makes the retailer better off? What are each party’s optimal decisions(pricing,

ordering etc.) under different situations? What kinds of factors (production cost, inventory

cost etc.) will affect optimal decisions?

Methodology

Centralised ModelAssumptions: 1.) In the centralized model, it is assumed that the retailer and

the supplier belong to the same company.2.) It is assumed that there is no set up cost for the supplier.3.) Payment of the wholesale price could be seen as an internal

revenue transfer, which will not influence the supply chain performance.

The expected supply chain profit is expressed as:

Where,Π=Expected profit of supply chainp=Selling priceQ=Order quantityX=DemandC=production cost per unitU=random variable of random yield

distribution

The optimal order quantity (Q)is given by equation :

The optimal production decision, Q0, decreases in the cost–profit ratio.

When the ratio is small, the supplier has an incentive to produce more to prevent underproduction.

Since, the loss of underproduction will be high when the profit margin is large, or when the cost of production is small as compared to the potential profit.

Unlike the classic supply chain literature, both the yield randomness and the demand uncertainty influence the optimal production decision in our model.

0

( ) ( )c

uG uQ F u dup

No Risk Sharing Model The retailer orders from the supplier and requires all

the ordered products delivered on time.

The retailer does not share the random yield risk with the supplier. Retailer faces only the risk from the market uncertainty.

If supplier’s yield is below the ordered quantity, we assume that there is an emergency production source with unit cost ce.Supplier Retailer Demand

q

xqQ uQ

Supplier Profit function:

The supplier profit function is concave in Q and Q* satisfy,

Q*(q)=K1 q where, K1 is a function in c, ce and f(.).

K1 is increasing in ce and decreasing in c and independent of w(wholesale price).

The higher the emergency cost is, the more the supplier produces; the higher the production cost is, the less the supplier produces.

Note that we assume that ce and w are independent.

( )se uqw c E q uQ cQ

/

0

( )q Q

e

cuf u du

c

Since, supplier always delivers q units the retailer faces a standard newsvendor problem assuming no holding costs the optimal order quantity satisfies,

If w is not exogenous then, supplier pricing problem becomes,

min( , )R xpE x q wq

( *)p w

G qp

max( ) * ( * *) *se u

wwq c E q uQ cQ

Underproduction Risk SharingURS-1

The retailer orders q, the supplier makes his optimal production decision Q.

At the end of the period, the supplier ships the minimum of the ordered quantity and the produced quantity.

Assuming an exogenous wholesale price w, this problem is again a two-stage decision problem.

The underproduction yield risk is shared between the supplier and the retailer, while the overproduction risk is carried only by the supplier.

The supplier profit function is,

The supplier profit function is concave in Q and Q* satisfy,

Q*(q)=K2,1 q where, K2,1 is a function in c, w and f(.).

K2,1 is increasing in w and decreasing in c.

The higher the wholesale price is, the more the supplier produces; the smaller the production cost is, the more the supplier produces.

max( ) ( )su

Qqw wE q uQ cQ

/

0

( )q Q

cuf u du

w

Knowing the supplier’s optimal response, the retailer maximizes his own profit function.

The actual shipped quantity from the supplier is min{q,uQ}.

So the expected profit of the retailer,

The ΠR is concave on q and q* can be solved from:

, min( , ) min( , )Ru x upE x uQ wE q uQ

1/

0 1/

min(1, )( ) ( ) ( ) ( )

ku

ukq k q

wE ukukg x f u dxdu g x f u dxdu

p

Supplier Retailer Demand

q

x

Min(q,uQ)Q uQ

When the supplier’s yield is less than the ordered quantity, it is assumed that the supplier uses an emergency source to fulfill the unmet order.

The supplier and the retailer share the emergency production cost by a certain fraction.

The retailer always get his order, but shares risk of under production.

For each unit of emergency production, the supplier pays ce(1-β) and retailer pays ceβ.

Underproduction Risk SharingURS-2

Supplier Retailer Demandx

q

q

Q uQMax(0,q-uQ)@ Ce

The supplier profit function is,

The supplier profit function is concave in Q and Q* satisfy,

Q*(q)=K2,2 q where, K2,2 is a function in c, ce, β and f(.).

K2,2 is increasing in ce and decreasing in c and β.

The higher the emergency cost is, the more the supplier produces; the smaller the production cost is, the more the supplier produces.

max( ) (1 ) ( )se u

Qqw c E q uQ cQ

/

0

( )(1 )

q Q

e

cuf u du

c

The retailer’s profit function is:

The ΠR is concave on q and q* can be solved from:

As expected emergency cost shared by retailer increases, his optimal order quantity decreases.

Depending on how large the fraction of emergency cost is shared the suppliers optimal quantity deviates from that in centralised model.

min( , ) ( )Rx e upE x q c E q uQ wq

(1 )( *)

e up w c E ukG q

p

When the supplier’s yield turns out to be more than the ordered quantity, the retailer shares the cost incurred by over production.

Constant production costs are assumed and extra units are bought at discounted price.

The retailer always get his order, if uQ<q then he satisfies demand from emergency sources.

Retailer buys all units from supplier and pays w1 for q units and w2 for rest of the units.

Over production Risk Sharing (ORS)

Supplier Retailer Demandx

q at w1Max(0,uQ-q) at w2

q

Q uQ

Max(0,q-uQ)@ Ce

The supplier profit function is,

The supplier profit function is concave in Q and Q* satisfy,

Q*(q)=K3 q where, K3 is a function in c, ce, w1, w2 and f(.).

K3 is increasing in ce and decreasing in c.

Given, w2µ <c < w1µ, if w1µ < c then, it would not be optimal for supplier to produce any .

1 2max( ) ( ) ( )se u u

Qqw c E q uQ w E uQ q cQ

/

2

1 20

( )q Q

e

c wuf u du

w c w

The retailer’s profit function is:

The ΠR is concave on q and q* can be solved from

Keeping the whole sale price w2 small prevents the supplier from producing excessively large number of units.

The pressure of delivering ordered quantity and overproduction risk sharing gives supplier incentive to produce more.

1/ 1/

1 1 2

0 1/ 0 1/

( ) ( ) ( ) ( ) ( ) ( 1) ( )k k

ukq k q k

p ukg x f u dxdu p g x f u dxdu ukw f u du w w uk f u du

, 1 2min ,max( , ) ( )Ru x upE x q uQ w q w E uQ q

When the supplier’s yield turns out to be more than the ordered quantity, the retailer shares the cost incurred by over production.

Constant production costs are assumed and extra units are bought at discounted price.

The retailer also shares underproduction risks if, uQ<q and the supplier does not incur emergency costs.

Retailer buys all units from supplier and pays w1 for q units and w2 for rest of the units.

Hybrid Risk Sharing (ORS)

Supplier Retailer Demandx

Min(q,uQ) at w1Max(0,uQ-q) at w2

q

Q uQ

Max(0,q-uQ)@ Ce

The supplier profit function is,

The supplier profit function is concave in Q and Q* satisfy,

Q*(q)=K4 q where, K4 is a function in c, w1, w2 and f(.).

K4 is increasing in w1 and decreasing in c.

If µ<c/w1 then K4 is decreases in w2 and if µ≥c/w2 then K4 increases in w2.

1 1 2max( ) ( ) ( )su u

Qqw w E q uQ w E uQ q cQ

/

2

1 20

( )q Q

c wuf u du

w w

The retailer’s profit function is:

The ΠR is concave on q and q* can be solved from

, 1 2min , ( ) ( )Ru x u upE x uQ w E q q uQ w E uQ q

1/

1 1 2

0 0 1/

( ) ( ) ( ) ( 1) ( )k

k

p ukG ukq f u du ukw f u du w w uk f u du

conclusionsFive contracts representing different situations in random yield risk have been considered:1.) Centralized supply chain2.)No-risk sharing contract(NRS)3.) Underproduction risk sharing

URS-I : Underproduction loss sharingURS-II : Emergency Cost Sharing

4.)Overproduction risk sharing contracts5.) Both risk shared by supplier and retailers (UORS)

The application of different contracts depends on the negotiating power of supply chain parties.

The supplier’s production quantity decision usually shows a linear relation with the retailer’s order quantity.Depending on the random yield risk sharing scheme, the linear coefficients are different in different contracts.

When the supplier shares more random yield risk, the supplier inputs more for the same ordered quantity.

The retailer’s order decision depends on both the supplier’s response and the risk sharing plan.

The random yield might help to enhance the whole chain profit by inducing higher delivered amount from the supplier.

Future Scope The models presented here have been solved only

for a two level supply chain, but in reality supply chain can have different levels. So these models can be solved for supply chains having more than two levels and a comparative study of different contracts can be done.

Here the stochastic proportional yield model is used. However, the input amount Q sometimes plays a role in yield distribution and could influence yield distribution. Further study can be done that deals with a generalized yield in the supply chain.

References Erdem, A.S., Ozekici, S., 2002. Inventory models with random yield in a

random environment. International Journal of Production Economics 78 (3), 239–253.

Gerchak, Y., 1992. Order point/order quantity models with random yield. International Journal of Production Economics 26, 297–298.

Gerchak, Y., Grosfeld-Nir, A., 1998. Multiple lot-sizing, and value of probabilistic information, in production to order of an uncertain size. International Journal of Production Economics 56–57, 191–197.

Gerchak, Y., Vickson, R.G., Parlar, M., 1988. Periodic review production models with variable yield and uncertain demand. IIE Transactions 20 (2), 144–150.

Gerchak, Y., Wang, Y., Yano, C.A., 1994. Lot sizing in assembly systems with random component yields. IIE Transaction 26 (2), 19–24.

Gerchak, Y., Tripathy, K., Wang, K., 1996. Co-production models with random functionality yields. IIE Transaction 28, 391–403.

Grosfeld-Nir, A., Gerchak, Y., 1996. Production to order with random yields: Single-stage multiple lot-sizing. IIE Transaction 28, 669–676.

Gurnani, H., Gerchak, Y., 2007. Coordination in decentralized assembly systems with uncertain component yields. European Journal of Operational Research 176, 1559–1576.

Gurnani, H., Akella, R., Lehoczky, J., 2000. Supply management in assembly systems with random yield and random demand. IIE Transaction 32, 701–714.

Henig, M., Gerchak, Y., 1990. The structure of periodic review policies in the presence of random yield. Operations Research 38 (4), 634–643.

Hillier, F.S., 1963. Reject allowance for job lot orders. Journal of Industrial Engineering 14, 311–316.

Jones, P.C., Lowe, T.J., Traub, R.D., Kegler, G., 2001. Matching supply and demand: The value of a second chance in producing hybrid seed corn. Manufacturing & Service Operations Management 3 (2), 122–137.

Kazaz, B., 2004. Production planning under yield and demand uncertainty with yield-dependent cost and price. Manufacturing & Service Operations Management 6 (3), 209–224.

Kurtulus, I.S., Pentico, D.W., 1988. Materials requirement planning when there is scrap loss. Production and Inventory Management Journal 29 (2), 18–21.

Yuanjie He, Jiang Zhang, 2007. Random yield risk sharing in two level supply chain. International Journal of Production Economics 112 769-781.

Lariviere, M.A., Porteus, E.L., 2001. Selling to the newsvendor: An analysis of price-only contracts. Manufacturing & Service Operations Management 3 (4), 293–305.

Levitan, R.E., 1960. The optimal reject allowance problem. Management Science 6 (5), 172–186.

Wang, Y., Gerchak, Y., 1996. Periodic review production models with variable capacity, random yield, and uncertain demand. Management Science 42 (1), 130–137.

Wang, Y., Gerchak, Y., 2000. Input control in a batch production system with lead times, due dates and random yields. European Journal of Operations Research 126 (2), 371–385.

Yano, C.A., Lee, H., 1995. Lot sizing with random yields: A review. Operations Research 43 (2), 311–334.