Newsvendor Problem

Transcript Newsvendor Problem - Department of Industrial Engineering

Supply Chain Contracts

Gabriela Contreras Wendy O’Donnell April 8, 2005

Outline

• • • • Introducing Contracts Example: ski jackets – Buy-back – Revenue-sharing – Quantity-flexibility Newsvendor Problem – Wholesale – Buy-back – Revenue-sharing – Quantity-flexibility Results for other problems and open questions

A contract provides the parameters within which a retailer places orders and the supplier fulfills them.

Example: Music store

• • • • Supplier’s cost c=$1.00/unit Supplier’s revenue w=$4.00/unit Retail price p=$10.00/unit Retailer’s service level CSL*=0.5

Question

What is the highest service level both the supplier and retailer can hope to achieve?

Example: Music store (continued)

• • • • Supplier’s cost c=$1.00/unit Supplier’s revenue w=$4.00/unit Retail price p=$10.00/unit Supplier & retailer’s service level CSL*=0.9

Characteristics of an Effective Contract:

• Replacement of traditional strategies • No room for improvement • Risk sharing • Flexibility • Ease of implementation

Why?

Sharing risk increase in order quantity increases supply chain profit

Types of Contracts:

• Wholesale price contracts • Buyback contracts • Revenue-sharing contracts • Quantity flexibility contracts

Outline

Example: Ski Jacket Supplier

• • • • Supplier cost c = $10/unit Supplier revenue w = $100/unit Retail price p = $200/unit Assume: – Demand is normal( m=1000,s=300) – No salvage value

Formulas for General Case

E[retailer profit] = p [ m    q ( X  q ) f ( X ) dX ]  wq E[supplier profit] = q(w-c) 3.

E[supply chain profit] = E[retailer profit] + E[supplier profit]

Results:

Optimal order quantity for retailer = 1,000 Retail profit = $76,063 Supplier profit = $90,000 Total supply chain profit = $166,063 Loss on unsold jackets: – For retailer = $100/unit – For supply chain = $10/unit

Optimal Quantities for Supply Chain:

• • • • When we use cost = $10/unit, supply chain makes $190/unit Optimal order quantity for retailer = 1,493 Supply chain profit = $183,812 Difference in supply chain profits = $17,749

Outline

Buy-Back Contracts

Supplier agrees to buy back all unsold goods for agreed upon price $b/unit

Change in Formulas:

E[retailer profit] = p [ m    q ( X  q ) f ( X ) dX ]  wq E[supplier profit] = q(w-c) – bE[overstock] + bE[overstock] 3.

E[overstock] = q  m    ( X  q ) q f ( X ) dX

Expected Results from Buy-back Contracts for Ski Example

Price w $100 $100 $100 $110 $110 $110 $120 $120 $120 Price b $ $ 30 $ 60 $ $ 78 $ 105 $ $ 96 $ 116 Order Size 1000 1067 1170 962 1191 1486 924 1221 1501 Profit $ 76,063 $ 80,154 $ 85,724 $ 66,252 $ 78,074 $ 86,938 $ 56,819 $ 70,508 $ 77,500 Returns 120 156 223 102 239 493 80 261 506 Profit $ 90,000 $ 91,338 $ 91,886 $ 96,230 $ 100,480 $ 96,872 $ 101,640 $ 109,225 $ 106,310 Chain Profits $ 166,063

$ 171,492 $ 177,610

$ 162,482

$ 178,555 $ 183,810

$ 158,459

$ 179,733 $ 183,810

Outline

Revenue-sharing Contracts

Seller agrees to reduce the wholesale price and shares a fraction f of the revenue

Change in formulas

• E[supplier profit]= (w-c)q+ f p(q-E[overstock]) • E[retailer profit]= (1 f )p(q-E[overstock])+v E[overstock]-wq

Expected results from revenue sharing contracts for ski example Wholesale Price w Revenue sharing Fraction,

Optimal Order Size Expected Overstock Retail Expected Profit Supplier. Expected Profit Expected Supply Chain Profit

$10 $10 0.3

1440 449 $124,273 $ 59,429 $183,702 0.5

1384 399 $ 84,735 $ 98,580 $183,315 $10 0.7

1290 317 $ 45,503 $136,278 $181,781 $10 0.9

1000 120 $ 7,606 $158,457 $166,063 $20 0.3

1320 342 $110,523 $ 71,886 $182,409 $20 0.5

1252 286 $ 71,601 $109,176 $180,777 $20 0.7

1129 195 $ 33,455 $142,051 $175,506

“Go Away Happy” “Guaranteed to be There”

Outline

Quantity-flexibility Contracts

• Retailer can change order quantity after observing demand • Supplier agrees to a full refund of d q units

Quantity-flexibility Contract for Ski Example

d 0 0.2

0.4

0 0.15

0.42

0 0.2

0.5

Price w $100 $100 $100 $110 $110 $110 $120 $120 $120 Order Size Purchase 1000 1000 1050 1024 1070 962 1014 1048 1011 962 1009 1007 924 1000 1040 924 1000 1005 Sales 880 968 994 860 945 993 838 955 994 Profit $ 76,063 $ 91,167 $ 97,689 $ 66,252 $ 78,153 $ 87,932 $ 56,819 $ 70,933 $ 78,171 Profit $ 90,000 $ 89,830 $ 86,122 $ 96,200 $ 99,282 $ 95,879 $ 101,640 $ 108,000 $ 105,640 Chain Profits $ 166,063 $ 180,997 $ 183,811 $ 162,452 $ 177,435 $ 183,811 $ 158,459 $ 178,933 $ 183,811

Outline

• •

Contracts and the Newsvendor Problem

One supplier, one retailer Game description:

Y Accept Contract?

N End Q Production Product Delivery Demand Recognition Transfer payments

Assumptions

• Risk neutral • Full information • Forced compliance

Profit Equations p= price per unit sold S(q)= expected sales c= production cost

p p r = pS(q) – T s = T – cq P( q ) = pS(q) – cq = p r + p s Proof:

Transfer Payment

What the retailer pays the supplier after demand is recognized T = wq

w = what the supplier charges the retailer per unit purchased

Outline

Wholesale Price Contract

Decide on q, w

Let w be what the supplier charges the retailer per unit purchased T w (q,w)=wq

Retailer’s profit function

p r = pS(q)-T

Supplier’s Profit Function

p s = (w-c)q

Results:

• Commonly used • Does not coordinate the supply chain • Simpler to administer

Outline

Buy-back Contracts

• Decide on q,w,b • Transfer payment T = wq – bI(q) = wq – b(q – S(q))

Claim

A contract coordinates retailer’s and supplier’s action when each firm’s profit with the contract equals a constant fraction of the supply chain profit. i.e. a Nash equilibrium is a profit sharing contract

Buy-back contracts coordinate if w & b are chosen such that:   ( 0 , 1 ] p  b =  p w b  b =  c

Recall: p r = pS(q) – T p r = pS(q) – wq – b(q – S(q)) = (p – b)S(q) – (w – b)q = P (q)

Recall: p s p s = T - cq = wq – b(q – S(q)) – cq = bS(q) + (w – b)q – cq = (1  )P( q)

Results

Since q 0 maximizes p (q), q 0 p r is the optimal quantity for both and p s And both players receive a fraction of the supply chain profit

Outline

Newsvendor Problem Revenue-Sharing Contracts

Decide on q, w, f

Transfer Payment

T r = wq + pS(q)

Retailer’s Profit

p r = pS(q)- T • For  Є (0,1], let f p=  p w=  c p r= P (q)

Similar to Buy-Back

From Previous Slide: p r (q,w r , f )= P (q) Recall from Buy-Back: p r (q,w r ,b)= P (q)

Outline

Quantity-flexibility Contracts

• Decide on q,w, d Supplier gives full refund on d q unsold units i.e. min{I, d q}

Expected # units retailer gets compensated for is I r I r = ( 1  d  q ) q F ( x ) dx Proof:

Retailer’s profit function p r = pS(q) – wq + w ( 1  d  q ) q F ( x ) dx

Optimal q satisfies: w = 1 p(1 – F(q)) – F(q) + F((1 – d )q)(1 – d ) If supplier plays this w, will the retailer play this q?

Only if retailer’s profit function is concave As long as w < p and w > 0

Supplier’s profit function p s = wq – w ( 1  q d  ) F ( q x ) dx What is supplier’s optimal q?

Key result • The supply chain is not coordinated if (1 – d ) 2 f((1 – d )q 0 ) > f(q 0 ) q 0 is the minimum

Result

• Supply chain coordination is not guaranteed with a quantity flexibility contract • • Even if optimal w(q) is chosen It depends on d & f(q)

Summary

You can coordinate the supply chain by designing a contract that encourages both players to always want to play q 0 , the optimal supply chain order quantity

Outline

Newsvendor with Price Dependent Demand

• Retailer chooses his price and stocking level • Price reflects demand conditions • Can contracts that coordinate the retailer’s order quantity also coordinate the retailer’s pricing?

• Revenue-sharing coordinates

Multiple Newsvendors

• • • • One supplier, multiple competing retailers Fixed retail price Demand is allocated among retailers proportionally to their inventory level Buy-back permits the supplier to coordinate the S.C.

Competing Newsvendors with Market Clearing Prices

• Market price depends on the realization of demand (high or low) & amount of inventory purchased • Retailers order inventory before demand occurs • After demand occurs, the market clearing price is determined • Buy-back coordinates the S.C.

Two-stage Newsvendor

• Retailer has a 2 nd opportunity to place an order • Buy-back • Supplier’s margin with later production < margin with early production

Open Questions

• Current contracting models assume on single shot contracting.

• Multiple suppliers competing for the affection of multiple retailers • Eliminate risk neutrality assumption • Non-competing heterogeneous retailers

Newsvendor Problem - Department of Industrial Engineering

Transcript Newsvendor Problem - Department of Industrial Engineering

Supply Chain Contracts

A contract provides the parameters within which a retailer places orders and the supplier fulfills them.

Newsvendor Problem

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