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# Supply Contract Models 2

## by TheSupplychainniche on May 24, 2010

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## Supply Contract Models 2Document Transcript

• Double Marginalization • “Classic double marginalization” result has a single supplier selling a product to a single retailer, who faces downward-sloping customer demand. When the retailer doesn’t consider the supplier’s profit margin while ordering, it will tend to order less than level that would maximize supplier profits. •q = quantity retailer orders from supplier, q ≥ 0 •p(q) = price at which retailer can sell q units, p(q) ≥ 0 There exists a maximum sales quantity q such that p( q ) = 0 ˆ ˆ ˆ Over [0, q ], assume p(q) is decreasing, concave, and ∈C 2 •c = production cost per unit for supplier, c ≤ p(0) •w = (wholesale) price per unit paid by retailer •Over one period, given c and p(q) are known, game follows: 1. Supplier chooses w 2. Retailer buys q 3. Retailer sells at p(q) •To analyze this and all subsequent games, follow these steps: 1. Find centralized solution, where a single agent controls all aspects of supply chain to maximize profits 2. Find decentralized solution, where players make decisions to maximize individual profits. 3. If (1) and (2) differ, modify profit equations to find a new decentralizes solution where the behavior more closely follows (1). 1
• Double Marginalization 1. Centrally controlled supply chain •Profits: Π ( q ) = ( p( q ) − c ) q As the retailer paying w to the supplier is a transfer of funds within the supply chain, doesn’t affect the whole chain’s profits ˆ •Since Π(q) is strictly concave in over [0, q ], there exists an optimal solution for the chain q which satisfies Π(q ) = 0: o o ( ) ( ) p q o − c + q o p′ q o = 0 (5.1) 2. Decentralized solution •Retailer’s profits: π r ( q ) = ( p( q ) − w) q •Again, profit equation strictly convex, so there exists a q such than * ( ) ( ) p q * − w + q * p′ q * = 0 (5.1*) •Since supplier will choose w > c in order to have a profit, comparing (5.1) and (5.1*) shows that q < q , meaning the retailer will order less than the * o system-wide optimal quantity whenever the supplier makes a profit. 2
• Double Marginalization 3. Investigation •Marginal cost pricing: setting w = c will allow for q = q , but will leave * o the supplier without any profits •Two-part tariff: set w = c but charge fixed fee of Π(q ), then retailer will o order q but will see no profits o •Profit-sharing contract: select 0 ≤ λ ≤ 1 where supplier earns λΠ(q) and retailer earns (1 - λ)Π(q). Since retailer no longer cares about wholesale price, will pick q to maximize profits. o 3
• Buy-back Contracts •Buy-back contract specifies a price b at which the supplier will purchase unsold goods from retailer. Additionally, assume no supplier receives any income from returned goods. •Single supplier and retailer •q = quantity retailer orders from supplier, q ≥ 0 •p = fixed price retailed charges per item, p > 0 •c = production cost per unit for supplier, c ≤ p •w = (wholesale) price per unit paid by retailer •φ(x) and Φ(x) = p.d.f. and c.d.f. of demand on retailer, where Φ(x) ∈C •Over one period, given c and Φ(x) are known, game follows: 1. Supplier sets w and b 2. Retailer selects amount q to order 3. Supplier produces q units at cost c per unit 4. Demand realized and unsold units returned to supplier 1. Centralized control •Profits:  q  Π ( q ) = −cq + p (1 − Φ ( q ) ) q + ∫ xφ ( x ) dx    0   = production costs + expected sales revenues Again, since w and b represent transfers within supply chain, overall profit does not depend on them. 4
• Buy-back Contracts •However, this is the traditional newsvendor problem which has an optimal order quantity q determined by: o ( ) Φ qo = p −c p (5.2) 2. Decentralized solution •Retailer profits:  q  q π r ( q ) = − wq + p (1 − Φ( q ) ) q + ∫ xφ ( x ) dx  + b ∫ ( q − x )φ ( x ) dx   0   0 = purchase cost + expected sales revenues + expected returns revenue •If p > w > b, π (q) is strictly concave and has an optimal solution q ∋ r * ( ) Φ q* = p − w p −b (5.3) •If w > c and b = 0, a comparison of (5.2) and (5.3) imply that q < q and * o the double marginalization situation occurs. 3. Investigation •q = q if w = c, but again, not attractive for supplier * o •(5.3) indicates that increasing b will increase q . In fact, q = q if * * o p − w p −c = (5.4) p −b p Let b( w) be the value of b which satisfies (5.4): ˆ w − c w−c b( w) = p ˆ  =  p − c   Φ(q o ) Supply chain profits will be maximized when w > c and b = b( w) , ˆ where b( w) is independent of the demand distribution. ˆ Supplier revenue: 5
• Buy-back Contracts q π s ( w, b, q ) = q( w − c ) − b ∫ xφ ( x ) dx 0 = sales revenue + expected return cost If b = b( w) , assume retailer will select q = q : ˆ o qo ( ) π s w, b( w) , q o = q o ( w − c ) − b( w) ∫ xφ ( x ) dx ˆ ˆ 0 ( ∂π s w, b( w) , q o ˆ ) p qo ∂w = qo − p −c ∫ xφ ( x ) dx 0 o q 1 = Φ (q o ) ∫ Φ( x ) dx 0 As wholesale price increases, supplier’s profits increase. If w = p - ε, where ε ≈ 0, the supplier takes almost all the supply chain profits, but the retailer will still order q , even as its profit margin shrinks to 0. o 6
• Quantity Discounts • Can mitigate double marginalization: > c q < q o Retailer pays w(q) where w′( q )  o = c q = q Can be shown that retailer will choose q since its marginal cost o equals that of the supply chain. Additionally, the supplier will earn a profit since the average wholesale price is > c. •Manage operating costs: If a supplier incurs a fixed cost K for producing any order, each unit o costs an average of K /q + c, which is decreasing in q. Quantity o discounts encourage the retailer to order more than they would otherwise (as they don’t see the additional cost). 7
• Competition in Supply Chain Inventory Game: Model •One supplier (referred to as stage 2 or player 2) and one retailer (stage 1/ player 1) •Time divided into infinitely many discrete periods •Consumer demand is stochastic, i.i.d. over all periods •Sequence of events within a period: 1. Shipments arrive 2. Orders submitted and shipped out 3. Consumer demand is realized 4. Holding and backorder penalties assessed •Lead time for order’s arrival: L periods between supplier and its source 2 L periods between supplier and retailer 1 •Any non-negative amount may be ordered •No fixed costs for placing or processing an order •Each player pays a constant price for each item ordered •Holding costs: Supplier pays h > 0 for each unit in-stock or in-transit Retailer 2 pays h + h per unit in inventory (h ≥ 0) 2 1 1 •Backlog: All orders are backlogged until filled (no demand is turned away): p = system-wide cost for backlogging an order α p = retailer’s cost to backlog an order 1 α p = supplier’s cost to backlog an order 2 α +α = 1, α , α ≥ 0. 1 2 1 2 •Demand: 8
• Competition in Supply Chain Inventory Game: Model D = random total demand over τ periods τ µ = mean total demand over τ periods τ φ τ and Φτ = p.d.f. and c.d.f. of demand over τ periods, where Φ (x) is a τ continuous, increasing, and differentiable function for all x ≥ 0, τ ≥ 1 Φ (0) = 0, so positive demand occurs in each period 1 •Local inventory variables for stage i and period t IT = in-transit inventory between stages i+1 and i it IL = inventory level at stage i minus all backorders it IP = IT + IL = inventory position it it it •Policy: Player i uses a base stock policy of ordering enough items to raise inventory position plus outstanding orders to level s ∈ [0, S], i where S is arbitrarily large When selecting its base stock level, each player is aware of all model parameters After selecting base stock levels, model extended over infinite horizon. 9
• Competition in Supply Chain Inventory Game: Model/Optimal Solution •Externalities: 1. Retailer ignores supplier’s backorder costs, so tends to carry too little inventory 2. Supplier ignores retailer’s backorder costs, so tends to carry too little inventory 3. Supplier ignores retailer’s holding costs so tends to carry too much inventory (supplier’s average delivery time decreases, raising retailer’s average inventory) Optimal Solution •Optimal solution for the supply chain minimizes the total average cost per period; it has been shown that a base stock policy produces the optimal solution. Traditional method allocates cost to firms and then minimizes each player’s new cost function. ˆ ( ) • G1o IL1t − D1 = retailer’s charge in period t G1o ( x ) = h1 [ x ] + + ( h2 + p ) [ x ] − ˆ = holding cost for inventory + backorder and order cost • G1o ( IP t ) = retailer’s expected charge in period t + L 1 1 [ ( G1o ( y ) = E G1o y − D L1 + 1 ˆ )] • s1 = retailer’s optimal base stock level found by minimizing G1o ( y ) : o ( ) Φ L1 + 1 s1 = o h2 + p h1 + h2 + p (5.5) 10
• Competition in Supply Chain Inventory Game: Optimal Solution/Game Analysis G1o ( y ) • G1 ( y ) = induced penalty function o ( { G1 ( y ) = G1o min s1 , y − G1o s1 o o o }) ( ) • G2 ( y ) = supplier’s charge in period t ˆo ( G2 ( y ) = h2 y − µ 1 + G1 ( y ) ˆo o ) = holding cost for inventory + induced penalty • G2 ( IP2t ) = supplier’s charge in period t o o ˆo [ ( G2 ( y ) = E G2 y + s1 − D L2 o )] • s2 = supplier’s optimal base stock level found by minimizing G2 ( y ) o o Game Analysis •H (s , s ) = player i’s expected per period cost using base stock levels i 1 2 s and s 1 2 •Best reply mapping for player i is a set-values relationship associating each strategy s with a subset of the decision space σ under the following rules: {s } j r1 ( s2 ) = 1 ∈ σ H1 ( s1 , s2 ) = min H1 ( x, s2 ) x ∈σ r2 ( s1 ) = {s 2 ∈ σ H1 ( s1 , s2 ) = min H ( s , x ) } x ∈σ 2 1 A pure strategy Nash equilibrium is a (s , s ) such that each player 1 * 2 * chooses a best reply to the other’s equilibrium base stock level: s ∈ r (s ) such and s ∈ r (s ) 2 * 2 1 * 1 * 1 2 * Retailer’s cost function: • ˆ ( ) G1 IL1t − D1 = retailer’s charge in period t 11
• Competition in Supply Chain Inventory Game: Optimal Solution/Game Analysis G1 ( y ) = ( h1 + h2 ) [ y ] + + α 1 p[ y ] − ˆ • G1 ( IP t ) = retailer’s expected charge in period t + L 1 1 [ ( G1 ( y ) = E G1 y − D L1 +1 ˆ )] ( ) = ( h1 + h2 ) y − µ L1 +1 + ( h1 + h2 + αp ) ∫ ∞ y ( x − y )φ L1 +1 ( x ) dx •After firms place orders in period t - L , the suppliers IP 2 2(t-L2) =s 2 •After inventory arrives in period t, IL2t = s2 − D L2 (as retailer has ordered D L2 over periods [t - L + 1, t]). If s2 − D L2 ≥ 0, the supplier can 2 completely fill the retailer's order for period t and IP t = s1 . If s2 − D L2 < 0, 1 the order cannot be completely filled and IP t = s1 + s2 − D L2 1 [ ( { H1 ( s1 , s2 ) = E G1 min s1 + s2 − D L2 , s1 }) ] ∞ =Φ L2 ( s2 ) G1 ( s1 ) + ∫ φ L2 ( x ) G1 ( s1 + s2 − x ) dx s2 12
• Competition in Supply Chain Inventory Game: Game Analysis Supplier’s cost function: • ˆ ( ) G2 IL1t − D1 = supplier's actual period t backorder cost G2 ( y ) = α 2 p[ y ] − ˆ • G2 ( IP t ) = supplier's expected period t + L backorder cost 1 1 [ ( G2 ( y ) = E G2 y − D L1 +1 ˆ )] • H 2 ( s1 , x ) = h2 µ L1 + h2 [ x ] + + G2 ( s1 + min{ x,0} ) ˆ [ ( H 2 ( s1 , s2 ) = E H 2 s1 , s2 − D L2 ˆ )] s2 = h2 µ L1 + h2 ∫ ( s2 − x )φ L2 ( x ) dx + 0 ∞ Φ L2 ( s2 ) G2 ( s1 ) + ∫ φ L2 ( x ) G2 ( s1 + s2 − x ) dx s2 where the first term is the expected holding cost for the units in-transit to the retailer, the second term is the expected cost for the supplier's inventory, and the final two terms are the supplier's expected backorder cost. 13
• Competition in Supply Chain Inventory Game: Game Analysis Equilibrium Analysis • Theorem 1: H (s , s ) is strictly convex in s and H (s , s ) is strictly 2 1 2 2 1 1 2 convex in s . 1 Since the cost functions are strictly convex, each player has a unique best reply to the other player's strategy. • Next two theorems show that when a player cares about backorder costs, each will maintain a positive base stock. Also, as one player reduces its base stock, the other will increase its base stock, but at a slower rate. Theorem 2: r (s ) is a function; when α = 0, r (s ) = 0; and when α > 0, 2 1 2 2 1 2 r (s ) > 0 and -1 < r' (s ) < 0. 2 1 2 1 Theorem 3: r (s ) is a function; when α = 0, r (s ) = 0; and when α > 0, 1 2 1 1 2 1 r (s ) > 0 and -1 < r' (s ) < 0. 1 2 1 2 • Theorem 4: (s , s ) is the unique Nash equilibrium. 1 * 2 * Figures show he best reply functions, Nash equilibrium, and optimal solution. • Theorem 5: Assuming α < 1, s + s < s + s . 1 1 * 2 * 1 o 2 o System optimal solution is not a Nash equilibrium whenever α < 1. 1 When α = 1, it may be one under a very special condition. 1 So, competitive selection of inventory policies almost always lead to a deterioration of supply chain performance. 14
• Competition in Supply Chain Inventory Game: Game Analysis 1.4 1.3 Supplier reaction 1.2 function Retailer reaction 1.1 function Nash equilibriu 1 m Optimal Solution 0.9 0.8 0.7 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 α = 0.3, p = 5, h1= h2 = 0.5, L1 = L2 = 1 1.4 1.3 Supplier reaction 1.2 function Retailer reaction 1.1 function Nash equilibrium 1 Optimal Solution 0.9 0.8 0.7 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 α = 0.9, p = 5, h1= h2 = 0.5, L1 = L2 = 1 15