1HW 3 Supply Chain¡ Stocking People magazinel p = $3.95, w = $1.50, c = $0.50, g = $2.00l Uniform demand [1000, 2000]¡ F(x) = (x-a)/(b-a); f(x) = 1/(b-a)¡ What is q, profit for M and S with no coordination? Expected overstock?l cu= 3.95+2.00-1.50l co= 1.50l We know optimal q satisfies:¡ F(q*) = cu/(cu+co) = 4.45/(4.45+1.5) = 0.7479 ¡ F(q) = (q-1000)/(2000-1000) à q*= 1748Decentralized Supply Chain¡ Seller’s Profit:l Π = p*(sales) – wq – g*(lost sales)l Sales = q(1-F(q)) + sq1000xf(x)dx= qF(q) - sq1000 F(x)dxl Lost sales = sq2000(x-q)f(x)dxl Π = 3.95(1468.25) – 1.50(1748) –2.00(31.75) = $3114.08¡ Manufacturer: Π = q(w-c) = 1748¡ Leftover Inventory = sq1000(q-x)f(x)dx = 279.75¡ Total SC profit = 3115.08 + 1748 = $4862.082Centralized Supply Chain¡ Order quantity?l cu= 3.95 + 2.00 – 0.50; co= 0.50l F(q*) = cu/(cu+co) = 5.45/(5.45+0.50) = 0.9159 à q^* = 1916¡ Profit?l Same as seller’s profit function but with different cost valuesl Π = p*(sales) – cq – g*(lost sales) = $4946.01Buyback¡ If there is a buyback price b = $0.75, does this coordinate? l In coordinating contract, b = p(1-α) and wb= p(1-α) + αcl b = 0.75 à α = 0.81013l To coordinate we need wb= 3.95(1-0.81) + 0.81(0.50) = $1.155¡ Here buyback without a changed w will not coordinate3Buyback¡ Seller’s q: l cu= 3.95 + 2.00 – 1.50; co= 1.50 – 0.75l F(q*) = 0.8558 à q*= 1856 ¡ Seller’s profit is same as before but now + b*(leftover inventory)l Π = p*[q(1-F(q)) + sq1000xf(x)dx ] – wq –g*sq2000(x-q)f(x)dx + b sq1000(q-x)f(x)dx = $3354¡ Manufacturer: Π = q(w-c) - b sq1000(q-x)f(x)dx = $1581¡ SC Profit = $4935¡ Supplier is better off and Manufacturer is worse off, but total SC is better offBuyback with Capacity¡ What if limited to capacity of 1748?l Since q*>1748, the seller will stock as much as possible until that point, so q = 1748l Seller: Π(q=1748,b=0.75) = $3323l Manufacturer: Π (q=1748,b=0.75) = $1538¡ Buyback is still a good idea for the supplier (although the M still loses some compared to the
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