Inventory Management: Cycle Inventory-II 【本著作除另有註明外，採取創用 CC 「姓名標示 －非商業性－相同方式分享」台灣 3.0 版授權釋出】創用 CC 「姓名標示 －非商業性－相同方式分享」台灣 3.0 版 第五單元： Inventory Management: Cycle Inventory-II 郭瑞祥教授 1

Lessons From Aggregation  Aggregation allows firm to lower lot size without increasing cost  Tailored aggregation is effective if product specific fixed cost is a large fraction of joint fixed cost  Complete aggregation is effective if product specific fixed cost is a small fraction of joint fixed cost 2

Holding Cycle Inventory for Economies of Scale ►F►Fixed costs associated with lots ►Q►Quantity discounts ►T►Trade Promotions 3

Quantity Discounts ►Lot size based ►Volume based  How should buyer react? How does this decision affect the supply chain in terms of lot sizes, cycle inventory, and flow time?  What are appropriate discounting schemes that suppliers should offer? 》 Based on total quantity purchased over a given period 》 Based on the quantity ordered in a single lot Total Material Cost C0C0 C1C1 C2C2 q1q1 q2q2 q3q3 Average Cost per Unit Quantity Purchased Order Quantity q1q1 q2q2 q3q3 > All units > Marginal unit 4

 Evaluate EOQ for price in range q i to q i+1, 》 Case 1:If q i  Q i < q i+1, evaluate cost of ordering Q i 》 Case 2:If Q i < q i, evaluate cost of ordering q i 》 Case 3:If Q i  q i+1, evaluate cost of ordering q i+1  Choose the lot size that minimizes the total cost over all price ranges. Evaluate EOQ for All Unit Quantity Discounts hC i DS QiQi 2  DC i hC i QiQi S D TC i  2 QiQi DC i+1 hC i S D TC i  q i+1 2 DC i hC i qiqi S D TC i  2 qiqi 5

Marginal Unit Quantity Discounts C0C0 C1C1 C2C2 q1q1 q2q2 q3q3 Marginal Cost per Unit Quantity Purchased Order Quantity q1q1 q2q2 q3q3 Total Material Cost If an order of size q is placed, the first q 1 -q 0 units are priced at C 0, the next q 2 -q 1 are priced at C 1, and so on. 6

Optimal lot size 2D(S+V i -q i C i ) hC i Qi=Qi= ►Evaluate EOQ for each marginal price C i (or lot size between q i and q i+1 ) Evaluate EOQ for Marginal Unit Discounts 》 Let V i be the cost of order q i units. Define V 0 = 0 and V i =C 0 (q 1 -q 0 )+C 1 (q 2 -q 1 )+ ‧‧‧ +C i-1 (q i -q i-1 ) 》 Consider an order size Q in the range q i to q i+1 Total annual cost = ( D/Q )S (Annual order cost) +[V i +(Q-q i )C i ] h/2 (Annual holding cost) + ( D/Q ) [V i +(Q-q i )C i ] (Annual material cost) 7

》 Evaluate EOQ for each marginal price C i –Case 1 :If q i  Q i < q i+1, calculate cost of ordering Q i –Case 2 and 3 : If Q i q i+1, the lot size in this range is either q i or q i+1 depending on which has the lower total cost 》 Choose the lot size that minimizes the total cost over all price ranges. Evaluate EOQ for Marginal Unit Discounts 2D(S+V i -q i C i ) hC i Qi=Qi= ► Evaluate EOQ for each marginal price Ci, D S D TC i +[ V i +(Q i -q i )C i ] + [ V i +(Q i -q i )C i ]  QiQi QiQi h 2           , 2 Min i i i i i i i i i V q Dh VS q D V q Dh VS q D TC 8

The Comparison between All Unit and Marginal Unit Quantity Discounts  The order quantity of all unit quantity discounts is less than the order quantity of marginal unit quantity discounts.  The marginal unit quantity discounts will further enlarge the cycle inventory and average flow time. 9

►Coordination: max total profits of suppliers and retailers Why Quantity Discounts? ►Quantity discounts are valuable only if they result in: ►Coordination in the supply chain ►Use price discrimination to max supplier’s profits 》 Quantity discounts for commodity products (in the perfect competition market, price is fixed) 》 Quantity discounts for products for which the firm has market power (in the oligopoly market, the determined price can influence demand) 》 Improved coordination in the supply chain 》 Extraction of surplus through price discrimination >Two-part tariffs > Volume discounts 10

►Assume the following data. –Retailer: D =120,000/year, S R =$100, h R =0.2, C R =$3 –Suplier: S S =$250, h S =0.2, C S =$2 ►Retailer cost ►Supplier’s cost is based on retailer’s optimal order size. 》 Supply chain total cost = 3,795+6,009=$9,804 Coordination for Commodity Products 6, ,0002 Q *  x xx  $3, , , TC  xx  x  $6, , , TC  xx  x  11

Coordination for Commodity Products ► Consider a coordinated order size=9,165. ► Coordination through all unit quantity discounts. –$3 for lots below 9,165 $ for lots of 9,165 or higher –Increase in retailer’s holding cost and order cost can be compensated by the reduction in material cost.  120,000( )=$264 –Decrease in supplier’s cost = supply chain savings = 903–264=$639 (can be further shared between two parties) 》 Supply chain total cost=4,059+5,106 =$9,165(decreased by $639) 2 9, ,000 Suppler's TC=250x + x0.2x2 =$5, , ,000 Suppler's TC=100x + x0.2x3 =$4,059 (Increased by $264) $3,795 (decreased by $903) $6,009 12

Coordination for Commodity Products  Since the price is determined by the market, supplier can use lot- size based quantity discounts to achieve coordination in supply chain and decrease supply chain cost.  In practice, the cycle inventory does not decrease in the supply chain because in most firms, marketing and sales department design quantity discounts independent of operations department who works on reducing the order cost.  In theory, if supplier reduces its setup or order cost, the discount it offers will change and the cycle inventory is expected to decrease.  Lot size-based quantity discounts will increase cycle inventory. 13

Quantity Discounts When Firm has Market Power RetailerSupplier p CRCR C S =$2 Demand =360,000-60,000p ►No inventory related costs. ►Assume the following data Demand curve = 360,000-60,000p (p is retailer’s sale price) C S = $2 (cost of supplier). ►Need to determine C R (Suppler’s charge on retailer) and p. 14

►Maximize individual profits and make pricing decision independently ►Demand = 360,000-60,000(5)=60,000 Profit for retailer = (5-4)(60,000)=$60,000 Profit for supplier = (4-2)(60,000)=$120,000 Profit for supply chain = 60, ,000=$180,000 RetailerSupplier p CRCR C S =$2 Demand =360,000-60,000p Scenario 1: No Coordination Profit R C R =4; p=5  2 CRCR  p  =(p-C R ) 0  pp  (Profit R ) 0  CRCR  (Profit S ) Profit S = (360,000-60,000p) 2 CRCR =(C R -2)[360,000-60,000( 3+ )] (C R -2)(360,000-60,000p) 15

Quantity Discounts When Firm has Market Power Demand =360,000-60,000p ►No inventory related costs. ►Assume the following data Demand curve = 360,000-60,000p (p is retailer’s sale price) C S = $2 (cost of supplier). ►Need to determine C R (Suppler’s charge on retailer) and p. CRCR Variation p Fix C S =$2 RetailerSupplier 16

How to increase the total profit through coordination ? ►Profit for supply chain ►Demand = 360,000-60,000(4)=120,000 ►Profit for supple chain = (4-2)(120,000)=$240,000 > $180,000 Maximize Supply Chain Profits  p=4 =(p-C s ) (360,000-60,000p) =(p-2) (360,000-60,000p) 0  pp  (Profit) Microsoft 。 17

Scenario 2: Coordination through Two-Part Tariff -I ►Supplier charges his entire profit as an up-front franchise fee. Supplier sells to the retailer at production cost (C S ). ► Proof:Assume demand function = a-bp (a, b are constants) Then retailer’s profit = (p-c R )(a-bp)-F (F: franchise fee) The supply chain’s profit = (p-c S )(a-bp) Maximize both profits will obtain 18

Scenario 2: Coordination through Two-Part Tariff-II ►Supplier charges his entire profit as an up-front franchise fee. Supplier sells to the retailer at production cost (C S ). ► Proof:Assume demand function = a-bp (a, b are constants) Then retailer’s profit = (p-c R )(a-bp)-F (F: franchise fee) The supply chain’s profit = (p-c S )(a-bp) Maximize both profits will obtain 19

Scenario 2: Coordination through Two-Part Tariff-III  C R = C S ►Supplier charges his entire profit as an up-front franchise fee. Supplier sells to the retailer at production cost (C S ). ► Proof:Assume demand function = a-bp (a, b are constants) Then retailer’s profit = (p-c R )(a-bp)-F (F: franchise fee) The supply chain’s profit = (p-c S )(a-bp) Maximize both profits will obtain P = + = + 2b2b a 2b2b a 2 CRCR 2 CSCS In our example, CR =CS =2, p =4, demand=120,000 Assume a franchise fee of 180,000 Retailer’s profit =(4-2)(120,000)-180,000=$60,000 (same as before) Supplier’s profit = F = $180,000 Supply chain’s profit = 60, ,000=$240,000 20

Scenario 3: Coordination through Volume- based Quantity Discounts In our example, design the volume discounts C R =$4 (for volume < 120,000) C R =$3.5 (for volume  120,000) To sell 120,000, the retailer sets price at p = 4.(from the demand function) Retailer’s profit =(4-3.5)(120,000)=$60,000 (same as before) Supplier’s profit = (3.5-2)(120,000) = $180,000 Supply chain’s profit = 60, ,000=$240,000 ► The two-part tariff is really a volume-based quantity discounts. ► Supplier offers the volume discounts at the break point of optimal demand. ► Supplier offers the discount price so that the retailer will have a profit  the profit of no coordination and no discount. 21

Lessons From Discounting Schemes  Lot size-based discounts increase lot size and cycle inventory in the supply chain.  Volume-based discounts with some fixed cost passed on to retailer are more effective in general  Lot size-based discounts are justified to achieve coordination for commodity products.  Volume-based discounts are better using rolling horizon to avoid the “hockey stick phenomenon”. 22

Price Discrimination to Max Supplier Profits ►Price discrimination is the practice which a firm charges differential prices to maximize profits. ►Price discrimination is also a volume-based discount scheme. ►Consider an example Demand curve (supplier sells to retailer)=200,000-50,000C R C S =2 Profit of supplier = (C R -2)(200,000-50,000C R ) ►What is the optimal “ fixed ” price C R to maximize profit ?  C R  $  0  CRCR  (Profit) Demand=200,000-50,000(3)=50,000 Profit =(3-2)(50,000)=$50,000 23

Demand Curve and Demand at Price of $3 Price p=4 p=3 p=2 100, ,000 Demand Marginal cost = $2 ► The fixed price of $3 does not maximize profits for the supplier. The profit is only the shaded area in the following figure. ► The supplier could obtain the entire area under the demand curve above his marginal cost of $2 (the triangle within the solid lines) by pricing each unit differently. 50,000 24

An Equivalent Two-Part Tariff to Price Discrimination ►The entire triangle under the demand curve (above the marginal cost of $2) = franchise fee = 1/2(4-2)(100,000)=$100,000 ►The selling price to retailer: C R =C S =2. ►Demand = 200,000-50,000(2)=100,000 Profit of supplier = F = $100,000 Price p=4 p=3 p=2 100, ,000 Demand Marginal cost = $2 50,000 Price p=4 p=3 p=2 100, ,000 Demand Marginal cost = $2 50,000 25

Demand Curve and Demand at Price of $3 ► The fixed price of $3 does not maximize profits for the supplier. The profit is only the shaded area in the following figure. ► The supplier could obtain the entire area under the demand curve above his marginal cost of $2 (the triangle within the solid lines) by pricing each unit differently. Price p=4 p=3 p=2 100, ,000 Demand Marginal cost = $2 50,000 26

Holding Cycle Inventory for Economies of Scale ►F►Fixed costs associated with lots ►Q►Quantity discounts ►T►Trade Promotions 27

Trade Promotion ►Goals: –Induce retailers to spur sales –Shift inventory from manufacture to retailer and the customer –Defend a brand against competition ►Retailer options: –Pass through some or all of the promotion to customers to spur sales –Pass through very little of the promotion to customers but purchase in greater quantity to exploit temporary reduction in price (forward buying) 28

Inventory Profile for Forward Buying t I(t)I(t) Q*Q* Q*Q* Q*Q* Q*Q* Q*Q* QdQd Q d : lot size ordered at the discount price Q * : EOQ at normal price 29

Forward Buying Decisions ►Goal: ►Assumptions: –Discount will only be offered once. –Order quantity Q d is a multiple of Q *. –The retailer takes no action to influence the demand. –Identify Q d that maximizes the reduction in total cost (material cost + order cost + holding cost) I(t)I(t) QdQd Q d : lot size ordered at the discount price Q * : EOQ at normal price t Q*Q* Q*Q* Q*Q* Q*Q* Q*Q* 30

Decision on Q * d ► Assume the following data Normal order quantity = EOQ = The discount = $d. The discounted material cost = $(C-d ) ►N►Now estimate the total cost of ordering Q d in the discount period TC(Q d ) = material cost + order cost + inventory cost =(C-d)Q d =(C-d)Q d + S + (Q d /D) 2 (C-d)h / 2D Note: Discount period =Q d /D hC 2DS Q* = + S+ Q d /2 (C-d)h [ Q d /D ] t I(t)I(t) Q*Q* Q*Q* Q*Q* Q*Q* Q*Q* QdQd Q d : lot size ordered at the discount price Q * : EOQ at normal price 31

Decision on Q * d 2hCDS=CD+ =CD hC 2DS +(D/ )S ►N►Now estimate the total cost of ordering Q * in the discount period Annual TC(Q * ) = material cost + order cost + inventory cost hC 2DS +  hC/2 Discount period TC (Q*) = Q d /D [Annual TC(Q*) ]=Q d /D [CD+ 2hCDS ] ►D►Define the cost reduction in the discount period F(Q d ) = TC(Q d ) – Discount period TC(Q * ) C-d  F(Q d ) QdQd =0  Q d = + CQ* [C-d]h dD ► Forward buy = Q d – Q* 32

Decision on Q * d 2hCDS=CD+ =CD hC 2DS +(D/ )S ►Now estimate the total cost of ordering Q * in the discount period Annual TC(Q * ) = material cost + order cost + inventory cost hC 2DS +  hC/2 Discount period TC (Q*) = Q d /D [Annual TC(Q*) ]=Q d /D [CD+ 2hCDS ] ►Define the cost reduction in the discount period F(Q d ) = TC(Q d ) – Discount period TC(Q * ) C-d  F(Q d ) QdQd =0  Q d = + CQ* [C-d]h dD ► Forward buy = Q d – Q* 33

Decision on Q * d 2hCDS=CD+ =CD hC 2DS +(D/ )S ►Now estimate the total cost of ordering Q * in the discount period Annual TC(Q * ) = material cost + order cost + inventory cost hC 2DS +  hC/2 Discount period TC (Q*) = Q d /D [Annual TC(Q*) ]=Q d /D [CD+ 2hCDS ] ►Define the cost reduction in the discount period F(Q d ) = TC(Q d ) – Discount period TC(Q * ) C-d  F(Q d ) QdQd =0  Q d = + CQ* [C-d]h dD ► Forward buy = Q d – Q* 34

Example Assume the following data without promotion.  D =120,000/year, C =$3, h =0.2, S =$100  then →Q* = 6,324  Cycle inventory = Q * /2 = 3,162  Average flow time = Q * /2D = (year) = (month). Q d = + = dD [C-d]h CD* C-d 0.15X120,000 [3-0.15][0.2] + 3(6,324) =38,236 ► Assume a promotion is offered (d =$0.15) Cycle inventory = Q d /2 = 19,118 Average flow time = Q d /2D = (year) = (month). ►Forward buy = 38,236 – 6,324 =31,912 ►Trade promotions generally result in reduced supply chain profits unless the trade promotions reduce demand fluctuations. ►Trade promotions lead to a significant increase in lot size and cycle inventory because of forward buying by the retailer. 35

►Assume demand function = a-bp (a, b are constants) Then retailer’s profit = [p-C R ][a-bp] Maximizing retailer’s profits will obtain If a discount d is offered, the new C R =C R -d Then the new ►Retailer’s optimal response to a discount is to pass only 50% of the discount to the customers. Promotion Pass through to Customers 2 CRCR 2b2b a  P = p 2 d p 2 d 2 CRCR 2b2b a p   36

 Demand curve at retailer: 300,000 – 60,000p ►Normal supplier price, C R = $3.00 ►Promotion discount = $0.15 ►Retailer only passes through half the promotion discount Example-I 》 Optimal retail price = $4.00 》 Customer demand = 60,000 37

 Demand curve at retailer: 300,000 – 60,000p ►Normal supplier price, C R = $3.00 ►Promotion discount = $0.15 ►Retailer only passes through half the promotion discount Example-II 》 Optimal retail price = $4.00 》 Customer demand = 60,000 38

DDemand curve at retailer: 300,000 – 60,000p ►N►Normal supplier price, C R = $3.00 ►P►Promotion discount = $0.15 ►R►Retailer only passes through half the promotion discount Example-III 》 Optimal retail price = $4.00 》 Customer demand = 60,000 》 Optimal retail price = $3.925 》 Customer demand = 64,500 》 Demand increases by only 7.5% 》 Cycle inventory increases significantly 39

Counter measures Trade Promotions Goal is to discourage forward buying in the supply chain – EDLP – Scan based promotions – Customer coupons Wikipedia Microsoft 。 40

Levers to Reduce Lot Sizes Without Hurting Costs Cycle Inventory Reduction ► Reduce transfer and production lot sizes ► Are quantity discounts consistent with manufacturing and logistics operations? re trade promotions essential? 》 Aggregate fixed cost across multiple products, supply points, or delivery points. 》 Volume discounts on rolling horizon 》 Two-part tariff 》 EDLP 》 Base on sell-thru rather than sell-in 41

頁碼作品授權條件作者 / 來源 17 本作品轉載自 Microsoft Office 2007 多媒體藝廊，依據 Microsoft 服 務合約及著作權法第 46 、 52 、 65 條合理使用。 Microsoft 服 務合約 40 本作品轉載自 WIKIPEDIA ( Mart_logo.svg) ，瀏覽日期 2012/2/20 。 Mart_logo.svg 40 本作品轉載自 Microsoft Office 2007 多媒體藝廊，依據 Microsoft 服 務合約及著作權法第 46 、 52 、 65 條合理使用。 Microsoft 服 務合約 40 本作品轉載自 Microsoft Office 2007 多媒體藝廊，依據 Microsoft 服 務合約及著作權法第 46 、 52 、 65 條合理使用。 Microsoft 服 務合約 版權聲明 42