1. Inventory Management: Cycle Inventory 【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 第四單元： Inventory Management: Cycle Inventory 郭瑞祥教授 1

2. Understocking: Demand exceeds amount available –Lost margin and future sales Overstocking: Amount available exceeds demand – Liquidation, Obsolescence, Holding Role of Inventory in the Supply Chain 2

3. ►Economies of scale ►Stochastic variability of supply and demand 》 Batch size and cycle time 》 Quantity discounts 》 Short term discounts / Trade promotions 》 Service level given safety inventory 》 Evaluating Service level given safety inventory Why hold inventory? 3

4. Improve Matching of Supply and Demand Improved Forecasting Reduce Material Flow Time Reduce Waiting Time Reduce Buffer Inventory Economies of Scale Supply / Demand Variability Seasonal Variability Cycle InventorySafety Inventory Seasonal Inventory Cost Efficiency Cost Efficiency Availability Responsiveness Availability Responsiveness Improve Matching of Supply and Demand Role of Inventory in the Supply Chain 4

5. Cycle Inventory Improve Matching of Supply and Demand Improved Forecasting Reduce Material Flow Time Reduce Waiting Time Reduce Buffer Inventory Economies of Scale Supply / Demand Variability Seasonal Variability Safety Inventory Seasonal Inventory Cost Efficiency Availability Responsiveness Cycle inventory is the average inventory that built up in the supply chain because a stage of the supply chain either produces or purchases in lots that are larger than those demanded by the customer. Cycle Inventory Inventory Time t Cycle inventory = lot size/2 = Q/2 Q 5

6. Little’s Law ►A►Average flow time = Average inventory / Average flow rate Average flow time resulting from cycle inventory Q: Lot size D: Demand per unit time ►F►For any supply chain, average flow rate equals the demand, = Cycle inventory / Demand = Q / 2D 6

7. Holding Cycle Inventory for Economies of Scale ►F►Fixed costs associated with lots ►Q►Quantity discounts ►T►Trade Promotions 7

8. Economics of Scale to Exploit Fixed Costs — Economic Order Quantity— »D= Annual demand of the product »S= Fixed cost incurred per order »C= Cost per unit »h=Holding cost per year as a fraction of product cost »H=Holding cost per unit per year =hC »Q=Lot size »n=Order frequency 8

9. Cost Lot Size Lot Sizing for a Single Product (EOQ) ► Annual order cost =(D/Q)S=ns Annual holding cost = (Q/2)H =(Q/2)hC Annual material cost = CD TC =CD + (D/Q)S + (Q/2)hC Holding Cost Material Cost Order Cost Total Cost 9

10. Cost Lot Size ► Annual order cost =(D/Q)S Annual holding cost = (Q/2)H =(Q/2)hc Annual material cost = CD ‧ TC =CD + (D/Q)S + (Q/2)hc Holding Cost Material Cost Order Cost Total Cost Lot Sizing for a Single Product (EOQ) Total annual cost, TC =CD + (D/Q)S + (Q/2)hc Optimal lot size, Q is obtained by taking the first derivative hC Q DS dQ TCd 0 2 )( 2    hC DS Q 2 *  Average flow time = Q*/2D S * DhC Q D n 2 *  10

11. Example 5 Demand ， D =1,000 units/month = 12,000 units/year 5 Fixed cost, S = $4,000/order 5 Unit cost, C = $500 5 Holding cost, h = 20% = 0.2 》 Optimal order size 》 Cycle inventory 》 Numbers of orders per year 》 Average flow time Q / 2D = 490 / 12000 =0.041 (year) =0.49(mounth) Q = = 980 0.2X500 2X12000X4000 Q/2 =490 D / Q = 12000 / 980 =12.24 11

12. Example - Continued 5 Demand ， D =1,000 units/month = 12,000 units/year 5 Fixed cost, S = $4,000/order 5 Unit cost, C = $500 5 Holding cost, h = 20% = 0.2 》 Optimal order size 》 Cycle inventory 》 Numbers of orders per year 》 Average flow time Q / 2D = 490 / 12000 =0.041 (year) =0.49(mounth) Q = = 980 0.2X500 2X12000X4000 Q/2 =490 D / Q = 12000 / 980 =12.24 ► If we want to reduce the optimal lot size from 980 to 200, then how much the order cost per lot should be. 2D hC(Q*) 2 S = 0.2X500X200 2 2X12000 == $166.7 > If we increase the lot size by 10% (from 980 to 1100), what the total cost would be. Annual cost = $ 98,636 (from $ 97,980)(an increase by only 0.6%) (Note: material cost is not included) Microsoft 。 CoolCLIPS Microsoft 。 12

13. Key Points from EOQ 5Total order and holding costs are relatively stable around the economic order quantity. A firm is often better served by ordering a convenient lot size close to the EOQ rather than the precise EOQ. 5 To reduce the optimal lot size by a factor of k, the fixed order cost S must be reduced by a factor of k 2. 5 If demand increases by a factor of k, the optimal lot size increases by a factor of. The number of orders placed per year should also increase by a factor of. Flow time attributed to cycle inventory should decrease by a factor of. k k k 13

14. Aggregating Multiple Products in a Single Order ► One of major fixed costs is transportation ► Ways to lower the fixed ordering and transportation costs: ► Ways to lower receiving or loading costs: 》 ASN (Advanced Shipping Notice) with EDI 》 Aggregating across the products from the same supplier 》 Single delivery from multiple suppliers 》 Single delivery to multiple retailers Microsoft 。 14

15. Example: Lot Sizing with Multiple Products 5Three computer models (L, M, H) are sold and the demand per year: –D L = 12,000; D M = 1,200; D H = 120 ► Common fixed (transportation) cost, S = $4,000 ► Holding cost, h = 0.2 ► Unit cost 》 C L = $500; C M = $500; C H = $500 ► Additional product specific order cost 》 s L = $1,000; s M = $1,000; s H = $1,000 L M H Microsoft 。 15

16. Delivery Options ► No aggregation ► Complete aggregation ► Tailored aggregation 》 Each product is ordered separately 》 All products are delivered on each truck 》 Selected subsets of products on each truck 16

17. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 1: No Aggregation Result LiteproMedproHeavypro Demand per year120001200120 Fixed cost / order$5,000 Optimal order size1,095346110 Order frequency11.0/year3.5/year1.1/year Annual holding cost$109,544$34,642$10,954 Annual total cost = $155,140 (no material cost) 17

18. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 2: Complete Aggregation 》 Combined fixed cost per order is given by 》 Let n be the number of orders placed per year. We have Total annual cost = Annual order cost + Annual holding cost = HML sss S S  * )]2/()2/()2/[()( * nhCDn Dn DnS HHMMLL  DS Q 2 *  * * 2S hCD D D n HHMMLL   n S DhC 2 *  18

19. Option 2: Complete Aggregation ► No aggregation ► Complete aggregation ► Tailored aggregation 》 Combined fixed cost per order is given by 》 Let n be the number of orders placed per year. We have Total annual cost = Annual order cost + Annual holding cost = HML sss S S  * )]2/()2/()2/[()( * nhCDn Dn DnS HHMMLL  * * 2S D D D n HHMMLL   n S DhC 2 *  19

20. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 2: Complete Aggregation Result LiteproMedproHeavypro Demand per year120001200120 Order frequency9.75/year Optimal order size1,23012312.3 Annual holding cost$61,512$6,151$615 Annual order cost = 9.75×$7,000 = $68,250 Annual total cost = $68,250+$61,512+$6,151+$615=$136,528 20

21. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 3: Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. ►Step 3: Recalculate order frequency of most frequently ordered product. ►Step 4: Identify ordering frequency of all products. ►Step 1: Identify most frequently ordered product. 21

22. Option 3: Tailored aggregation ► No aggregation ► Complete aggregation ► Tailored aggregation ►A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. ►Step 3: Recalculate order frequency of most frequently ordered product. ►Step 4: Identify ordering frequency of all products. 0.11 1.1 3.5,,0.11 )(2    n H n M n L sS L D L hC L n ►Step 1: Identify most frequently ordered product. } 2(S+s i ) { i hC i D i nini Maxn  22

23. Option 3: Tailored aggregation ► No aggregation ► Complete aggregation ► Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. >Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. ►Step 3: Recalculate order frequency of most frequently ordered product. ►Step 4: Identify ordering frequency of all products. 0.11 1.1 3.5,,0.11 )(2    n H n M n L sS L D L hC L n ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. 2.4 7.7 2  H M MM M n s DhC n  55.4  H m  24.1    7.7/0.11/  MM nnm ►Step 1: Identify most frequently ordered product. n=  hC i D i 2si2si 23

24. Option 3: Tailored aggregation ► No aggregation ► Complete aggregation ► Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. ►Step 3: Recalculate order frequency of most frequently ordered product. >Step 4: Identify ordering frequency of all products. ►Step 1: Identify most frequently ordered product. 0.11 1.1 3.5,,0.11 )(2    n H n M n L sS L D L hC L n >Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. 2.4 7.7 2  H M MM M n s DhC n   7.7/0.11/  MM nnm ►Step 3: Recalculate order frequency of most frequently ordered product. )] / ( [ 2     ii iii msS mDhC n  55.4  H m  24.1  TC= order cost + holding cost Derivation of n 24

25. Option 3: Tailored aggregation ► No aggregation ► Complete aggregation ► Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. >Step 3: Recalculate order frequency of most frequently ordered product. ►Step 4: Identify ordering frequency of all products. ►Step 1: Identify most frequently ordered product. 0.11 1.1 3.5,,0.11 )(2    n H n M n L sS L D L hC L n >Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. 2.4 7.7 2  H M MM M n s DhC n   7.7/0.11/  MM nnm >Step 3: Recalculate order frequency of most frequently ordered product. )] / ( [ 2     ii iii msS mDhC n Derivation of n    += 2n2n i i hC i D i M i mimi n nS TC= order cost + holding cost  (hC i ) i nisinisi i 2n i DiDi nS TC=( )+ + sisi  55.4  H m  24.1    7.7/0.11/  MM nnm 25

26. Option 3: Tailored aggregation > No aggregation > Complete aggregation > Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. >Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. >Step 3: Recalculate order frequency of most frequently ordered product. >Step 4: Identify ordering frequency of all products. ►Step 1: Identify most frequently ordered product. 0.11 1.1 3.5,,0.11 )(2    n H n M n L sS L D L hC L n >Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. 2.4 7.7 2  H M MM M n s DhC n   7.7/0.11/  MM nnm ►Step 3: Recalculate order frequency of most frequently ordered product. )] / ( [ 2     ii iii msS mDhC n Derivation of n    += 2n2n i i    0 n TC hC i D i M i mimi n nS TC= order cost +  (hC i ) i nisinisi i 2n i DiDi nS TC=( )+ + sisi  =0  S+  i mimi sisi  2n22n2 i hC i D i M i holding cost  )] / ( [ 2     ii iii msS mDhC n 26

27. Option 3: Tailored aggregation ► No aggregation ► Complete aggregation ► Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. >Step 3: Recalculate order frequency of most frequently ordered product. >Step 4: Identify ordering frequency of all products. ►Step 1: Identify most frequently ordered product. 0.11 1.1 3.5,,0.11 )(2    n H n M n L sS L D L hC L n ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. 2.4 7.7 2  H M MM M n s DhC n   7.7/0.11/  MM nnm ►Step 3: Recalculate order frequency of most frequently ordered product. )] / ( [ 2     ii iii msS mDhC n =11.47 L  55.4  H m  24.1  Derivation of n    += 2n2n i i    0 n TC hC i D i M i mimi n nS TC= order cost +  (hC i ) i nisinisi i 2n i DiDi nS TC=( )+ + sisi  =0  S+  i mimi sisi  2n22n2 i hC i D i M i holding cost  )] / ( [ 2     ii iii msS mDhC n Microsoft 。 27

28. Option 3: Tailored aggregation > No aggregation > Complete aggregation > Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. >Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. >Step 3: Recalculate order frequency of most frequently ordered product. >Step 4: Identify ordering frequency of all products. 0.11 1.1 3.5,,0.11 )(2    n H n M n L sS L D L hC L n ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. 2.4 7.7 2  H M MM M n s DhC n   7.7/0.11/  MM nnm ►Step 3: Recalculate order frequency of most frequently ordered product. )] / ( [ 2     ii iii msS mDhC n =11.47 ►Step 4: n L =11.47/year, n M =11.47/2=5.74/year, n H =11.47/5=2.29/year. ►Step 1: Identify most frequently ordered product.  55.4  H m  24.1  28

29. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 3: Tailored Aggregation Result LiteproMedproHeavypro Demand per year120001200120 Order frequency11.47/year5.74/year2.29/year Optimal order size1,04620952 Annual holding cost$52,310$10,453$2620 Annual order cost = nS + n L s L + s M s M + n H s H =$65,380 Annual total cost = $130,763 Complete aggregation (Annual total cost) =$136,528 29

30. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 3: Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. ►Step 3: Recalculate order frequency of most frequently ordered product. ►Step 4: Identify ordering frequency of all products. ─ A fixed cost of (S+si) is allocated to each product i, and        )(2 frequency order frequently most The i ii i i sS DhC nMaxn ►Step 1: Identify most frequently ordered product. 30

31. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 3: Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. ►Step 3: Recalculate order frequency of most frequently ordered product. ►Step 4: Identify ordering frequency of all products. 2s i hC i D i  nini  mimi  n / n i  mimi  mimi  ►Step 1: Identify most frequently ordered product. 31

32. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 3: Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. ►Step 3: Recalculate order frequency of most frequently ordered product. ►Step 4: Identify ordering frequency of all products. )] / ( [ 2     ii iii msS mDhC n ►Step 1: Identify most frequently ordered product. 32

33. ► No aggregation ► Complete aggregation ► Tailored aggregation Option 3: Tailored aggregation A heuristic that yields an ordering policy whose cost is close to optimal. ►Step 2: Identify frequency of other products as a multiple of the order frequency of the most frequently ordered product. ►Step 3: Recalculate order frequency of most frequently ordered product. ►Step 4: Identify ordering frequency of all products. n i =n/m i ►Step 1: Identify most frequently ordered product. 33

34. Impact of Product Specific Order Cost Product specific order cost =$1,000 Product specific order cost =$3,000 No aggregation$155,140$183,564 Complete aggregation$136,528$186,097 Tailored aggregation$130,763$165,233 S s ? ? 34

35. Lessons From Aggregation 5Aggregation allows firm to lower lot size without increasing cost 5 Tailored aggregation is effective if product specific fixed cost is large fraction of joint fixed cost 5 Complete aggregation is effective if product specific fixed cost is a small fraction of joint fixed cost 35

36. ►Economies of scale ►Stochastic variability of supply and demand Why hold inventory? 》 Batch size and cycle time 》 Quantity discounts 》 Short term discounts / Trade promotions 36

37. 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 > All units > Marginal unit 》 Based on the quantity ordered in a single lot 37

38. All Unit Quantity Discounts C0C0 C1C1 C2C2 q1q1 q2q2 q3q3 Average Cost per Unit Quantity Purchased Total Material Cost Order Quantity q1q1 q2q2 q3q3 If an order that is at least as large as q i but smaller than q i+1 is placed, then each unit is obtained at the cost of C i. 38

39. ►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 hC i qiqi S D TC i  2 qiqi DC i+1 hC i S D TC i  q i+1 2 Order Quantity Total Cost Lowest cost in the range EOQ i qiqi q i+1 Total Cost Lowest cost in the range EOQ i qiqi q i+1 Order Quantity Total Cost Lowest cost in the range EOQ i qiqi q i+1 Order Quantity Total Cost Lowest cost in the range EOQ i qiqi q i+1 39

40. 5Assume the all unit quantity discounts 5Based on the all unit quantity discounts, we have 5If i = 0, evaluate Q 0 as Since Q 0 > q 1, we set the lost size at q 1 =5,000 and the total cost Example Order QuantityUnit Price 0-5,000$ 3.00 5,000-10,000$ 2.96 10,000 or more$ 2.92 D = 120,000/ year S = $100/lot h = 0.2 q 0 =0, q 1 =5,000, q 2 =10,000 C 0 =$3.00, C 1 =$2,96, C 2 =$2.92 hC 0 DS Q0Q0 2  = 6,324 C0C0 C1C1 C2C2 q1q1 q2q2 q3q3 Average Cost per Unit Quantity Purchased 40

41. Evaluate EOQ for All Unit Quantity Discounts hC i DS QiQi 2  5Evaluate 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 5Choose the lot size that minimizes the total cost over all price ranges. DC i+1 hC i S D TC i  q i+1 2 DC i hC i qiqi S D TC i  2 qiqi DC i hC i QiQi S D TC i  2 QiQi 41

42. DC 1 hC 1 q1q1 S D TC 0  2 q1q1 5Assume the all unit quantity discounts 5Based on the all unit quantity discounts, we have 5If i = 0, evaluate Q 0 as Since Q 0 > q 1, we set the lost size at q 1 =5,000 and the total cost Example Order QuantityUnit Price 0-5,000$ 3.00 5,000-10,000$ 2.96 10,000 or more$ 2.92 D = 120,000/ year S = $100/lot h = 0.2 q 0 =0, q 1 =5,000, q 2 =10,000 C 0 =$3.00, C 1 =$2,96, C 2 =$2.92 hC 0 DS Q0Q0 2  = 6,324 = $359,080 42

43. 5Assume the all unit quantity discounts 5Based on the all unit quantity discounts, we have 5If i = 0, evaluate Q 0 as Since Q 0 > q 1, we set the lost size at q 1 =5,000 and the total cost Example Order QuantityUnit Price 0-5,000$ 3.00 5,000-10,000$ 2.96 10,000 or more$ 2.92 D = 120,000/ year S = $100/lot h = 0.2 q 0 =0, q 1 =5,000, q 2 =10,000 C 0 =$3.00, C 1 =$2,96, C 2 =$2.92 hC 0 DS Q0Q0 2  = 6,324 = $359,080 DC 1 hC 1 q1q1 S D TC 0  2 q1q1 43

44. All Unit Quantity Discounts Total Material Cost C0C0 C1C1 C2C2 q1q1 q2q2 q3q3 Average Cost per Unit Quantity Purchased Order Quantity q1q1 q2q2 q3q3 If an order is placed that is at least as large as q i but smaller than q i+1, then each unit is obtained at a cost of C i. 44

45. Example Order QuantityUnit Price 0-5,000$ 3.00 5,000-10,000$ 2.96 10,000 or more$ 2.92 D = 120,000/ year S = $100/lot h = 0.2 q 0 =0, q 1 =5,000, q 2 =10,000 C 0 =$3.00, C 1 =$2,96, C 2 =$2.92 hC 0 DS Q0Q0 2  = 6,324 = $359,080 5Assume the all unit quantity discounts 5Based on the all unit quantity discounts, we have 5If i = 0, evaluate Q 0 as Since Q 0 > q 1, we set the lost size at q 1 =5,000 and the total cost DC 1 hC 1 q1q1 S D TC 0  2 q1q1 45

46. Example - Continued 5For i = 1, we obtain Q 1 = 6,367 Since 5,000 < Q 1 <10,000, we set the lot size at Q 1 = 6,367. 5Observe that the lowest total cost is for i = 2. The optimal lot size = 10,000 (at the discount price of $2.92) 5For i = 2, we obtain Q 2 = 6,410 Since Q 2 < q 2, we set the lot size at q 2 =10,000. = $358,969 DC 1 hC 1 Q1Q1 S D TC 1  2 Q1Q1 46

47. DC 2 hC 2 q2q2 S D TC 2  2 q2q2 = $358,969 DC 1 hC 1 Q1Q1 S D TC 1  2 Q1Q1 Example - Continued 5For i = 1, we obtain Q 1 = 6,367 Since 5,000 < Q 1 <10,000, we set the lot size at Q 1 = 6,367. ►Observe that the lowest total cost is for i = 2. The optimal lot size = 10,000 (at the discount price of $2.92) 5For i = 2, we obtain Q 2 = 6,410 Since Q 2 < q 2, we set the lot size at q 2 =10,000. = $354,520 47

48. Example - Continued 5For i = 1, we obtain Q 1 = 6,367 Since 5,000 < Q 1 <10,000, we set the lot size at Q 1 = 6,367. ►Observe that the lowest total cost is for i = 2. The optimal lot size = 10,000 (at the discount price of $2.92) 5For i = 2, we obtain Q 2 = 6,410 Since Q 2 < q 2, we set the lot size at q 2 =10,000. = $354,520 = $358,969 DC 2 hC 2 q2q2 S D TC 2  2 q2q2 DC 1 hC 1 Q1Q1 S D TC 1  2 Q1Q1 48

49. Example - Continued 5For i = 1, we obtain Q 1 = 6,367 Since 5,000 < Q 1 <10,000, we set the lot size at Q 1 = 6,367. ►Observe that the lowest total cost is for i = 2. The optimal lot size = 10,000 (at the discount price of $2.92) 5For i = 2, we obtain Q 2 = 6,410 Since Q 2 < q 2, we set the lot size at q 2 =10,000. DC 1 hC 1 Q1Q1 S D TC 1  2 Q1Q1 = $358,969 = $354,520 DC 2 hC 2 q2q2 S D TC 2  2 q2q2 49

50. The Impact of All Unit Discounts on Supply Chain ►In the above example ►If the fixed ordering cost S = $4, ►All unit quantity discounts encourage retailers to increase the size of their lots. ►This also increases cycle inventory and average flow time. 》 The optimal order size = 6,324 when there is no discount. 》 The quantity discounts result in a higher order size = 10,000. 》 The optimal order size without discount = 1,265 》 The optimal order size with all unit discounts = 10,000 50

51. 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. 51

52. 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) + (Q/2) ‧ h ‧ [ V i +(Q-q i )C i ] / Q (Annual holding cost) + D ‧ [ V i +(Q-q i )C i ] / Q(Annual material cost) 52

53. 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) + (Q/2) ‧ h ‧ [ V i +(Q-q i )C i ] / Q (Annual holding cost) + D ‧ [ V i +(Q-q i )C i ] / Q(Annual material cost) 53

54. 5Assume the all unit quantity discounts 5q 0 =0, q 1 =5,000, q 2 =10,000 C 0 =$3.00, C 1 =$2,96, C 2 =$2.92 V 0 =0 ; V 1 =3(5,000-0)=$15,000 V 2 =3(5,000-0)+2.96(10,000-5,000)=$29,800 5If i = 0, evaluate Q 0 as Since Q 0 > q 1, we set the lost size at q 1 =5,000 and the total cost 2D(S+V 0 -q 0 C 0 ) hC 0 Q0=Q0= Example Order QuantityUnit Price 0-5,000$ 3.00 5,000-10,000$ 2.96 10,000 or more$ 2.92 D = 120,000/ year S = $100/lot h = 0.2 = 6,324 $363,900 D S D TC 0 +[ V 1 +(q 1 -q 1 )C 1 ] + [ V 1 +(q 1 -q 1 )C 1 ]=  q1q1 q1q1 h 2 54

55. 5For i = 1, evaluate Since Q 1 > q 2, we evaluate the cost of ordering q 2 =10,000 5For i = 2, evaluate 5 Optimal order size = 16,961 Example - Continued 2D(S+V1-q1C1)2D(S+V1-q1C1) hC 1 Q1=Q1= = 11,028 $361,780 2D(S+V 2 -q 2 C 2 ) hC 2 Q2=Q2= = 16,961 $360,365 D S D TC 2 +[ V 2 +(Q 2 -q 2 )C 2 ] + [ V 2 +(Q 2 -q 2 )C 2 ]=  Q2Q2 Q2Q2 h 2 D STC 1 +[ V 2 +(q 2 -q 2 )C 2 ] + [ V 2 +(q 2 -q 2 )C 2 ]=  q2q2 h 2 D q2q2 55

56. 頁碼作品授權條件作者 / 來源 12 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 12, 14 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 12, 14 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 12 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 12 CoolCLIPS 。本作品轉載自 CoolCLIPS 網站 （ http://dir.coolclips.com/Popular/World_of_Industry/Food/Shopping_cart_full_of_ groceries_vc012266.html ），瀏覽日期 2011/12/28 。依據著作權法第 46 、 52 、 65 條合理使用。 http://dir.coolclips.com/Popular/World_of_Industry/Food/Shopping_cart_full_of_ groceries_vc012266.html 12 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 14 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 56

57. 頁碼作品授權條件作者 / 來源 15, 27 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 15 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 15 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 57