1. Managing Economies of Scale in a Supply Chain Cycle Inventory11

2. MARGINAL UNIT QUANTITY DISCOUNT
Marginal unit quantity discounts are also referred to as multiblock tariffs. In this case, the pricing schedule contains specified break points q0, q1, ... , qr. It is not the average cost of a unit but the marginal cost of a unit that decreases at a breakpoint (in contrast to the all unit discount scheme). If an order of size q is placed, the first q1 – q0 units are priced at C0, the next q2 – q1 are priced at C1 and so on.

3. MARGINAL UNIT QUANTITY DISCOUNT
The marginal cost per unit varies with the quantity purchased, as shown in Figure 11-4.
FIGURE 11-4

4. MARGINAL UNIT QUANTITY DISCOUNT
Faced with such a pricing schedule, the retailer's objective is to decide on a lot size that maximizes profits or, equivalently, minimizes material, order, and holding costs. The solution procedure evaluates the optimal lot size for each marginal price Ci (this forces a lot size between qi and qi + l) and then settles on the lot size that minimizes the overall cost.

5. MARGINAL UNIT QUANTITY DISCOUNT
For each value of i, 0 ≤ i ≤ r, let Vi be the cost of ordering qi units. Define V0 = 0 and Vi for 0 ≤ i ≤ r as follows: For each value of i, 0 ≤ i ≤ r – 1, consider an order of size Q in the range qi to qi + l units; that is, qi + l ≥ Q ≥ qi. The material cost of each order of size Q is given by Vi + ( Q – qi) Ci.

6. MARGINAL UNIT QUANTITY DISCOUNT
The various costs associated with such an order are as follows:

7. MARGINAL UNIT QUANTITY DISCOUNT
The total annual cost is the sum of the three costs and is given by The optimal lot size for this price range is obtained by taking the first derivative of the total cost with respect to the lot size and setting it equal to 0. This results in an optimal lot size for this price range of

8. MARGINAL UNIT QUANTITY DISCOUNT
Observe that the optimal lot size is obtained using a formula very much like the EOQ formula (Equation 11.5), except that the presence of the quantity discount has the effect of raising the fixed cost per order by Vi – qiCi (from S to S + Vi – qiCi).

9. MARGINAL UNIT QUANTITY DISCOUNT
There are three possible cases for Qi: 1. qi ≤ Qi ≤ qi + l 2. Qi < qi 3. Qi > qi + l

10. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount Let us return to DO from Example Assume that the manufacturer uses the following marginal unit discount pricing schedule:

11. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount This implies that if an order is placed for 7,000 bottles, the first 5,000 are at a unit cost of $3.00, with the remaining 2,000 at a unit cost of $2.96. Evaluate the number of bottles that DO should order in each lot.

12. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount In this case we have

13. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount For i = 0, evaluate Q0 (using Equation 11.14) as follows:
Because 6,324 > q 1 = 5,000, we evaluate the cost of ordering lots of q 1 = 5,000 (we do
not consider lots of 0). The total annual cost of ordering 5,000 bottles per lot is as follows
(set 0 = 5,000 and i = 1):

14. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount Because 6,324 > q1 = 5,000, we evaluate the cost of ordering lots of q1 = 5,000 (we do not consider lots of 0). The total annual cost of ordering 5,000 bottles per lot is as follows (set Q = 5,000 and i = 1):

15. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount For i = 1, evaluate Q1 using Equation as follows:
Because 11,028 > q2 = 10,000, we evaluate the cost of ordering lots of q2 = 10,000 (the
cost of ordering lots of 5,000 has already been evaluated). The total annual cost of ordering
10,000 bottles per lot is as follows (set 0 = 10,000 and i = 2):

16. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount Because 11,028 > q2 = 10,000, we evaluate the cost of ordering lots of q2 = 10,000 (the cost of ordering lots of 5,000 has already been evaluated). The total annual cost of ordering 10,000 bottles per lot is as follows (set Q = 10,000 and i = 2):

17. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount Because $361,780 < $363,900, it is less expensive to order in lots of 10,000 than in lots of 5,000. If the lot size is to be 10,000 units or less, we are better off ordering 10,000 units per lot. Now we investigate the cost of ordering in lots larger than 10,000 units; that is, i = 2. For i = 2, evaluate Q2 using Equation as follows:

18. MARGINAL UNIT QUANTITY DISCOUNT
Example 11-8: Marginal Unit Quantity Discount The total annual cost of ordering 16,961 bottles per lot is as follows (set Q = 16,961 and i = 2): DO minimizes its total cost by ordering in lots of 16,961 bottles. This is much larger than the optimal lot size of 6,324 in the case where the manufacturer does not offer any discount.
WHY QUANTITY DISCOUNTS?

19. MARGINAL UNIT QUANTITY DISCOUNT
If the fixed cost of ordering is $4, the optimal lot size for DO is 15,755 with the discount compared to a lot size of 1,265 without the discount. This discussion demonstrates that there can be significant order sizes and thus cycle inventory in the absence of any formal fixed ordering costs as long as quantity discounts are offered.
In many
supply chains, quantity discounts contribute more to cycle inventory than fixed ordering
costs. This forces us once again to question the value of quantity discounts in a supply
chain.

20. MARGINAL UNIT QUANTITY DISCOUNT
Thus, quantity discounts lead to a significant buildup of cycle inventory in a supply chain. In many supply chains, quantity discounts contribute more to cycle inventory than fixed ordering costs. This forces us once again to question the value of quantity discounts in a supply chain.

21. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
Best Buy sells three models of computers, the Litepro, the Medpro, and the Heavypro. Annual demands for the three products are DL = 12,000 for the Litepro, DM = 1,200 units for the Medpro, and DH = 120 units for the Heavypro. Each model costs Best Buy $500. A fixed transportation cost of $4,000 is incurred each time an order is delivered.
For each
model ordered and delivered on the same truck, an additional fixed cost of $1 ,000 is
incurred for receiving and storage. Best Buy incurs a holding cost of 20 percent. Evaluate
the lot sizes that the Best Buy manager should order if lots for each product are ordered
and delivered independently. Also evaluate the annual cost of such a policy.

22. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
For each model ordered and delivered on the same truck, an additional fixed cost of $1,000 is incurred for receiving and storage. Best Buy incurs a holding cost of 20 percent. Evaluate the lot sizes that the Best Buy manager should order if lots for each product are ordered and delivered independently. Also evaluate the annual cost of such a policy.

23. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
Demand, DL = 12,000/year, DM = 1,200/year, DH = 120/year
Common order cost, S = $4,000
Product-specific order cost, SL = $1,000, SM = $1,000, SH = $1,000
Holding cost, h = 0.2
Unit cost, CL = $500, CM = $500, CH = $500

24. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
Because each model is ordered and delivered independently, a separate truck delivers each model. Thus, a fixed ordering cost of $5,000 ($4,000 + $1 ,000) is incurred for each product delivery.

25. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
The optimal ordering policies and resulting costs for the three products (when the three products are ordered independently) are evaluated using the EOQ formula (Equation ) and are shown in Table 11-1.

26. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
Litepro
Medpro
Heavypro
Demand per year
12,000
1,200
120
Fixed cost/order
$5,000
Optimal order size
1,095
346
110
Cycle inventory
548
173 55
Annual holding cost
$54,772
$17,321
$5,477
Order frequency
11.0/year
3.5/year
1.1/year
Annual ordering cost
Average flow time
2.4 weeks
7.5 weeks
23.7 weeks
Annual cost
$109,544
$34,642
$10,954
TABLE 11-1

27. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
The Litepro model is ordered 11 times a year, the Medpro model is ordered 3.5 times a year, and the Heavypro model is ordered 1.1 times each year. The annual ordering and holding cost Best Buy incurs if the three models are ordered independently turns out to be $155,140.

28. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
Independent ordering ignores the opportunity to aggregate orders. Thus, the product managers at Best Buy could potentially lower costs by combining orders on a single truck. We next consider the scenario in which all three products are ordered and delivered each time an order is placed.

29. Example 11-3: Multiple Products with Lots Ordered and Delivered Independently
Discussion Problem W03-07:
We add Tinypro. Its demand, product specific cost and unit cost are 20,000 units, $1,500 and $300. The holding cost h = 20 percent is same. Work on the Excel Sheet “Example 11-3_add | Chapter 11- examples 3-5_addition.xlsx”. Fill in the blue cell (B21) and the purple cells (C21:F21) beside and the green cells (D23:F23) below "Tinypro".
LOT SIZING WITH MULTIPLE PRODUCTS OR CUSTOMERS

30. Example 11-4: Products Ordered and Delivered Jointly
In Example 11-4 we consider the case in which the product managers at Best Buy order all three models each time they place an order (see worksheet Example 11-4). Consider the Best Buy data in Example The three product managers have decided to aggregate and order all three models each time they place an order. Evaluate the optimal lot size for each model.

31. Example 11-4: Products Ordered and Delivered Jointly
Because all three models are included in each order, the combined order cost is
The optimal order frequency is obtained using Equation 11.7 and is given by

32. Example 11-4: Products Ordered and Delivered Jointly
Thus, if each model is to be included in every order and delivery, the product managers at
Best Buy should place 9.75 orders each year. In this case the ordering policies and costs
are as shown in Table 11-2.

33. Example 11-4: Products Ordered and Delivered Jointly
Litepro
Medpro
Heavypro
Demand per year (D)
12,000
1,200
120
Order frequency (n∗)
9.75/year
Optimal order size (D/n∗)
1,230
123
12.3
Cycle inventory
615
61.5
6.15
Annual holding cost
$61,512
$6,151
$615
Average flow time
2.67 weeks
TABLE 11-2

34. Example 11-4: Products Ordered and Delivered Jointly
Because 9.75 orders are placed each year and each order costs a total of $7,000, we have
The annual ordering and holding cost, across the three sizes, of the aforementioned policy is given by
Annual order cost = 9.75 x 7,000 = $68,250
Annual ordering
and holding cost = $61,512 + $6,151 + $615 + $68,250
= $136,528

35. Example 11-4: Products Ordered and Delivered Jointly
Observe that the product managers at Best Buy lower the annual cost from $155,140 to $136,528 by ordering all products jointly. This represents a decrease of about 12 percent.

36. Example 11-4: Products Ordered and Delivered Jointly
Discussion Problem W03-08:
We add Tinypro. Its demand, product specific cost and unit cost are 20,000 units, $1,500 and $300. The holding cost h = 20 percent is same. Work on the Excel Worksheet “Example 11-4_add | Chapter 11-examples 3-5_addition.xlsx”. Change the blue cell (B17) beside "Litepro". Fill in the grey cell (B22), the purple cells (C22,E22) beside and the green cells (D24:F24) below "Tinypro". (Do not use the old “Joint order cost” (D12) but the new “Joint order cost” (I12).)
LOT SIZING WITH MULTIPLE PRODUCTS OR CUSTOMERS

37. Example 11-4: Products Ordered and Delivered Jointly
Because all three models are included in each order, the combined order cost is
𝑆∗=𝑆+𝑠𝐿+𝑠𝑀+𝑠𝐻+𝑠𝑇=$8,500
The optimal order frequency is obtained using Equation 11.7 and is given by
𝑛∗= × × × ×60 2×8500

38. Example 11-5: Aggregation with Capacity Constraint
In Example 11.5, we consider optimal aggregation of orders or deliveries in the presence of capacity constraints. W.W. Grainger sources from hundreds of suppliers and is considering the aggregation of inbound shipments to lower costs.
Truckload shipping costs $500 per truck along with
$100 per pickup. Average annual demand from each supplier is 10,000 units. Each unit
costs $50 and Grainger incurs a holding cost of 20 percent. What is the optimal order frequency
and order size if Grainger decides to aggregate four suppliers per truck? What is
the optimal order size and frequency if each truck has a capacity of 2,500 units?

39. Example 11-5: Aggregation with Capacity Constraint
Truckload shipping costs $500 per truck along with $100 per pickup. Average annual demand from each supplier is 10,000 units. Each unit costs $50 and Grainger incurs a holding cost of 20 percent.

40. Example 11-5: Aggregation with Capacity Constraint
What is the optimal order frequency and order size if Grainger decides to aggregate four suppliers per truck? What is the optimal order size and frequency if each truck has a capacity of 2,500 units?

41. Example 11-5: Aggregation with Capacity Constraint
In this case, W.W. Grainger has the following inputs:
Demand per product, Di = 10,000
Holding cost, h = 0.2
Unit cost per product, Ci = $50
Common order cost, S = $500
Supplier-specific order cost, si = $100

42. Example 11-5: Aggregation with Capacity Constraint
The combined order cost from four suppliers is given by
From Equation 11.8, the optimal order frequency is

43. Example 11-5: Aggregation with Capacity Constraint
It is thus optimal for Grainger to order times per year. The annual ordering cost per
supplier is

44. Example 11-5: Aggregation with Capacity Constraint
The quantity ordered from each supplier is Q = 10,000/14.91 = 671 units per order. The
annual holding cost per supplier is
Annual holding cost per supplier

45. Example 11-5: Aggregation with Capacity Constraint
This policy, however, requires a total capacity per truck of 4 x 671 = 2,684 units. Given a truck capacity of 2,500 units, the order frequency must be increased to ensure that the order quantity from each supplier is 2,500/4 = 625.

46. Example 11-5: Aggregation with Capacity Constraint
Thus, W.W. Grainger should increase the order frequency to 10,000/625 = 16. This action will increase the annual order cost per supplier to $3,600 and decrease the annual holding cost per supplier to $3,125.

47. Example 11-5: Aggregation with Capacity Constraint
Discussion Problem W03-09:
Change the number of suppliers per truck in Cell B8 from 2 to 6 and see how the quantity order / supplier and the optimal order frequency changes in Cells B22, 23 for the capacity constrained case. Work on the Excel Worksheet “Example 11-5 | Chapter 11-examples 3-5_addition.xlsx”.
LOT SIZING WITH MULTIPLE PRODUCTS OR CUSTOMERS

48. Managing Uncertainty in a Supply Chain Safety Inventory12
Managing Uncertainty in a Supply Chain Safety Inventory

49. Evaluating Safety Inventory Given a Replenishment Policy
Example 12-1: Evaluating safety inventory given an inventory policy Assume that weekly demand for phones at B&M Computer World is normally distributed, with a mean of 2,500 and a standard deviation of 500. The manufacturer takes two weeks to fill an order placed by the B&M manager. The store manager currently orders 10,000 phones when the inventory on hand drops to 6,000.

50. Evaluating Safety Inventory Given a Replenishment Policy
Example 12-1: Evaluating safety inventory given an inventory policy Evaluate the safety inventory carried by B&M and the average inventory carried by B&M. Also evaluate the average time spent by a phone at B&M.

51. Evaluating Safety Inventory Given a Replenishment Policy
Average demand per week, D = 2,500 Standard deviation of weekly demand, sD = 500 Average lead time for replenishment, L = 2 weeks Reorder point, ROP = 6,000 Average lot size, Q = 10,000
Safety inventory, ss = ROP – DL = 6,000 – 5,000 = 1,000
Cycle inventory = Q/2 = 10,000/2 = 5,000

52. Evaluating Safety Inventory Given a Replenishment Policy
Average inventory = cycle inventory + safety inventory = 5, ,000 = 6,000 Average flow time = average inventory/throughput = 6,000/2,500 = 2.4 weeks

53. Evaluating Safety Inventory Given a Replenishment Policy
Discussion Problem W04-05: Evaluating safety inventory given an inventory policy Assume that weekly demand for phones at B&M Computer World is normally distributed, with a mean of 2,500 and a standard deviation of 500. The manufacturer takes two weeks to fill an order placed by the B&M manager. The store manager currently orders 10,000 phones when the inventory on hand drops to 6,000.

54. Evaluating Cycle Service Level Given a Replenishment Policy
We now illustrate this evaluation in Example (see worksheet Example 12-2). Example 12-2: Evaluating cycle service level given a replenishment policy Weekly demand for phones at B&M is normally distributed, with a mean of 2,500 and a standard deviation of 500. The replenishment lead time is two weeks.

55. Evaluating Cycle Service Level Given a Replenishment Policy
Example 12-2: Evaluating cycle service level given a replenishment policy Assume that the demand is independent from one week to the next. Evaluate the CSL resulting from a policy of ordering 10,000 phones when there are 6,000 phones in inventory. In this case, we have
Q = 10,000, ROP = 6,000, L = 2 weeks
D = 2,500/week, sD = 500

56. Evaluating Cycle Service Level Given a Replenishment Policy
Example 12-2: Evaluating cycle service level given a replenishment policy Observe that B&M runs the risk of stocking out during the two weeks between when an order is placed and when the replenishment arrives. Thus, whether a stockout occurs or not depends on the demand during the lead time of two weeks.

57. Evaluating Cycle Service Level Given a Replenishment Policy
Example 12-2: Evaluating cycle service level given a replenishment policy Because demand across time is independent, we use Equation 12.2 to obtain demand during the lead time to be normally distributed with a mean of DL and a standard deviation of 𝜎𝐿, where

58. Evaluating Cycle Service Level Given a Replenishment Policy
Example 12-2: Evaluating cycle service level given a replenishment policy Using Equation 11.4, the CSL is evaluated as
CSL = F(ROP, DL, sL) = NORMDIST(ROP, DL, sL, 1)
= NORMDIST(6,000, 5,000, 707, 1) = 0.92

59. Evaluating Cycle Service Level Given a Replenishment Policy
Example 12-2: Evaluating cycle service level given a replenishment policy A CSL of 0.92 implies that in 92 percent of the replenishment cycles, B&M supplies all demand from available inventory. In the remaining 8 percent of the cycles, stockouts occur and some demand is not satisfied because of the lack of inventory.

60. Evaluating Cycle Service Level Given a Replenishment Policy
Example 12-2: Evaluating cycle service level given a replenishment policy A CSL of 0.92 implies that in 92 percent of the replenishment cycles, B&M supplies all demand from available inventory. In the remaining 8 percent of the cycles, stockouts occur and some demand is not satisfied because of the lack of inventory.

61. Evaluating Cycle Service Level Given a Replenishment Policy
Discussion Problem W04-06: Evaluating cycle service level given a replenishment policy Weekly demand for phones at B&M is normally distributed, with a mean of 2,500 and a standard deviation of 500. The replenishment lead time is two weeks. Assume that the demand is independent from one week to the next. Evaluate the CSL resulting from a policy of ordering 10,000 phones when there are 6,000 phones in inventory.

62. EVALUATING SAFETY INVENTORY GIVEN DESIRED CYCLE SERVICE LEVEL
Example 12-3: Evaluating safety inventory given a desired cycle service level Weekly demand for Lego at a Wal-Mart store is normally distributed, with a mean of 2,500 boxes and a standard deviation of 500. The replenishment lead time is two weeks. Assuming a continuous-review replenishment policy, evaluate the safety inventory that the store should carry to achieve a CSL of 90 percent.

63. EVALUATING SAFETY INVENTORY GIVEN DESIRED CYCLE SERVICE LEVEL
Example 12-3: Evaluating safety inventory given a desired cycle service level
D = 2,500/week, sD = 500, CSL = 0.9, L = 2 weeks

64. EVALUATING SAFETY INVENTORY GIVEN DESIRED CYCLE SERVICE LEVEL
Discussion Problem W04-7: Evaluating safety inventory given a desired cycle service level Weekly demand for Lego at a Wal-Mart store is normally distributed, with a mean of 2,500 boxes and a standard deviation of 500. The replenishment lead time is two weeks. Assuming a continuous-review replenishment policy, evaluate the safety inventory that the store should carry to achieve a CSL of 90 percent.