Discussion

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11-‹#›

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Cycle inventory = lot size

2 = Q 2

Average flow time = average inventory average flow rate

= cycle inventory

demand = Q

2D

WACC = E

D+ E (Rf + β ×MRP)+

D D+ E

Rb(1– t)

Pretax WACC = after-tax WACC / (1– t)

Annual material cost =CD

Number of orders per year = D Q

Annual ordering cost = D Q ⎛

⎝ ⎜

⎞

⎠ ⎟S

Annual holding cost = Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟H =

Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟hC

Total annual cost, TC =CD+ D Q ⎛

⎝ ⎜

⎞

⎠ ⎟S +

Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟hC

Chapter 11 • Managing Economies of Scale in a Supply Chain: Cycle Inventory 277

The purchasing manager makes the lot-sizing decision to minimize the total cost the store incurs. He or she must consider three costs when deciding on the lot size:

• Annual material cost • Annual ordering cost • Annual holding cost

Because purchase price is independent of lot size, we have

The number of orders must suffice to meet the annual demand D. Given a lot size of Q, we thus have

(11.3)

Because an order cost of S is incurred for each order placed, we infer that

(11.4)

Given a lot size of Q, we have an average inventory of Q/2. The annual holding cost is thus the cost of holding Q/2 units in inventory for one year and is given as

The total annual cost, TC, is the sum of all three costs and is given as

Figure 11-2 shows the variation in different costs as the lot size is changed. Observe that the annual holding cost increases with an increase in lot size. In contrast, the annual ordering cost declines with an increase in lot size. The material cost is independent of lot size because we have assumed the price to be fixed. The total annual cost thus first declines and then increases with an increase in lot size.

From the perspective of the manager at Best Buy, the optimal lot size is one that minimizes the total cost to Best Buy. It is obtained by taking the first derivative of the total cost with respect to Q and setting it equal to 0 (see Appendix 11A at the end of this chapter). The optimal lot size

Total annual cost, TC = CD + aD Q bS + aQ

2 bhC

Annual holding cost = aQ 2 bH = aQ

2 bhC

Annual ordering cost = aD Q bS

Number of orders per year = D Q

Annual material cost = CD

Cost

Total Cost

Holding Cost

Ordering Cost

Material Cost

Lot Size FIGURE 11-2 Effect of Lot Size on Costs at Best Buy

M11_CHOP3952_05_SE_C11.QXD 11/15/11 7:39 PM Page 277

Optimal lot size, Q* = 2DS hC

n* = D Q *

= DhC 2S

Optimal order size = Q* = 2 ×12,000 × 4,000

0.2 × 500 = 980

Cycle inventory = Q * 2

= 980

2 = 490

Number of orders per year = D Q *

= 12.24

Annual ordering and holding cost = D Q * S +

Q * 2

⎛

⎝ ⎜

⎞

⎠ ⎟hC = 97,980

Average flow time = Q * 2D

= 490

12,000 = 0.041= 0.49 month

Annual inventory-related costs = D Q * S +

Q * 2

⎛

⎝ ⎜

⎞

⎠ ⎟hC = 250,000

S = hC(Q*)2

2D = 0.2×500×2002

2×12,000 =166.7

QP = 2DS

(1– D / P)hC

D QP ⎛

⎝ ⎜

⎞

⎠ ⎟S

(1– D / P) QP

2

⎛

⎝ ⎜

⎞

⎠ ⎟hC

Annual holding cost = DLhCL

2n + DMhCM

2n + DHhCH

2n

S* = S + sL + sM + sH

Annual order cost = S * n

Total annual cost = DLhCL

2n + DMhCM

2n + DHhCH

2n +S * n

n* = DLhCL + DMhCM + DHhCH

2S *

n* = DihCii=1

k ∑ 2S *

S* = S + sA + sB + sC = $7,000 per order

n* = 12,000×100+1,200×100+120×100

2×7,000 = 9.75

S* = S + s1 + s2 + s3 + s4 = $900 per order

n* = D1hC1i=1

4 ∑ 2S *

= 4×10,000×0.2×50

2×900 =14.91

Annual order cost = 14.91× 900

4 = $3,354

= hCiQ 2

= 0.2×50× 671 2

= $3,355

ni = hCiDi 2(S + si)

ni = hCiDi 2si

mi = n /ni ⎡ ⎢⎢

⎤ ⎥⎥

n = hCimiDi=1

l ∑

2 S + si /mii=1 l

∑( )

TC = nS + nisi i=1

l

∑ + Di 2ni

⎛

⎝ ⎜

⎞

⎠ ⎟hC1

i−1

l

∑

nL = hCLDL 2(S + sL)

=11.0

nM = hCMDM 2(S + sM )

= 3.5

nL = hCHDH 2(S + sH)

=1.1

n =11.0

nM = hCMDM

2sM = 7.7 and nH =

hCHDH 2sH

= 2.4

mM = n

nM

⎡

⎢

⎢ ⎢

⎤

⎥

⎥ ⎥ =

11.0 7.7

⎡

⎢ ⎢

⎤

⎥ ⎥= 2 and mH =

n

nH

⎡

⎢

⎢ ⎢

⎤

⎥

⎥ ⎥ =

11.0 2.4

⎡

⎢ ⎢

⎤

⎥ ⎥= 5

n =11.47

nL =11.47/ yr

nM =11.47/2= 5.74/ yr

nH =11.47/5 = 2.29/ yr

nS +nLsL +nMsM +nHsH = $65,383.5

Qi = 2DS hCi

qi ≤ Qi < qi+1

Qi < qi

Qi ≥ qi+1

Total annual cost, TCi = D Qi

*

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟S +

Qi *

2

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟hCi + DCi

C* = 1 D DCr +

DS qr

+ h 2 qrCr – 2hDSCr

⎛

⎝ ⎜

⎞

⎠ ⎟

Q0 = 2DS hC0

= 6,324; Q1 = 2DS hC1

= 6,367; Q2 = 2DS hC2

= 6,410

Q1 * = Q1 = 6,367; Q2

* = q2 = 10,000

TC1 = D Q1

*

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟S +

Q1 *

2

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟hC1 + DC1 = $358,969; TC2 = $354,520

Q2 * =10,000

Vi =C0(q1 –q0)+C1(q2 –q1)+...+Ci–1(qi –qi–1)

Annual order cost = D Q ⎛

⎝ ⎜

⎞

⎠ ⎟S

Annual holding cost = Vi + (Q – qi)Ci⎡⎣ ⎤⎦h / 2

Annual materials cost = D Q Vi + (Q – qi)Ci⎡⎣ ⎤⎦

= D Q ⎛

⎝ ⎜

⎞

⎠ ⎟S + Vi +(Q –qi)Ci⎡⎣ ⎤⎦h / 2

+ D Q Vi +(Q –qi)Ci⎡⎣ ⎤⎦

Optimal lot size for Ci is Qi = 2D(S +Vi – qiCi)

hCi

If qi ≤ Qi ≤ qi+1 then set Qi * = Qi

If Qi < qi then set Qi * = qi

If Qi > qi+1 then set Qi * = qi+1

TCi = D Qi *

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟S + Vi +(Qi

* –qi)Ci ⎡ ⎣

⎤ ⎦h / 2+

D Qi * Vi +(Qi

* –qi)Ci ⎡ ⎣

⎤ ⎦

V0 = 0; V1 = 3(5,000 – 0) = $15,000 V2 = 3(5,000 – 0) + 2.96(10,000 – 5,000) = $29,800

Q0 = 2D(S +V0 –q0C0)

hC0 = 6,324

Q1 = 2D(S +V1 –q1C1)

hC1 =11,028

Q2 = 2D(S +V2 –q2C2)

hC2 =16,961

Q0 * = q1 = 5,000 because Q0 = 6,324 > 5,000

Q1 * = q2 = 10,000; Q2 = Q2 = 16,961

TC0 = D Q0 *

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟S + V0 +(Q0

* –q0)C 0 ⎡ ⎣

⎤ ⎦h / 2+

D Q0 * V0 +(Q0

* –q0)C0 ⎡ ⎣

⎤ ⎦= $363,900

TC2 = D Q2 *

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟S + V2 +(Q2

* –q2)C 2 ⎡ ⎣

⎤ ⎦h / 2+

D Q2 * V2 +(Q2

* –q2)C2 ⎡ ⎣

⎤ ⎦= $360,365

TC1 = D Q1 *

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟S + V1 +(Q1

* –q1)C1 ⎡ ⎣

⎤ ⎦h / 2+

D Q1 * V1 +(Q1

* –q1)C1 ⎡ ⎣

⎤ ⎦= $361,780

QR = 2DSR hRCR

= 2×120,000×100

0.2×3 = 6,324

Annual cost for DO = D QR

⎛

⎝ ⎜

⎞

⎠ ⎟SR +

QR 2

⎛

⎝ ⎜

⎞

⎠ ⎟hRCR = $3,795

Annual cost for manufacturer = D QR

⎛

⎝ ⎜

⎞

⎠ ⎟SM +

QR 2

⎛

⎝ ⎜

⎞

⎠ ⎟hMCM = $6,009

= D Q ⎛

⎝ ⎜

⎞

⎠ ⎟SR +

Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟hRCR +

D Q ⎛

⎝ ⎜

⎞

⎠ ⎟SM +

Q 2

⎛

⎝ ⎜

⎞

⎠ ⎟hMCM

Q* = 2D(SR +SM ) hRCR +hMCM

= 9,165

Annual cost for DO = D Q * ⎛

⎝ ⎜

⎞

⎠ ⎟SR +

Q * 2

⎛

⎝ ⎜

⎞

⎠ ⎟hRCR = $4,059

Annual cost for manufacturer = D Q * ⎛

⎝ ⎜

⎞

⎠ ⎟SM +

Q * 2

⎛

⎝ ⎜

⎞

⎠ ⎟hMCM = $5,106

ProfR = (p–CR)(360,000–60,000p) ProfM = (CR –CM )(360,000–60,000p)

p = 3+ CR 2

ProfM = (CR –CM ) 360,000–60,000 3+ CR 2

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜⎜

⎞

⎠ ⎟⎟

= (CR –2)(180,000–30,000CR)

p = 3+ CM 2

= 3+ 2 2 = $4

Qd = dD

(C –d)h + CQ * C –d

Forward buy = Qd – Q *

Qd = dD

(C –d)h + CQ * C –d

= 0.15×120,000

(3.00–0.15)×0.20 + 3×6,324 3.00–0.15

= 38,236

Chapter 11 • Managing Economies of Scale in a Supply Chain: Cycle Inventory 307

• For customers with a longer reorder interval than the supplier, make the customer’s reorder interval an integer multiple of the supplier’s interval and synchronize replenishment at the two stages to facilitate cross-docking. In other words, a supplier should cross-dock all orders from customers who reorder less frequently than the supplier.

• For customers with a shorter reorder interval than the supplier, make the supplier’s reorder interval an integer multiple of the customer’s interval and synchronize replenishment at the two stages to facilitate cross-docking. In other words, a supplier should cross-dock one out of every k shipments to a customer who orders more frequently than the supplier, where k is an integer.

• The relative frequency of reordering depends on the setup cost, holding cost, and demand at different parties.

Whereas the integer policies discussed above synchronize replenishment within the supply chain and decrease cycle inventories, they increase safety inventories , because of the lack of flexi- bility with the timing of a reorder, as discussed in Chapter 12. Thus, these polices make the most sense for supply chains in which cycle inventories are large and demand is relatively predictable.

11.7 SUMMARY OF LEARNING OBJECTIVES

1. Balance the appropriate costs to choose the optimal lot size and cycle inventory in a supply chain. Cycle inventory generally equals half the lot size. Therefore, as the lot size grows, so does the cycle inventory. In deciding on the optimal amount of cycle inventory, the supply chain

Group of Customers

Stage 1

Stage 2

Stage 3

Stage 4

Stage 5

FIGURE 11-8 A Multiechelon Distribution Supply Chain

Key Point

Integer replenishment policies can be synchronized in multiechelon supply chains to keep cycle inventory and order costs low. Under such policies, the reorder interval at any stage is an integer multiple of a base reorder interval. Synchronized integer replenishment policies facilitate a high level of cross-docking across the supply chain.

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