managing economies of scale in the supply chain: cycle inventory

Managing Economies of Scale in the Supply Chain: Cycle Inventory

Fall, 2014

Supply Chain Management:Strategy, Planning, and Operation

Chapter 10

Byung-Hyun Ha

2

Contents

Introduction

Economies of scale to exploit fixed costs

Economies of scale to exploit quantity discount

Short-term discounting: trade promotions

Managing multiechelon cycle inventory

3

Cycle inventory

Notation D: demand per unit time Q: quantity in a lot or batch size (order quantity)

Cycle inventory management (basic) Determining optimal order quantity Q* that minimizes total

inventory cost, with demand D given

Introduction

inventorylevel

time

4

Basic analysis of cycle Average inventory level (cycle inventory) = Q/2 Average flow time = Q/2D

Little’s law: (arrival rate) = (avg. number in system)/(avg. flow time)

Example• D = 2 units/day, Q = 8 units

• Average inventory level

• (7 + 5 + 3 + 1)/4 = 4 = Q/2

• Average flow time

• (0.25 + 0.75 + 1.25 + 1.75 + 2.25 + 2.75 + 3.25 + 3.75)/8 = 2 = Q/2D

Introduction

5

Introduction

Costs that influence total cost by order quantity C: (unit) material cost ($/unit)

• Average price paid per unit purchased

Quantity discount

H: holding cost ($/unit/year)• Cost of carrying one unit in inventory for a specific period of time

• Cost of capital, obsolescence, handling, occupancy, etc.

• H = hC

Related to average flow time

S: ordering cost ($/order)• Cost incurred per order

• Assuming fixed cost regardless of order quantity

• Cost of buyer time, transportation, receiving, etc.

10.2 Estimating cycle inventory-related costs in practice SKIP!

6

Assumptions Constant (stable) demand, fixed lead time, infinite time horizon

Cycle optimality regarding total cost Order arrival at zero inventory level is optimal. Identical order quantities are optimal.

Introduction

?

?

7

Introduction

Determining optimal order quantity Q* Economy of scale vs. diseconomy of scale, or Tradeoff between total fixed cost and total variable cost

Q1

Q2

?

D

D

8

Economies of Scale to Exploit Fixed Costs

Lot sizing for a single product Economic order quantity (EOQ) Economic production quantity (EPQ)

• Production lot sizing

Lot sizing for multiple products Aggregating multiple products in a single order Lot sizing with multiple products or customers

9

Economic Order Quantity (EOQ)

Assumptions Same price regardless of order quantity

Input D: demand per unit time, C: unit material cost S: ordering cost, H = hC: holding cost

Decision Q: order quantity

• D/Q: average number of orders per unit time

• Q/D: order interval

• Q/2: average inventory level

Total inventory cost per unit time (TC)TO: total order costTH: total holding costTM: total material costhC

QS

Q

DTMTHTOTC

2)(

10


Total cost by order quantity Q

Optimal order quantity Q* that minimizes total cost

Opt. order frequency

Avg. flow time

TC

QQ*

hCQ

SQ

DTC

2

hCDSTC

hC

DSQ

2

2

*

*

S

DhC

Q

Dn

2*

DhC

S

D

Q

22

*

11


Robustness around optimal order quantity (KEY POINT) Using order quantity Q' = Q* instead of Q*

0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0

1/2( + 1/)

1.250 1.133 1.064 1.025 1.006 1.000 1.017 1.057 1.113 1.178 1.250

TC*

TC' = 1.25TC*

10.5 2

*1

2

1

2TChC

QS

Q

DCT

12


Robustness regarding input parameters Mistake by indentifying ordering cost S' = S instead of real S

• Misleading to

0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0

1/2( + 1/) 1.061 1.033 1.016 1.006 1.001 1.000 1.004 1.014 1.028 1.043 1.061

TC*

10.5 2

TC' = 1.061TC*

Robustness?

*ββ22

QhC

SD

hC

SDQ

*

β

1β

2

1TCCT

13


Sensitivity regarding demand (KEY POINT) Demand change from D to D1 = kD

Opt. order frequency

Avg. flow time

*1

*1

*1*1

222

222

TCkhCDSkhCkDSShCDTC

QkhC

DSk

hC

kDS

hC

SDQ

*1*1

1*1 2

nkS

hCD

Q

Dn

D

Q

khCD

S

D

Q

2

1

22

*

11

*1

14


Effect of reducing order quantity Using order quantity Q' = Q* instead of Q* (revisited)

Reducing flow time by reducing ordering cost (KEY POINT) Efforts on reducing S to S1 = S Hoping Q1

* = kQ*

How much should S be reduced? (What is ?)

= k2 (ordering cost must be reduced by a factor of k2)

**1*1 γ

2γ

γ22QkQ

hC

DS

hC

SD

hC

DSQ

*1

2

1

2TChC

QS

Q

DCT

15

Production of lot instead of ordering P: production per unit time

Total cost by production lot size Q

Optimal production quantity Q* When P goes to infinite, Q* goes to EOQ.

Economic Production Quantity (EPQ)

Q

x(P – D)

D

Q/P Q/D – Q/P = Q(1/D – 1/P)

1/(D/Q) = Q/D

hCQ

P

DS

Q

DTC

21

hCPD

DSQ

1

2*

16

Aggregating Products in a Single Order

Multiple products m products D: demand of each product S: ordering cost regardless of aggregation level All the other parameters across products are the same.

All-separate ordering

All-aggregate ordering

Impractical supposition (for analysis purpose)

hCDSTC

hC

DSQ

i

i

2

2

*

*

hCDSmSSTC

hC

DSmSSQ

2

2

*

*

hCDSmhCmDSASTC

hC

DSm

hC

mDSASQ

22

22

*

*

17

Lot Sizing with Multiple Products

Multiple products with different parameters m products Di, Ci, hi: demand, price, holding cost fraction of product i

S: ordering cost each time an order is placed• Independent of the variety of products

si: additional ordering cost incurred if product i is included in order

Ordering each products independently?

Ordering all products jointly Decision

• n: number of orders placed per unit time

• Qi = Di /n: order quantity of item i

Total cost and optimal number of orders

nChDnsSTCm

iiii

m

ii 2

11

m

ii

m

iiii sSChDn

11

* 2

18


Example 10-3 and 10-4 Input

• Common transportation cost, S = $4,000

• Holding cost fraction, h = 0.2

Ordering each products independently• ITC* = $155,140

Ordering jointly• n* = 9.75

• JTC* = $136,528

i LE22B LE19B LE19A

Di

Ci

si

12,000$500

$1,000

1,200$500

$1,000

120$500

$1,000

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000

160,000

180,000

200,000

LE22B LE19B LE19A

i LE22B LE19B LE19A

Qi*ni* T

Ci*

1,09511.0

$109,544

3463.5

$34,642

1101.1

$10,954

i LE22B LE19B LE19A

Qi* 1,230 123 12.3

19


How does joint ordering work? Reducing fixed cost by enjoying robustness around optimal orde

r quantity

Is joint ordering is always good? No!

Then, possible other approaches? Partially joint

• NP-hard problem (i.e., difficult)

Number of all possible ways

• http://en.wikipedia.org/wiki/Bell_number

A heuristic algorithm• Subsection: “Lots are ordered and delivered jointly for a selecte

d subset of the products”

• SKIP!

http://en.wikipedia.org/wiki/Bell_number

http://en.wikipedia.org/wiki/Bell_number

20

Exploiting Quantity Discount

Total cost with quantity discount

Types of quantity discount Lot size-based

• All unit quantity discount

• Marginal unit quantity discount

Volume-based

Decision making we consider Optimal response of a retailer Coordination of supply chain

TO: total ordering costTH: total holding costTM: total material cost

DChCQ

SQ

DTMTHTOTC

2

21

Pricing schedule Quantity break points: q0, q1, ..., qr , qr+1

• where q0 = 0 and qr+1 =

Unit cost Ci when qi Q qi+1, for i=0,...,r

• where C0 C1 Cr

It is possible that qiCi (qi + 1)Ci

All Unit Quantity Discount

averagecost

per unit

C0

C1

C2

Cr

...

q0 q1 q2 q3 qr

...

...

1

32

21

10

2

1

0 0

,

,

,

,

rrr qQq

qQq

qQq

qQq

C

C

C

C

C

22

Solution procedure1. Evaluate the optimal lot size for each Ci.

2. Determine lot size that minimizes the overall cost by the total cost of the following cases for each i.

• Case 1: qi Qi* qi+1 , Case 2: Qi* qi , Case 3: qi+1 Qi*


ii hC

DSQ

2*

23

Example 10-7 r = 2, D = 120,000/year S = $100/lot, h = 0.2 Q* = 10,000

335,000

340,000

345,000

350,000

355,000

360,000

365,000

370,000

375,000

380,000

385,000

390,000

0 2000 4000 6000 8000 10000 12000 14000 16000


i 0 1 2

qi

Ci

0$3.00

5,000$2.96

10,000 $2.92

24


Example 10-7 (cont’d) Sensitivity analysis

• Optimal order quantity Q* with regard to ordering cost

(no discount)C = $3

(discount)

(original) S = $100/lot 6,324 10,000

(reduced) S' = $4/lot 1,256 10,000

25

Marginal Unit Quantity Discount

Pricing schedule Quantity break points: q0, q1, ..., qr , qr+1

• where q0 = 0 and qr+1 =

Marginal unit cost Ci when qi Q qi+1, for i=0,...,r

• where C0 C1 Cr

Price of qi units

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

Ordering Q units Suppose qi Q qi+1 .

marginalcost

per unitC0

C1

C2

Cr

...

q0 q1 q2 q3 qr

...

...

iii

iii

CqQVQ

D

hCqQV

SQ

D

TMTHTOTC

2

26

Example 10-8 r = 2, D = 120,000/year S = $100/lot, h = 0.2 Q* = 16,961

350,000

355,000

360,000

365,000

370,000

375,000

380,000

385,000

390,000

395,000

0 4000 8000 12000 16000 20000 24000 28000


i 0 1 2

qi

Ci

Vi

0$3.00

$0

5,000$2.96

$15,000

10,000 $2.92

$29,800

27


Example 10-8 (cont’d) Sensitivity analysis

• Optimal order quantity Q* with regard to ordering cost

Higher inventory level (longer average flow time)

(no discount)C = $3

(discount)

(original) S = $100/lot 6,324 16,961

(reduced) S' = $4/lot 1,256 15,775

28

Why Quantity Discount?

1. Improve coordination to increase total supply chain profit Each stage’s independent decision making for its own profit

• Hard to maximize supply chain profit (i.e., hard to coordinate)

How can a manufacturer control a myopic retailer?• Quantity discounts for commodity products

• Quantity discounts for products for which firm has market power

2. Extraction of surplus through price discrimination Revenue management (Ch. 15)

Other factors such as marketing that motivates sellers Munson and Rosenblatt (1998)

Manufacturer(supplier)

Retailer customers

supply chain

29

Coordination for Total Supply Chain Profit

Quantity discounts for commodity products Assumptions

• Fixed price and stable demand fixed total revenue

Max. profit min. total cost

Example case• Two stages with a manufacture (supplier) and a retailer


Retailer customers

D = 120,000SR = 100hR = 0.2CR = 3

SS = 250hS = 0.2CS = 2

30


Quantity discounts for commodity products (cont’d) (a) No discount

• Retailer’s (local) optimal order quantity ( supply chain’s decision)

• Q(a) = (2120,000100/0.23)1/2 = 6,325

• Total cost (without material cost)

• TC0(a) = TCS

(a) + TCR(a) = $6,008 + $3,795 = $9,803

() Minimum total cost, TC*, regarding supply chain (coordination)• Q* = 9,165

• TC0* = TCS* + TCR* = $5,106 + $4,059 = $9,165

• Dilemma?

• Manufacturer saving by $902, but retailer cost increase by $264

• How to coordinate (decision maker is the retailer)?

RRSSRS

RRRSSSRS

2

22

ChChQ

SSQ

D

ChQ

SQ

DCh

QS

Q

DTCTCTC

165,9

2

RRSS

RS*

ChCh

SSDQ

31


Quantity discounts for commodity products (cont’d) (b) Lot size-based quantity discount offering by manufacturer

• Price schedule of CR

• q1 = 9,165, C0 = $3, C1 = $2.9978

• Retailer’s (local) optimal order quantity (considering material cost)

• Q(b) = 9,165

• Total cost (without material cost)

• TC0(b) = TCS

(b) + TCR(b) = $5,106 + $4,057 = $9,163

• Savings (compared to no discount)

• Manufacturer: $902

• Retailer: $264 (material cost) – $262 (inventory cost) = $2

KEY POINT• For commodity products for which price is set by the market, manuf

acturers with large fixed cost per lot can use lot size-based quantity discounts to maximize total supply chain profit.

• Lot size-based discount, however, increase cycle inventory in the supply chain.

32


Quantity discounts for commodity products (cont’d) (c) Other approach: setup cost reduction by manufacturer

• Retailer’s (local) optimal order quantity

• Q(c) = Q(a) = 6,325

• Total cost (without material cost): no need to discount!

• TC0(c) = TCS

(c) + TCR(c) = $3,162 + $3,795 = $6,957

Same with optimal supply chain cost when material cost is considered

Expanding scope of strategic fit• Operations and marketing departments should be cooperate!


Retailer customers

D = 120,000SR = 100hR = 0.2CR = 3

S'S = 100hS = 0.2CS = 2

33


Quantity discounts for products with market power Assumption

• Manufacturer’s cost, CS = $2

• Customer demand depending on price, p, set by retailer

• D = 360,000 – 60,000p

Profit depends on price.


Retailer customers

D = 360,000 – 60,000p

CR = ?CS = 2 p = ?

34


Quantity discounts for products with market power (cont’d) (a) No coordination (deciding independently)

• Manufacturer’s decision on CR

• Expected retailer’s profit, ProfR

» ProfR = (p – CR)(360 – 60p)

• Retailer’s optimal price setting (behavior) when CR is given

» p = 3 + 0.5CR

• Demand by p (supplier’s order quantity)

» D = 360 – 60p = 180 – 30CR

• Expected manufacturer’s profit, ProfS

» ProfS = (CR – CS)(180 – 30CR)

CR(a) that maximizes ProfS (manufacturer’s decision)

» CR(a) = $4

• Retailer’s decision on p(a) with given CR(a)

• p(a) = $5 (D(a) = 360,000 – 60,000p(a) = 60,000)

• Supply chain profit, Prof0(a)

• Prof0(a) = ProfS

(a) + ProfR(a) = $120,000 + $60,000 = $180,000

35


Quantity discounts for products with market power (cont’d) () Coordinating supply chain

• Optimal supply chain profit, Prof0*

• Prof0 = (p – CS)(360 – 60p)

• p* = $4

• D* = 120,000

• Prof0* = $240,000

Double marginalization problem (local optimization)

• But how to coordinate?

• i.e., ProfS* = ?, ProfR* = ?

36


Quantity discounts for products with market power (cont’d) Two pricing schemes that can be used by manufacturer

• (b) Two-part tariff

• Up-front fee $180,000 (fixed) + material cost $2/unit (variable)

• Retailer’s decision

» ProfR = (p – CR)(360 – 60p) – 180,000

» p(b) = 3 + 0.5CR = $4

• Prof0(b) = Prof S

(b) + ProfR(b) = $180,000 + $60,000 = $240,000

Retailer’s side: larger volume more discount

• (c) Volume-based quantity discount

• Pricing schedule of CR

» q1 = 120,000, C0 = $4, C1 = $3.5

• p(c) = $4

• Prof0(c) = ProfS

(c) + ProfR(c) = $180,000 + $60,000 = $240,000

37


Quantity discounts for products with market power (cont’d) KEY POINT

• For products for which the firm has market power, two-part tariffs or volume-based quantity discounts can be used to achieve coordination in the supply chain and maximizing supply chain profits.

• For those products, lot size-based discounts cannot coordinate the supply chain even in the presence of inventory cost.

• In such a setting, either a two-part tariff or a volume-based quantity discount, with the supplier passing on some of its fixed cost to the retailer, is needed for the supply chain to be coordinated and maximize profits.

Lot size-based vs. volume-based discount Lot size-based: raising inventory level suitable for supplier’s

high setup cost

Hockey stick phenomenon & rolling horizon-based discount

38

Short-term Discounting: Trade Promotion

Trade promotion by manufacturers Induce retailers to use price discount, displays, or advertising to

spur sales. Shift inventory from manufactures to retailers and customers. Defend a brand against competition.

Retailer’s reaction? 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 during the promotion period to exploit the temporary reduction in price.

• Forward buy demand variability increase inventory & flow time increase supply chain profit decrease

39


Analysis Determining order quantity with discount Qd

• Unit cost discounted by d (C' = C – d)

Assumptions• Discount is offered only once.

• Customer demand remains unchanged.

• Retailer takes no action to influence customer demand.

Q*

Qd

Qd/D 1 – Qd/D

40


Analysis (cont’d) Optimal order quantity without discount Q* = (2DS/hC)1/2

Optimal total cost without discount TC* = CD + (2DShC)1/2

Total cost with Qd

Example 10-9 C = $3 Q* = 6,324 d = $0.15 Qd* = 38,236 (forward buy: 31,912 500%)

KEY POINT Trade promotions lead to a significant increase in lot size and cy

cle inventory, which results in reduced supply chain profits unless the trade promotion reduces demand fluctuation.

dC

CQ

dCh

dDQ

D

QTCQdC

D

QdCh

QSTC

d

dd

dd

**

* 12

41


Retailer’s action of passing discount to customers Example 10-10 Assumptions

• Customer demand: D = 300,000 – 60,000p

• Normal price: CR = $3

• Ignoring all inventory-related cost

Analysis• Retailer’s profit, ProfR

• ProfR = (p – CR)(300 – 60p)

• Retailer’s optimal price setting with regard to CR

• p = 2.5 + 0.5CR

(a) No discount: CR = $3

• p(a) = $4, D(a) = 60,000

(b) Discount: C'R = $2.85

• p(b) = $3.925, D(b) = 64,500 p(a) – p(b) = 0.075 < 0.15 = CR – C'R

42


Retailers’ response to short-term discount Insignificant efforts on trade promotion, but High forward buying

• Not only by retailers but also by end customers

• Loss to total revenue because most inventory could be provided with discounted price

KEY POINT Trade promotions often lead to increase of cycle inventory in

supply chain without a significant increase in customer demand.

43


Some implications Motivation for every day low price (EDLP) Suitable to

• high elasticity goods with high holding cost

• e.g., paper goods

• strong brands than weaker brand (Blattberg & Neslin, 1990)

Competitive reasons Sometimes bad consequence for all competitors

Discount by not sell-in but sell-through• Scanner-based promotion

44

Managing Multiechelon Cycle Inventory

Configuration Multiple stages and many players at each stage

General policy -- synchronization Integer multiple order frequency or order interval Cross-docking

(Skip!)

managing economies of scale in the supply chain: cycle inventory

Documents

order intervalq2

order quantityinputd

order quantitydq

order frequencyavg

total order costth

total fixed cost

identical order quantities

unit material cost s