Determining the Optimal Level of Product Availability Spring, 2014 Supply Chain Management:...

Determining the Optimal Level of Product Availability

Spring, 2014

Supply Chain Management:Strategy, Planning, and Operation

Chapter 12

Byung-Hyun Ha

2

Contents

Introduction

Factors affecting the optimal level of product availability

Managerial levers to improve supply chain profitability

Setting product availability for multiple products under capacity constraints

Desired cycle service level for continuously stocked items

Setting optimal levels of product availability in practice

3

Introduction

Product availability i.e., customer service level Affecting supply chain responsiveness Measurement: cycle service level, fill rate

Trade-off in high levels of product availability Increased responsiveness and higher revenues Increased inventory levels and higher costs

Related to profit objectives, strategy, competitiveness e.g., Nordstrom, power plants, supermarkets, online retailers

Optimization problem? What is the optimal level of fill rate or cycle service level that will

result in maximum supply chain profits?

4

Factors Affecting the Optimal Availability

Assumptions Seasonal (i.e., perishable) items (news-vendor model)

• Unit cost c, unit retail price p, salvage value s (s < c < p)

Probabilistic demand, D Single order

Expected profit when O units are ordered Discrete case

• Demand will be x with probability Pr(D=x)

Continuous case• Probability density function of D: f(x)

cOxpOxxOspxOEPOx

O

x

10

DD PrPr

cOxxfpOxxfxOspxOEPO

O

d)(d)(

0

5


Expected profit when O units are ordered (cont’d) Normally distributed demand D

= E(D), 2 = Var(D)

• where

• F(x) = Pr(D x).

• fS(x) is probability density function of a standard normal random variable.

OcpOfspOFOsp

cOOFpOOFsOOfOFsp

cOxxfpOxxfsOxxxfsp

cOxxfpOxxfxOspxOEP

O

OO

O

O

σμσμ

1σμσμ

d)(d)(d)(

d)(d)(

S

S

6


Expected profit when O units are ordered (cont’d) Example 12-1

• Normally distributed demand with = 350 and = 100

• p = $250, c = $100, s = $80

0

10000

20000

30000

40000

50000

60000

70000

0 100 200 300 400 500 600 700

EP(O)

7


Expected profit when O units are ordered (cont’d) Example 12-1'

• Normally distributed demand with = 350 and = 100

• p = $250, c' = $200, s = $80

0

5000

10000

15000

20000

25000

0 100 200 300 400 500 600 700

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Expected overstock by order quantity O

Expected understock by order quantity O

Example 12-1 (cont’d) Expected overstock and understock

Expected profit by EO(O)

O

x

xxOOEO0

DPr O

xxfxOOEO0

d

1Ox

xOxOEU DPr

O

xxfOxOEU d

)(d)(

d)(

d)(d)(

0

0

0

OEOspOcpxxfxOspOcp

cOpOxxfxOsxOp

cOxxfpOxxfxOspxOEP

O

O

O

O

0

100

200

300

400

0 100 200 300 400 500 600 700

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Optimal Cycle Service Level

Notation O*: Optimal order size that maximizes expected profit CSL* = F(O*) ; optimal cycle service level

Analysis

cOxxfpOxxfsOxxxfspOEPO

OO

d)(d)(d)(

00

cOFpOsF

cxxfpxxfs

cOfpOxxfpOfsOxxfsOfOspOEPO

O

O

O

O

)()(

d)(d)(

)(d)()(d)(d

d

1

0

0

**

** )()(

CSLsp

cpOF

cOFpOsF

01

10


Example 12-1 (cont’d) Normally distributed demand with = 350 and = 100 p = $250, c = $100, s = $80

CSL* = (p – c)/(p – s) = 150/170 = 88%

O* = + FS –1(0.88) = 350 + 1.18100 = 468

• where FS(x) = Pr(Z x) with a standard normal random variable Z.

Expected profit? Expected overstock and understock?

Example: discrete case Pr(D = Di) = pi

p = $125, c = $80, s = $20

CSL* = (p – c)/(p – s) = 45/105 = 43%

O* = 10,000 or 12,000?

i pi Di F(Di)

1 0.1 8,000 0.1

2 0.2 10,000 0.3

3 0.4 12,000 0.7

4 0.3 14,000 1.0

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Marginal costs Co = c – s ; cost of overstocking by one unit

Cu = p – c ; cost of understocking by one unit

Expected profit by using marginal costs


O

O

O

O

O

O

xxfOxCxxfxOCEC

xxfOxcpxxfxOscxxxfcp

cOxxfpOxxfxOspxOEP

d)(d)(

d)(d)(d)(

d)(d)(

u0ou

00

0

D

Ox

O

12

Alternative (marginal) analysis for CSL* Effect of purchasing extra unit (i.e., ordering O + 1 units)

• Marginal benefit: (1 – F(O))Cu

• Marginal cost: F(O)Co

Possible interpretation• 1 – F(O) = Pr(demand is larger than O)

1 – F(O) = Pr(additional unit will be sold)

• F(O) = Pr(demand is equal to or smaller than O)

F(O) = Pr(additional unit will not be sold)


Ox

1 – F(O)

F(O)

O

13

Alternative (marginal) analysis for CSL* (cont’d) Effect of purchasing extra unit (i.e., ordering O + 1 units)

• Marginal benefit: (1 – F(O))Cu

• Marginal cost: F(O)Co

• Marginal contribution: (1 – F(O))Cu – F(O)Co

Expected marginal contribution must be 0 at CSL* = F(O*)• (1 – CSL*)Cu = CSL*Co

Impact of marginal cost change e.g., Nordstorm and discount store


ouuo

o

uoou

u*

CCCC

C

CCsp

cp

CC

CCSL

1

111

1

1

14


Example: discrete case (cont’d) p = $125, c = $80, s = $20, Co = $60, Cu = $45

CSL* = Cu /(Cu + Co) = 45/105 = 43%

O* = 12,000

k pk Dk 1 – F(Dk)Marginal

benefit F(Dk)Marginal

costMarginal

contribution

1 0.1 8,000 0.9 40.5 0.1 6.0 34.5

2 0.2 10,000 0.7 31.5 0.3 18.0 13.5

3 0.4 12,000 0.3 13.5 0.7 42.0 –28.5

4 0.3 14,000 0.0 0.0 1.0 60.0 –60.0

n

kiiki

k

iiikk pDDCpDDCECDEP

11uou )(D

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One-time Order with Quantity Discount

Assumptions Discounted cost cd when O K

Solution procedure Using c, p, and s, evaluate CSL* and O*. Using cd, p, and s, evaluate CSLd* and Od*.

• Revise Od* regarding K.

Select O* or Od* that maximizes the expected profit.

Example 12-3 p = 200, c = 50, s = 0 = 150, = 40

CSL* = 0.75, O* = 177 K = 200, cd = 45

CSLd* = 0.78, Od* = 180

Order by 200.0

5000

10000

15000

20000

25000

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

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Managerial Levers to Improve Profitability

Increasing salvage value CSL* , Profitability e.g., Sport Obermeyer

Decreasing stockout e.g., McMaster-Carr and W.W. Grainger

cOOEOsOEUpEp

cOxxfxOsxxfOxpxxxfp

cOxxfpOxxfxOspxOEP

O

O

O

O

)()(

d)(d)(d)(

d)(d)(

D

00

0

sp

cp

CC

CCSL

ou

u*

17


Reducing demand uncertainty (“Improved forecast” in our textbook NO!) Example 12-6

• Normally distributed demand D

• E(D) = 350, Var(D) = 2

• p = $250, c = $100, s = $80

O*Expected

OverstockExpected U

nderstockExpected

Profit

150 526 186.7 8.6 $47,469

120 491 149.3 6.9 $48,476

90 456 112.0 5.2 $49,482

60 420 74.7 3.5 $50,488

30 385 37.3 1.7 $51,494

0 350 0.0 0.0 $52,500

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Quick response Reducing replenishment lead times

• Possibly being able to cope with demand change

Setting• p = $150, c = $40, s = $30, CSL* = 0.92

• 14 weeks in season

• Normally distributed weekly demand with mean 20 and S.D. 15

Ordering policies1. Single order for covering entire season’s demand

2. Two orders at beginning of season and at beginning of 8th week

Scenarios1. Unchanged demand

2. Reduced demand uncertainty from 8th week

• S.D. of weekly demand changes to 3

19


Quick response (cont’d) Unchanged demand

• Single order

• O* = 358

• E.P. = $29,767, E.O. = 79.8

• Two orders

• O1* = 195, EO1(O1*) = 56.4, E(O2*) = 195 – 56.4 = 138.6

• E.P. = $14,670 + $1056.4 + $14,670 = $29,904, E.O. = 56.4

Reduced demand uncertainty from 8th week• Single order

• O = 358 (at this time, demand change cannot be expected)

• E.P. = $29,973, E.O. = 79.8

• Two orders

• O1* = 195, EO1(O1*) = 56.4, E(O2*) = 151 – 56.4 = 94.6

• E.P. = $14,670 + $1056.4 + $15,254 = $30,488, E.O. = 11.3

KEY POINT: quick response multiple order E.P. , E.O.

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Postponement Delay of product differentiation until closer to sale of product

• Accurate by aggregate forecast and close-to-sale forecast

• Imposing associated cost

Example: aggregating products with equal demand• Setting

• 4 products, each with normally distributed demand (1,000, 500)

• p = $50, s = $10

• No postponement (c = $20)

• For each product

» CSL* = 0.75, O* = 1,337, E.P. = $23,644

• E.P. (for all): $94,576

• Postponement (c = $22)

• Aggregate demand: (4,000, 1,000)

• CSL* = 0.70, O* = 4,524, E.P. = $98,092

21


Postponement (cont’d) Example: including a product with dominant demand

• Setting

• Demand of dominant products: (3,100, 800)

• Demand of 3 other products: (300, 200)

• p = $50, s = $10

• No postponement (c = $20)

• E.P. = $102,205

• Postponement (c = $22)

• E.P. = $99,872

22


Postponement (cont’d) Example: tailored postponement

• Lying somewhere between two extremes

• No analytical solution for evaluating optimal decision

• Using simulation

• Setting: same as ‘equal demand’ example

Ordering Policy

AverageProfit

AverageOverstock

AverageUnderstockO1 OA

01,337

700800900900

1,0001,0001,1001,100

4,5240

1,8501,550

9501,050

850950550650

97,84794,377

102,730104,603101,326101,647100,312100,951

99,180100,510

5101,369

308427607664815803

1,0261,008

210282168170266230195149211185

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Tailored sourcing Using combination of two supply sources

• One focusing on cost but unable to handle uncertainty well

• The other focusing on flexibility but at a higher cost

Types• Volume-based tailored sourcing

• e.g., Benetton with overseas production

• Product-based tailored sourcing

• e.g., Levi Strauss

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Multi. Products under Capacity Constraint

Input For product i, pi, ci, and si

Each product’s demand distribution Fi(x)

Production capacity B

Decision Qi: production quantity of product i

Optimization model max. i EPi(Qi)

s.t. i Qi B

s.t. Qi 0

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Multi. Products under Capacity Constraint

Expected marginal contribution of product i with quantity Qi

MCi(Qi) = pi(1 – Fi(Qi)) + siFi(Qi) – ci

Solution procedure1. Qi = 0 for all products i.

2. If no MCi(Qi) is positive, then stop.

3. Let j be the product with the highest MCi(Qi).

4. Qj Qj + 1

5. If i Qi < B, then go to Step 2; otherwise, stop.

26

CSL for Continuously Stocked Items

Assumptions Cycle inventory (ordered repeatedly) D: average demand per unit time, H: holding cost Q: order quantity

All out-of-stock is backlogged Discount by for each backlogged item (c' = c – )

• Cost of overstocking by one unit: Co = HQ/D

• Cost of understocking by one unit: Cu = – HQ/D

Example 12-4: Imputing cost of stockout form inventory policy• Mean & std. dev. of demand during lead time: DL, L

• D = 5,200, C = $3, H = $0.6, Q = 400, ROP = 300

CSL* = FS((ROP – DL)/L) = 0.9998 unit/.$*

82301

DCSL

HQ

D

HQDHQ

CC

CCSL 1

ou

u*

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CSL for Continuously Stocked Items

All out-of-stock is lost-sales Lost-sales cost per unit: cL

• Cost of overstocking by one unit: Co = HQ/D

• Cost of understocking by one unit: Cu = cL

Example 12-5• Q = 400, D = 52,000, H = $0.6, cL = $2

• CSL* = 0.98

CLS will be higher if sales are lost than if sales are backlogged, in general.

HQDc

HQ

DHQc

DHQ

DHQc

c

CC

CCSL

LLL

L

ou

u* 11

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Setting Optimal Availability in Practice

Use an analytical framework to increase profits

Beware of preset levels of availability

Use approximate costs because profit-maximizing solutions are quite robust

Estimate a range for the cost of stocking out

Ensure levels of product availability fit with strategy

Determining the Optimal Level of Product Availability Spring, 2014 Supply Chain Management:...

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