0. Determining the Optimal Level of Product Availability
Spring, 2014
Supply Chain Management: Strategy, Planning, and Operation
Chapter 12
Byung-Hyun Ha

1. 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

2. 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?

3. 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)

4. Factors Affecting the Optimal Availability
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.

5. Factors Affecting the Optimal Availability
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
EP(O)

6. Factors Affecting the Optimal Availability
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

7. Factors Affecting the Optimal Availability
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)

8. Optimal Cycle Service Level
Notation
O*: Optimal order size that maximizes expected profit
CSL* = F(O*) ; optimal cycle service level
Analysis

9. Optimal Cycle Service Level
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) = 100 = 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 2
0.2
10,000
0.3 3
0.4
12,000
0.7 4
14,000
1.0

10. Optimal Cycle Service Level
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 x O

11. Optimal Cycle Service Level
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) O
1 – F(O)
F(O) x O

12. Optimal Cycle Service Level
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

13. Optimal Cycle Service Level
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 cost
Marginal contribution 1
0.1
8,000
0.9
40.5
6.0
34.5 2
0.2
10,000
0.7
31.5
0.3
18.0
13.5 3
0.4
12,000
42.0
–28.5 4
14,000
0.0
1.0
60.0
–60.0

14. 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.

15. Managerial Levers to Improve Profitability
Increasing salvage value
 CSL* , Profitability 
e.g., Sport Obermeyer
Decreasing stockout
e.g., McMaster-Carr and W.W. Grainger

16. Managerial Levers to Improve Profitability
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 Overstock
Expected Understock
Expected 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
350
0.0
$52,500

17. Managerial Levers to Improve Profitability
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 policies
1. Single order for covering entire season’s demand
2. Two orders at beginning of season and at beginning of 8th week
Scenarios
1. Unchanged demand
2. Reduced demand uncertainty from 8th week
S.D. of weekly demand changes to 3

18. Managerial Levers to Improve Profitability
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 + $10 $14,670 = $29,904, E.O. = 56.4
Reduced demand uncertainty from 8th week
O = 358 (at this time, demand change cannot be expected)
E.P. = $29,973, E.O. = 79.8
O1* = 195, EO1(O1*) = 56.4, E(O2*) = 151 – 56.4 = 94.6
E.P. = $14,670 + $10 $15,254 = $30,488, E.O. = 11.3
KEY POINT: quick response  multiple order  E.P. , E.O. 

19. Managerial Levers to Improve Profitability
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

20. Managerial Levers to Improve Profitability
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

21. Managerial Levers to Improve Profitability
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
Average Profit
Average Overstock
Average Understock O1 OA
1,337
700
800
900
1,000
1,100
4,524
1,850
1,550
950
1,050
850
550
650
97,847
94,377
102,730
104,603
101,326
101,647
100,312
100,951
99,180
100,510
510
1,369
308
427
607
664
815
803
1,026
1,008
210
282
168
170
266
230
195
149
211
185

22. Managerial Levers to Improve Profitability
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

23. 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

24. Multi. Products under Capacity Constraint
Expected marginal contribution of product i with quantity Qi
MCi(Qi) = pi(1 – Fi(Qi)) + siFi(Qi) – ci
Solution procedure
1. 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.

25. 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 = HQ/D
Cost of understocking by one unit: Cu =  – HQ/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) =

26. 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 = HQ/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.

27. 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