Determining the Optimal Level
of Product Availability
Spring, 2014
Supply Chain Management:
Strategy, Planning, and Operation
Chapter 12
Byung-Hyun Ha
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
1
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?
2
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)
O
EPO     px  sO  x   Pr D  x  
x 0

 pO  PrD  x   cO
x O 1
 Continuous case
• Probability density function of D: f(x)
O

0
O
EPO    px  sO  x  f ( x)dx   pO  f ( x)dx  cO
3
Factors Affecting the Optimal Availability
 Expected profit when O units are ordered (cont’d)
 Normally distributed demand D
•  = E(D), 2 = Var(D)
EPO   

 px  sO  x   f ( x)dx  O pO  f ( x)dx  cO

O
O
O



O
  p  s  xf ( x)dx  sO  f ( x)dx  pO  f ( x)dx  cO
  p  s μ F O   σ f S O  μ  σ   sO  F O   pO  1  F O   cO
  p  s μ  O   F O    p  s  σ f S O  μ  σ    p  c O
• where
• F(x) = Pr(D  x).
• fS(x) is probability density function of a standard normal random
variable.
4
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
70000
60000
50000
EP(O)
40000
30000
20000
10000
0
0
100
200
300
400
500
600
700
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' = $200, s = $80
25000
20000
15000
10000
5000
0
0
100
200
300
400
500
600
700
6
Factors Affecting the Optimal Availability
 Expected overstock by order quantity O
O
EOO    O  x  Pr D  x 
EOO   O  x  f x dx
O
0
x 0
 Expected understock by order quantity O

EU O    x  O  Pr D  x 

EU O   x  O f x dx
O
x O 1
400
 Example 12-1 (cont’d)
300
 Expected overstock and understock
200
100
0
 Expected profit by EO(O)
EPO   
O
0
0
100
200
300
400
500
600
700

 px  sO  x   f ( x)dx  O pO  f ( x)dx  cO
   pO  x   sO  x   f ( x)dx  pO  cO
O
0
  p  c O   p  s  O  x   f ( x)dx   p  c O   p  s EO(O)
O
0
7
Optimal Cycle Service Level
 Notation
 O*: Optimal order size that maximizes expected profit
 CSL* = F(O*) ; optimal cycle service level
 Analysis
O
O

0
0
O
EPO   p  s  xf ( x)dx  sO  f ( x)dx  pO f ( x)dx  cO
O

d
EPO    p  s O  f O   s  f ( x)dx  sO  f (O)  p  f ( x)dx  pO  f (O)  c
0
O
dO
O

0
O
 s  f ( x)dx  p  f ( x)dx  c
 sF (O)  p1  F (O)   c


sF (O* )  p 1  F (O* )  c  0
 
 F O* 
pc
 CSL*
ps
8
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) = 350 + 1.18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
0.1
2
0.2
10,000
0.3
3
0.4
12,000
0.7
4
0.3
14,000
1.0
9
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
EPO   
O
0

 px  sO  x   f ( x)dx  O pO  f ( x)dx  cO

O

0
0
O
  p  c  xf ( x)dx  c  s  O  x  f ( x)dx   p  c  x  O  f ( x)dx
O

0
O
 Cu E D  Co  O  x  f ( x)dx  Cu  x  O  f ( x)dx
O
O
x
10
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)
O
x
11
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
 CSL* 
Cu
Co
pc
1
1


 1
 1
C u  Co p  s 1  C o C u
Co  C u
1  C u Co
 Impact of marginal cost change
 e.g., Nordstorm and discount store
12
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
EPDk   Cu E (D)  Co  Dk  Di  pi  Cu
i 1
n
 D  D  p
i  k 1
i
k
i
k
pk
Dk
1 – F(Dk)
Marginal
benefit
F(Dk)
Marginal
cost
Marginal
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
13
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.
25000
20000
15000
10000
5000
0
0
20
40
60
80 100 120 140 160 180 200 220 240 260 280 300
14
Managerial Levers to Improve Profitability
 Increasing salvage value
 CSL* , Profitability 
 e.g., Sport Obermeyer
EPO   
O
0
Cu
pc

C u  Co p  s
CSL* 

 px  sO  x   f ( x)dx  O pO  f ( x)dx  cO


O
0
O
0
 p  xf ( x)dx  p  x  O  f ( x)dx  s  O  x  f ( x)dx  cO
 p  E D  p  EU (O)  s  EO(O)  cO
 Decreasing stockout
 e.g., McMaster-Carr and W.W. Grainger
15
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
0
350
0.0
0.0
$52,500
16
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
17
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56.4 + $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 + $1056.4 + $15,254 = $30,488, E.O. = 11.3
 KEY POINT: quick response  multiple order  E.P. , E.O. 
18
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
19
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
20
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
O1
OA
Average
Profit
0
1,337
700
800
900
900
1,000
1,000
1,100
1,100
4,524
0
1,850
1,550
950
1,050
850
950
550
650
97,847
94,377
102,730
104,603
101,326
101,647
100,312
100,951
99,180
100,510
Average
Overstock
Average
Understock
510
1,369
308
427
607
664
815
803
1,026
1,008
210
282
168
170
266
230
195
149
211
185
21
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
22
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
23
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.
24
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
 CSL* 
Cu
  HQ D
HQ

 1
C u  Co

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
HQ
 CSL* = FS((ROP – DL)/L) = 0.9998   
1  CSL D
*
 $230.8 / unit
25
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
Cu
cL
HQ D
HQ
 CSL 
 L
 1 L
 1
Cu  Co c  HQ D
c  HQ D
Dc L  HQ
*
 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.
26
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
27