# Single Period Inventory Models Yossi Sheffi Mass Inst of Tech Cambridge, MA

```Single Period Inventory Models
Yossi Sheffi
Mass Inst of Tech
Cambridge, MA
Outline
 Single period inventory decisions
 Calculating the optimal order size
 Numerically
 Using simulation
 Analytically
 The profit function
 For specific distributions
 Level of Service
 Extensions:
 Fixed costs
 Risks
 Initial inventory
 Elastic demand
Single Period Ordering






Selling Magazines
 Weekly demand:
90
57
99
79
48 87
86 95
73 92
79 105
78
67
66
76
58
89
67
73
71 102 87
70 113 52
89 87 64
78 50 107
 Total: 4023 magazines
 Average: 77.4 Mag/week
 Min: 51; max: 113 Mag/week
66
84
70
80
79
62
54
78
97
91
67
51
75
71
88
79
89
66
62
80
Detailed Histogram
100%
4
90%
Frequency (Wks/Yr)
70%
60%
50%
2
40%
30%
1
20%
10%
0
0%
40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120
Demand (Mag/Wk)
Average=77.4 Mag/wk
Cumm Freq. (Wks/Yr)
80%
3
16
100%
14
90%
80%
70%
10
60%
8
50%
6
40%
30%
4
20%
90
10
0
11
0
12
0
13
0
M
or
e
80
0%
70
0
60
10%
50
2
40
Frequency (Wks/Yr)
12
Demand (Mag/week)
Cummulative Frequency
Histogram
The Ordering Decision
 Assume: each magazine sells for: \$15
 Cost of each magazine: \$8
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
d/wk Prob.
40
0.00 \$140
50
0.04 \$140
60
0.10 \$140
70
0.21 \$140
80
0.29 \$140
90
0.19 \$140
100
0.10 \$140
110
0.06 \$140
120
0.02 \$140
130
0.00 \$140
Exp. Profit: \$140
\$210
\$210
\$210
\$210
\$210
\$210
\$210
\$210
\$210
\$210
\$210
\$280
\$280
\$280
\$280
\$280
\$280
\$280
\$280
\$280
\$280
\$280
\$200
\$350
\$350
\$350
\$350
\$350
\$350
\$350
\$350
\$350
\$350
\$120
\$270
\$420
\$420
\$420
\$420
\$420
\$420
\$420
\$420
\$414
\$40
\$190
\$340
\$490
\$490
\$490
\$490
\$490
\$490
\$490
\$464
-\$40
\$110
\$260
\$410
\$560
\$560
\$560
\$560
\$560
\$560
\$482
-\$120
\$30
\$180
\$330
\$480
\$630
\$630
\$630
\$630
\$630
\$457
-\$200
-\$50
\$100
\$250
\$400
\$550
\$700
\$700
\$700
\$700
\$403
-\$280
-\$130
\$20
\$170
\$320
\$470
\$620
\$770
\$770
\$770
\$334
-\$360
-\$210
-\$60
\$90
\$240
\$390
\$540
\$690
\$840
\$840
\$257
-\$440
-\$290
-\$140
\$10
\$160
\$310
\$460
\$610
\$760
\$910
\$177
-\$520
-\$370
-\$220
-\$70
\$80
\$230
\$380
\$530
\$680
\$830
\$97
-\$600
-\$450
-\$300
-\$150
\$0
\$150
\$300
\$450
\$600
\$750
\$17
-\$680
-\$530
-\$380
-\$230
-\$80
\$70
\$220
\$370
\$520
\$670
-\$63
Order:
Expected Profits
\$600
\$500
Profit
\$400
\$300
\$200
\$100
\$0
-\$100
20
40
60
80
100
120
Order Size
140
160
Optimal Order
(Analytical)
 The optimal order is Q*
 At Q* - what is the probability of selling one more
magazine ?
 The expected profit from ordering the (Q*+1)st
magazine is:

If demand is high and we sell it:


(REV-COST) x Pr( Demand is higher than Q*)
If demand is low and we are stuck:

(-COST) x Pr( Demand is lower or equal to Q*)
 The optimum is where the total expected profit
from ordering one more magazine is zero:

(REV-COST) x Pr( Demand &gt; Q*) – COST x Pr( Demand ≤ Q*)
=0
Pr( Demand ≤ Q*) =
REV-COST
REV
Optimal Order
REV-COST 15 − 8
=
= 0.47
REV
15
100%
14
90%
80%
12
70%
10
60%
8
50%
6
40%
30%
4
20%
90
10
0
11
0
12
0
13
0
M
or
e
80
0%
70
0
60
10%
50
2
Demand (Mag/week)
Cummulative Frequency
Pr( Demand ≤ Q*) =
16
40
Frequency (Wks/Yr
The “critical ratio”:
Salvage Value
Salvage value = \$4/Mag. Critcal Ratio =
REV − COST 15 − 8
=
= 0.64
REV − SLV
15 − 4
\$600
\$500
40%
30%
4
20%
Demand (Mag/week)
13
0
M
or
e
90
10
0
11
0
12
0
80
0%
70
0
60
10%
50
2
Order Size
16
0
6
\$0
14
0
50%
12
0
8
\$100
10
0
60%
80
10
40
Frequency (Wks/Yr
70%
\$200
60
80%
12
\$300
40
14
90%
20
100%
Cummulative Frequency
16
Profit
\$400
The Profit Function
 Revenue from sold items
 Revenue or costs associated with unsold
items. These may include revenue from
salvage or cost associated with disposal.
 Costs associated with not meeting
customers’ demand. The lost sales cost can
include lost of good will and actual
penalties for low service.
 The cost of buying the merchandise in the
first place.
The Profit Function
∞
E [Sales ] = Qi
∫
Q
f ( x )dx +
X =Q
∫
x if ( x )dx
x =0
Q
E [Unsold ] =
∫ (Q − x )if ( x )dx = Q − E [Sales]
x =0
∞
E [Lost Sales ] =
∫
( x − Q )if ( x )dx = μ − E [Sales ]
X =Q
E [Profit ] = R iE [Sales ] + S iE [Unsold ] − L iE [Lost Sales ] − C iQ
The Profit Function – Simple Case
E [Profit ] = R iE [Sales ] − C iQ
Optimal Order:
d
E [Profit ] = (1 − F (Q ))iR − C = 0
dQ
d
E [Sales ] = 1 − F (Q )
dQ
F [Q *] =
R −C
R
and:
⎡R − C ⎤
Q * = F −1 ⎢
⎥
⎣ R ⎦
Level of Service
 Cycle Service – The probability that
there will be a stock-out during a
cycle

 Fill Rate - The probability that a
specific customer will encounter a
stock-out

REV=\$15
COST=\$8
Level of Service
100%
90%
80%
Service Level
70%
60%
50%
40%
30%
Cycle Service
20%
Fill Rate
10%
0%
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
Order
Normal Distribution of Demand
X ~ N ( μ ,σ )
E [sales ] = Q − σ i( ziΦ( z ) + φ ( z ))
z=
Q−μ
σ
E[Profit ] = ( R − C ) • Q − R • σ ⋅ [ z • Φ ( z ) + φ ( z )]
\$500
⎛ R−C ⎞
Q* = NORMINV ⎜
⎟=
⎝ R ⎠
\$300
\$200
⎛ 15 − 8 ⎞
= NORMINV ⎜
⎟ = 76 Mags
⎝ 15 ⎠
\$100
Order Size
-\$200
16
0
14
0
12
0
10
0
80
60
-\$100
40
\$0
20
Expected Profit
\$400
REV=\$15
COST=\$8
Incorporating Fixed Costs
With fixed costs of \$300/order:
\$600
\$500
\$400
\$200
\$100
-\$300
-\$400
-\$500
160
150
140
130
120
Order Size
110
90
80
70
60
50
100
-\$200
40
-\$100
30
\$0
20
Expected Profi
\$300
REV=\$15
COST=\$8
Risk of Loss
Probability of Loss
1.00
0.80
0.60
F=\$300
0.40
F=0
0.20
0.00
20
40
60
80
100
Order Quantity
120
140
160
Ordering with Initial Inventory
 Given initial Inventory: Q0, how to order?
1. Calculate Q* as before
2. If Q0 &lt; Q*, order (Q*&lt; Q0 )
3. If Q0 ≥ Q*, order 0
 Cost of initial inventory
 With fixed costs, order only if the expected
profits from ordering are more than the ordering
costs
1. Set Qcr as the smallest Q such that
with Qcr]&gt;E[Profits with Q*]-F
2. If Q0 &lt; Qcr, order (Q*&lt; Q0 )
3. If Q0 ≥ Qcr, order 0
E[Profits
Ordering with Fixed
Costs and Initial Inventory
REV=\$15
COST=\$8
Example: F = \$150
\$600
Expected Profit
\$500
\$400
\$300
\$200
\$100
\$0
Initial Inventory
20 30 40 50 60 70 80 90 100 110 120 130 140 150 160
-\$100
•If initial inventory is LE 46, order up to 80
•If initial inventory is GE 47, order nothing
Elastic Demand
⎝
P
\$600
\$500
\$400
Profit
 &micro; =D(P); σ = f(&micro;)
 Procedure:
1. Set P
2. Calculate &micro;
3. Calculate σ
P −C ⎞
4. Q * = F −1 ⎛⎜
⎟
Expected Profit Function
\$300
\$200
\$100
\$0
10
15
20
25
Price
⎠
5. Calculate optimal expected profits as a
function of P.
P* = \$22
Rev =
\$15
Cost=
\$8
&micro;(p)=165-5*p
σ=
&micro;/2
30
Q*= 65 Mag
&micro;(p)=56 Mag
σ= 28
Exp. Profit=\$543
Elastic Demand:
Numerical Optimization
Screenshots removed due to copyright
restrictions.
Any Questions?
?
? ??
?
?
Yossi Sheffi
```