Single Period Inventory Models
Yossi Sheffi
Mass Inst of Tech
Cambridge, MA
Outline
† Single period inventory decisions
† Calculating the optimal order size
„ Numerically
† Using spreadsheet
† 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
(Spreadsheet)
† 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 > 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 < Q*, order (Q*< 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]>E[Profits with Q*]-F
2. If Q0 < Qcr, order (Q*< 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
† µ =D(P); σ = f(µ)
† Procedure:
1. Set P
2. Calculate µ
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
µ(p)=165-5*p
σ=
µ/2
30
Q*= 65 Mag
µ(p)=56 Mag
σ= 28
Exp. Profit=$543
Elastic Demand:
Numerical Optimization
Screenshots removed due to copyright
restrictions.
Any Questions?
?
? ??
?
?
Yossi Sheffi