HEURISTICS FOR DYNAMIC SCHEDULING
OF
MULTI-CLASS BASE-STOCK CONTROLLED SYSTEMS
Bora KAT and Zeynep Müge AVŞAR
Department of Industrial Engineering
Middle East Technical University
Ankara TURKEY
1
OUTLINE
Two-class base-stock controlled systems
Related work in the literature
Analysis
Model to maximize aggregate fill rate
Solution approach
Structure of the optimal dynamic (state-dependent) scheduling policy
Heuristics to approximate the optimal policy
Numerical Results
Symmetric case (equal demand rates)
Asymmetric case
Optimality of the policy
to minimize inventory investment subject to aggregate fill rate constraint
to maximize aggregate fill rate under budget constraint
Conclusion and future work
2
SYSTEM CHARACTERISTICS
Single facility to process items
Exponential service times
Poisson demand arrivals
No set-up time
Backordering case
Preemption allowed
Each item has its own queue
managed by base-stock policies
Performance measure: aggregate fill rate
over infinite horizon
3
TWO-CLASS BASE-STOCK CONTROLLED SYSTEM
1
ni :
items of type i in process
ni : items of type i in stock
k i : backorders of type i
S i : base-stock level for type i
i : demand rate for type i
 : service rate
ni  ni  Si  ki for i  1, 2
2
4
RELATED WORK IN THE LITERATURE
•Zheng and Zipkin, 1990.
A queuing model to analyze the value of centralized inventory information.
•Ha, 1997.
Optimal dynamic scheduling policy for a make-to-stock production system.
•van Houtum, Adan and van der Wal, 1997.
The symmetric longest queue system.
•Pena-Perez and Zipkin, 1997.
Dynamic scheduling rules for a multi-product make-to-stock queue.
•Veatch and Wein, 1996.
Scheduling a make-to-stock queue: Index policies and hedging points.
•de Vericourt, Karaesmen and Dallery, 2000.
Dynamic scheduling in a make-to-stock system: A partial charact. of optimal policies.
•Wein, 1992.
Dynamic scheduling of a multi-class make-to-stock queue.
•Zipkin, 1995.
Perf. analysis of a multi-item production-inventory system under alternative policies.
•Bertsimas and Paschalidis, 2001.
Probabilistic service level guarantees in make-to-stock manufacturing systems.
•Glasserman, 1996.
Allocating production capacity among multiple products.
5
•Veatch and de Vericourt, 2003.
Zero Inventory Policy for a Two-Part-Type Make-To-Stock Production System.
RELATED WORK IN THE LITERATURE
Zheng and Zipkin, 1990.
A queuing model to analyze the value of centralized inventory information.
• symmetric case: identical demand (Poisson) and service (exponential) rates,
identical inventory holding and backordering costs
• base-stock policy employed, preemption allowed
• main results on the LQ (longest queue) system
• closed form steady-state distribution of the difference between the two queue lengths
• closed form formulas for the first two moments of the marginal queue lengths
• a recursive scheme to calculate joint and marginal distributions of the queue lengths
• it is analytically shown that LQ is better than FCFS discipline
under the long-run average payoff criterion
• alternative policy: specify (2S-1) as the maximum total inventory to stop producing
imposing a maximum of S on each individual inventory
• -policy as an extension of LQ for the asymmetric case,
extension of the recursive scheme to calculate steady-state probabilities
6
RELATED WORK IN THE LITERATURE
Ha, 1997.
Optimal dynamic scheduling policy for a make-to-stock production system.
• two item types allowing preemption,
demand (Poisson) and service (exponential) rates, and
inventory holding and backordering costs to be different
• perf. criterion: expected discounted cost over infinite horizon
• equal service rates: characterizing the optimal policy by two switching curves
(base-stock policy, together with a switching curve, for a subset of initial inv. levels)
• different service rates: optimal to process the item with the larger  index
when both types are backordered
• heuristics:
static priority () rule
dynamic priority ( modified  and switching) rules
7
RELATED WORK IN THE LITERATURE
van Houtum et al., 1997.
The symmetric longest queue system.
• symmetric multi-item case:
identical demand (Poisson) and service (exponential) rates,
base-stock policy employed, not preemptive
• performance measure is fill rate (the cost formulation used is the same)
• investigating (approximating) the performance of the LQ policy
with two variants: threshold rejection and threshold addition (to find bounds)
8
MODEL

total cost
long - run 1 
 length of time horizon
cost :

 :

c(n1 , n2 )  w11n1  S1   w21n2  S2 
f m n1 , n2   cn1 , n2  
weigh ted average of fill rates
weight : wi 
i
?
1  2
1


f m 1 n1  1, n2   2 f m 1 n1 , n2  1  min  f m 1 n1  1, n2 , f m 1 n1 , n2  1



f 0 n1 , n2   0
where   1  2    1.
9
SOLUTION APPROACH
• Value iteration
n2
to solve for long-run avg. payoff
N+m
1  lim  f m n1 , n2   f m 1 n1 , n2 
m 
N+m-1
weighted average of fill rates
• No truncation
f 0 n1 , n2   0 for (n1 , n2 )  0,1,..., N  m
2
f1 n1 , n2  for (n1 , n2 )  0,1,..., N  m  1
2
N

f m n1 , n2  for (n1 , n2 )  0,1,..., N
2
N
N+m-1
N+m
10
n1
OPTIMAL SCHEDULING POLICY: Symmetric, Finite-Horizon Case
n2
n2
S2
S2
S1
n1
n2
n2
S2
S2
S1
n1
S1
n1
S1
n1
11
OPTIMAL SCHEDULING POLICY: Symmetric, Infinite-Horizon Case
n2
process type 1
A(n1)
process type 2
S2
B(n1)
S1
n1
12
OPTIMAL SCHEDULING POLICY: Symmetric Case
n2
n2
A(n1)
PROCESS
TYPE 1
A(n1)
PROCESS
TYPE 1
PROCESS
TYPE 2
S2
PROCESS
TYPE 2
B(n1)
PROCESS
TYPE 1
S1
PROCESS
TYPE 2
S2
PROCESS
TYPE 2
PROCESS
TYPE 1
n1
r0.4, S1=S2=9
S1
B(n1)
n1
r0.9, S1=S2=9
13
STRUCTURE OF THE OPTIMAL POLICY: Symmetric Case
Cost: c(n1,n2)
n2
n2
SQ
S2
w2
1
0
w1
S2
B(n1)
LQ
S1
n1
S1
n1
14
STRUCTURE OF THE OPTIMAL POLICY: Symmetric Case
n2
Region I
n1  S1 and n2  S 2
none in stockout
LQ to avoid stockout
for the item with higher risk
IV
III
I
II
S2
Region II
n1  S1 and n2  S 2
type 1 in stockout
B(n1): threshold
to be away from region III
and to reach region I
Region III
n1  S1 and n2  S 2
both types in stockout
SQ to eliminate stockout
for more promising type
S1
n1
S1
n1
n2
S2
Region IV
n1  S1 and n2  S 2
type 2 in stockout
B(n1): threshold
to be away from region III
and to reach region I
15
HEURISTICS TO APPROXIMATE OPTIMAL POLICY: Symmetric Case
(to approximate curve B)
Heuristic 1
n1  S1  1 S 2  n2

  1
2
n2

 process type 1
 process type 2
Heuristic 2
SQ
n1  S1  1



S 2  n2
2
 process type 1
 process type 2
S2
S2-n2
Best performance by heuristic 2.
LQ
n1-(S1-1)
S1
(n1,n2)
n1
Heuristics 1 and 2 perform
almost equally well.
16
HEURISTICS TO APPROXIMATE OPTIMAL POLICY: Symmetric Case
n2
n2
n2
SQ
SQ
S2
SQ
2S
S2
LQ
LQ
LQ
S1
n1
Heuristic 3
S1
Heuristic 4
2S
n1
n1
Heuristic 5
steady-state probability distribution
performs better than heuristics 3 and 4
by Zheng-Zipkin’s algorithm in LQ region
for large r.
and then proceeding recursively in SQ region.
17
NUMERICAL RESULTS: Symmetric Case
Fill Rate (%)
S
r
0.7
0.8
0.9
Optimal
LQ
FCFS
Heuristic 1
Heuristic 2
Heuristic 3
Heuristic 4
Heuristic 5
Optimal
LQ
FCFS
Heuristic 1
Heuristic 2
Heuristic 3
Heuristic 4
Heuristic 5
Optimal
LQ
FCFS
Heuristic 1
Heuristic 2
Heuristic 3
Heuristic 4
Heuristic 5
1
53.72
43.81
46.15
53.72
53.72
53.72
53.72
53.72
44.97
30.81
33.33
44.97
44.97
44.97
44.97
44.97
35.46
16.23
18.18
35.46
35.46
35.46
35.46
35.46
2
76.26
70.25
71.01
76.06
76.26
75.03
74.68
76.06
66.00
53.95
55.56
65.92
66.00
62.94
64.74
65.92
53.20
31.12
33.06
53.20
53.17
46.65
52.48
53.20
3
87.73
84.70
84.39
87.60
87.72
87.03
85.11
87.43
78.14
69.88
70.37
77.95
78.14
75.62
75.57
77.95
63.86
43.80
45.23
63.79
63.74
56.36
62.11
63.79
4
93.75
92.26
91.59
93.67
93.74
93.40
91.09
93.48
85.86
80.48
80.25
85.69
85.85
84.15
82.52
85.57
71.24
54.31
55.19
71.08
71.13
64.48
68.50
71.08
6
98.43
98.07
97.56
98.40
98.43
98.35
96.89
98.31
94.12
91.91
91.22
93.98
94.12
93.41
90.94
93.83
81.26
69.93
70.00
80.97
81.19
76.61
77.24
80.90
8
99.61
99.53
99.29
99.61
99.61
99.59
98.98
99.57
97.58
96.67
96.10
97.49
97.58
97.29
95.41
97.39
87.71
80.26
79.92
87.37
87.66
84.64
83.38
87.25
11
99.95
99.94
99.89
99.95
99.95
99.95
99.83
99.95
99.36
99.13
98.84
99.34
99.36
99.29
98.43
99.29
93.47
89.50
89.00
93.17
93.44
91.83
89.74
93.05
15
100.00
100.00
99.99
100.00
100.00
100.00
99.99
100.00
99.89
99.85
99.77
99.89
99.89
99.88
99.65
99.88
97.19
95.48
95.07
97.01
97.17
96.48
94.72
96.92
18
r  0.90
Fill Rate (%)
NUMERICAL RESULTS: Symmetric Case
100.00
90.00
80.00
70.00
60.00
50.00
40.00
30.00
20.00
10.00
0.00
Optimal
LQ
FCFS
Heuristic 1
Heuristic 2
Heuristic 3
Heuristic 4
Heuristic 5
1
2
3
4
6
8
11
15
S
19
r  0.90
NUMERICAL RESULTS: Symmetric Case
100.00
Optimal
Fill Rate (%)
90.00
80.00
LQ
70.00
FCFS
60.00
Heuristic 1
50.00
Heuristic 2
40.00
30.00
Heuristic 3
20.00
Heuristic 4
10.00
Heuristic 5
0.00
1
2
3
4
6
8
11
15
S
20
THREE TYPES OF ITEMS: Symmetric Case
S
Fill Rate (%), r=0.75
LQ
FCFS
1
2
3
4
6
8
11
50.000
75.000
87.500
93.750
98.438
99.609
99.951
S
Fill Rate (%), r=0.90
LQ
FCFS
1
2
3
4
6
8
11
25.000
43.750
57.813
68.359
82.202
89.989
95.776
46.323
74.281
88.324
94.849
99.037
99.824
99.986
21.359
40.856
56.293
67.943
82.901
90.906
96.460
46.366
74.322
88.354
94.873
99.048
99.828
99.987
21.421
40.955
56.406
68.049
82.987
90.975
96.519
Heuristic 2
60.735
81.520
91.476
96.199
99.282
99.870
99.990
60.752
81.546
91.497
96.216
99.294
99.873
99.991
Heuristic 2
47.887
66.429
76.568
83.138
91.045
95.249
98.147
47.920
66.460
76.600
83.186
91.097
95.278
98.171
Simulation results for LQ policy and Heuristic 2
21
1  22
w1  w2
THE OPTIMAL SCHEDULING POLICY: Asymmetric Case
n2
n2
A(n1)
A(n1)
PROCESS
TYPE 1
PROCESS
TYPE 1
PROCESS
TYPE 2
PROCESS
S2 TYPE 2
PROCESS
TYPE 2
PROCESS
S2 TYPE 2
B(n1)
PROCESS
TYPE 1
S1
PROCESS
TYPE 1
n1
r0.4, S1=S2=8
B(n1)
S1
n1
r0.9, S1=S2=8
22
HEURISTIC 1 TO APPROXIMATE OPTIMAL POLICY: Asymmetric Case
(to approximate curves A and B)
1  2
w1  w2
Region I
n2
IV
III
n1-(S1-1)
S1  n1
1
(n1,n2)

S 2  n2
2
 process type 1

 process type 2
n1  S1  1 S 2  n2

  1
2
 process type 1
Region II
n2-(S2-1)

 process type 2
S2
S2-n2
(n1,n2)
S1-n1
Region III
S2-n2
I
n1-(S1-1)
S1
(n1,n2)
II
n1
n1  S1  1 n2  S 2  1
 process type 1

  1
  2

Curve A approximated for regions I and III is the diagonal when 1=2.
 process type 2
23
HEURISTIC 2 TO APPROXIMATE OPTIMAL POLICY: Asymmetric Case
1  2
w1  w2
Region I
n2
IV
III
n1-(S1-1)
S1  n1
1
(n1,n2)

S 2  n2
2

 process type 1
 process type 2
Region II
n2-(S2-1)
n1  S1  1



S 2  n2
2
 process type 1
 process type 2
S2
S2-n2
(n1,n2)
S1-n1
Region III
S2-n2
I
n1-(S1-1)
S1
(n1,n2)
II
n1
n1  S1  1 n2  S 2  1
 process type 1

  1
  2

 process type 2
24
1  22
w1  w2
NUMERICAL RESULTS: Asymmetric Case
Fill Rate (%)
S
r
1 2 2
Optimal
FCFS
Heuristic
Heuristic
0.7 Delta 1
Delta 2
Delta 3
Delta 5
Delta 8
Optimal
FCFS
Heuristic
Heuristic
0.8 Delta 1
Delta 2
Delta 3
Delta 5
Delta 8
Optimal
FCFS
Heuristic
Heuristic
0.9 Delta 1
Delta 2
Delta 3
Delta 5
Delta 8
1
2
1
2
1
2
1
2
3
4
6
8
11
15
56.21
47.69
56.16
56.16
45.11
46.77
47.77
48.78
49.31
48.76
35.06
48.72
48.72
32.05
33.86
35.16
36.83
38.12
41.12
19.64
41.07
41.07
17.07
18.48
19.64
21.48
23.45
77.12
71.90
77.00
76.90
70.09
70.19
71.14
72.12
72.66
67.89
57.23
67.65
67.54
53.97
54.49
55.99
57.96
59.52
56.93
35.14
56.76
56.45
31.26
32.00
33.59
36.16
38.97
87.94
84.54
87.71
87.71
84.38
83.64
83.47
84.18
84.58
78.93
71.44
78.51
78.54
69.68
69.05
69.29
71.04
72.44
65.78
47.42
65.31
65.20
43.77
43.57
44.30
47.02
50.02
93.72
91.30
93.57
93.60
91.99
91.46
90.77
90.72
90.98
86.17
80.68
85.81
85.93
80.26
79.63
78.91
79.51
80.63
72.25
57.19
71.54
71.78
54.23
53.85
53.61
55.40
58.27
98.37
97.11
98.31
98.32
97.94
97.81
97.52
96.74
96.62
94.19
90.86
93.93
94.06
91.74
91.47
90.93
89.55
89.71
81.72
71.27
80.88
81.40
69.83
69.55
69.06
68.09
69.76
99.59
98.99
99.57
99.58
99.48
99.45
99.38
99.09
98.62
97.60
95.52
97.46
97.54
96.57
96.48
96.24
95.37
94.16
88.00
80.43
87.22
87.78
80.17
80.01
79.65
78.49
77.35
99.95
99.78
99.95
99.95
99.93
99.93
99.92
99.88
99.75
99.37
98.39
99.33
99.35
99.09
99.07
99.01
98.75
98.02
93.64
88.71
93.07
93.51
89.44
89.37
89.19
88.48
86.72
100.00
99.97
100.00
100.00
100.00
100.00
100.00
99.99
99.98
99.90
99.57
99.89
99.89
99.85
99.84
99.83
99.79
99.65
97.27
94.38
96.96
97.21
95.45
95.43
95.35
95.04
94.17
n2
0.
1
0
0.
S2
S1
n1
Instead of LQ policy,
delta policy (Zheng-Zipkin):
for 1>2,
process type 2 when n2-n1>.
25
1  22
w1  w2
NUMERICAL RESULTS: Asymmetric Case
Fill Rates (%)
100.00
optimal
80.00
fcfs
60.00
h1
40.00
h2
20.00
delta3
0.00
1
2
3
4
6
8
11
15
S
26
1  22
w1  w2
NUMERICAL RESULTS: Asymmetric Case
Fill Rates (%)
100.00
optimal
80.00
fcfs
60.00
h1
40.00
h2
20.00
delta3
0.00
1
2
3
4
6
8
11
15
S
27
1  22
w1  w2
NUMERICAL RESULTS: Asymmetric Case
Weighted Cost Function
Fill Rate (%)
r
0.7
0.8
0.9
S
optimal
FCFS
h1
h1-mult
h2
h2-mult
delta1
delta2
delta3
delta5
delta8
optimal
FCFS
h1
h1-mult
h2
h2-mult
delta1
delta2
delta3
delta5
delta8
optimal
FCFS
h1
h1-mult
h2
h2-mult
delta1
delta2
delta3
delta5
delta8
1
51.66
44.84
51.54
50.98
51.54
50.98
40.08
41.18
41.84
42.52
42.87
42.69
32.47
42.50
41.80
42.50
41.80
28.03
29.24
30.11
31.22
32.08
32.95
17.86
32.73
32.22
32.74
32.22
14.72
15.65
16.43
17.65
18.97
2
74.98
68.92
74.41
74.71
74.52
74.92
66.37
65.81
66.44
67.10
67.46
64.62
53.85
63.39
64.46
63.36
64.52
50.30
50.02
51.02
52.33
53.37
52.01
32.27
48.98
51.79
49.29
52.01
28.66
28.70
29.76
31.47
33.34
3
86.96
82.17
86.63
86.86
86.66
86.95
82.18
80.86
80.25
80.73
80.99
77.84
68.14
76.67
77.75
76.71
77.84
66.99
65.53
65.08
66.25
67.18
64.79
43.95
60.70
64.42
61.30
64.78
41.44
40.40
40.37
42.18
44.18
4
93.26
89.63
92.98
93.09
93.00
93.15
90.77
89.92
88.80
88.44
88.61
85.89
77.80
84.99
85.74
85.04
85.80
78.40
77.18
75.71
75.62
76.37
73.24
53.47
69.59
72.82
70.10
73.21
52.25
51.14
49.96
50.68
52.59
n2
6
98.24
96.38
98.11
98.14
98.13
98.16
97.59
97.39
96.96
95.83
95.61
94.20
88.97
93.71
93.99
93.76
94.06
90.92
90.41
89.46
87.22
87.03
83.56
67.63
81.19
83.17
81.41
83.48
68.48
67.72
66.50
63.98
64.72
8
99.56
98.70
99.49
99.49
99.50
99.51
99.38
99.34
99.23
98.82
98.18
97.62
94.40
97.32
97.42
97.37
97.48
96.22
96.04
95.63
94.29
92.49
89.48
77.24
87.90
89.08
88.03
89.30
79.26
78.80
77.96
75.59
73.01
11
99.95
99.71
99.93
99.93
99.93
99.93
99.92
99.92
99.91
99.85
99.67
99.38
97.92
99.25
99.27
99.28
99.30
98.99
98.95
98.85
98.46
97.43
94.48
86.36
93.52
94.04
93.65
94.26
88.96
88.73
88.29
86.90
83.98
15
100.00
99.96
99.99
99.99
100.00
100.00
100.00
100.00
99.99
99.99
99.98
99.90
99.43
99.86
99.87
99.87
99.88
99.83
99.82
99.81
99.74
99.54
97.64
92.95
97.12
97.31
97.22
97.46
95.24
95.15
94.96
94.36
92.95
1
w2
S2
0
w1
S1
wi 
n1
i
1  2
Indices used for the heuristics
are multiplied by the respective wi.
Heur-mult: index1 is multiplied
also by (1-r).
(adjustment for high r)
28
I
OPTIMAL POLICY for min  ci Si
 i 1

I
wi FRi (S1 ,..., S I )   , max  wi FRi ( S1 ,..., S I )

i 1

 i 1
I

c
S

Budget


i i
i 1

I
Minimum Base-Stock Levels to satisfy Target Fill Rate
FCFS

0.90
0.95
0.99
r
0.25
0.50
0.60
0.75
0.90
0.95
0.25
0.50
0.60
0.75
0.90
0.95
0.25
0.50
0.60
0.75
0.90
0.95
S1 S2
2
3
3
5
12
24
2
3
4
6
15
30
3
5
6
10
23
47
1
2
3
5
11
23
2
3
4
6
15
30
3
4
6
9
23
46
FR(%) exp.inv. S1
91.837 1.347 2
92.593 2.037 3
92.128 2.309 3
92.224 3.617 5
90.001 7.450 12
90.470 14.905 23
97.959 1.837 2
96.296 2.519 3
96.626 3.275 4
95.334 4.570 6
95.071 10.722 15
95.034 20.972 30
99.708 2.834 3
99.177 4.004 4
99.380 5.255 6
99.194 8.012 9
99.010 18.545 23
99.046 37.091 46
LQ
S2
1
2
3
4
11
23
2
3
4
6
15
29
3
4
5
9
22
45
FR(%) exp.inv. S1
91.807 1.346 2
92.936 2.030 2
92.681 2.296 3
90.259 3.127 4
90.502 7.405 9
90.284 14.399 18
98.085 1.836 2
96.741 2.513 3
97.167 3.267 4
95.990 4.552 6
95.482 10.693 13
95.006 20.462 24
99.755 2.834 3
99.052 3.504 4
99.258 4.754 5
99.278 7.509 8
99.065 18.040 20
99.033 36.090 40
Heuristic 2
S2
1
2
3
4
9
18
2
3
3
5
12
24
2
4
5
8
20
40
FR(%)
92.462
90.139
93.616
90.458
90.000
90.858
98.115
96.989
95.837
95.864
95.214
95.075
99.025
99.113
99.042
99.007
99.014
99.073
exp.inv.
1.346
1.562
2.313
2.745
5.545
10.844
1.836
2.517
2.789
4.108
8.533
15.850
2.335
3.505
4.259
6.526
15.617
30.810
Heuristic 3
S1 S2
2
2
3
5
11
21
2
3
4
6
14
27
3
4
5
9
21
43
1
2
3
4
10
21
2
3
3
5
13
27
2
4
5
8
21
43
FR(%)
92.462
90.070
93.493
92.136
90.924
90.897
98.115
96.971
95.764
95.532
95.177
95.081
99.025
99.109
99.026
99.194
99.007
99.047
exp.inv.
1.346
1.555
2.304
3.138
6.523
12.632
1.836
2.516
2.784
4.078
9.278
18.112
2.335
3.504
4.257
7.014
16.557
33.619
Lower expected inventory levels under heuristic policies when r is not too small.
29
COMPARISON OF THE POLICIES in terms of fill rate, exp. backorders and inventories
r=0.75
S
Fill Rate (%)
FCFS
LQ
Heur.2
Expected Backorders
FCFS
LQ
Heur.2
1
2
3
4
6
8
11
15
0.40000
0.64000
0.78400
0.87040
0.95334
0.98320
0.99637
0.99953
0.90000
0.54000
0.32400
0.19440
0.06998
0.02519
0.00544
0.00071
0.37500
0.62813
0.78380
0.87589
0.95990
0.98720
0.99771
0.99977
0.49435
0.71433
0.83442
0.90458
0.96898
0.99007
0.99822
0.99982
0.87500
0.50312
0.28692
0.16281
0.05200
0.01652
0.00295
0.00030
0.99435
0.66484
0.40970
0.24512
0.08156
0.02632
0.00472
0.00047
Expected Inventory
FCFS
LQ
Heur.2
0.40000
1.04000
1.82400
2.69440
4.56998
6.52519
9.50544
13.50071
0.37500
1.00313
1.78692
2.66281
4.55200
6.51652
9.50295
13.50030
0.49435
1.16484
1.90970
2.74512
4.58156
6.52632
9.50472
13.50047
r=0.90
S
Fill Rate (%)
FCFS
LQ
Heur.2
Expected Backorders
FCFS
LQ
Heur.2
1
2
3
4
6
8
11
15
0.18182
0.33058
0.45229
0.55187
0.70002
0.79918
0.89001
0.95071
3.68182
3.01240
2.46469
2.01656
1.34993
0.90367
0.49495
0.22180
0.16227
0.31117
0.43797
0.54307
0.69933
0.80257
0.89505
0.95482
0.35459
0.53169
0.63745
0.71125
0.81191
0.87658
0.93436
0.97173
3.66223
2.97341
2.41139
1.95446
1.28298
0.84188
0.44743
0.19260
3.85455
3.35688
2.90969
2.50083
1.79612
1.25659
0.71384
0.32378
Expected Inventory
FCFS
LQ
Heur.2
0.18182
0.51240
0.96469
1.51656
2.84993
4.40367
6.99495
10.72180
0.16227
0.47344
0.91141
1.45448
2.78299
4.34190
6.94744
10.69262
0.35459
0.85691
1.40970
2.00085
3.29614
4.75660
7.21385
10.82379
30
COMPARISON OF THE POLICIES in terms of fill rate, exp. backorders and inventories
Fill Rate (%), r=0.90
Expected Backorders, r=0.90
1.00
0.95
0.90
0.85
0.80
0.75
0.70
4.00
3.00
FCFS
LQ
Heur.2
0.65
0.60
0.55
0.50
0.45
0.40
FCFS
LQ
Heur.2
2.00
1.00
0.35
0.30
0.00
1
2
3
4
6
8
11
1
15
2
3
4
6
8
11
15
S
S
Expected Inventory, r=0.90
11.00
10.00
9.00
8.00
FCFS
LQ
7.00
6.00
5.00
Heur.2
4.00
3.00
2.00
1.00
0.00
1
2
3
4
6
8
11
15
S
31
MODEL TO INCORPORATE SET-UP TIME
f
g
sf
sg
g
: minimum cost while processing items of type 1
: minimum cost while processing items of type 2
: minimum cost while setting up the facility for type 1
: minimum cost while setting up the facility for type 2
: set-up rate
f m n1 , n2   cn1 , n2 
 1 min  f m 1 n1  1, n2 , sg m1 n1  1, n2 
 2 min  f m 1 n1 , n2  1, sg m 1 n1 , n2  1
  min  f m 1 n1  1, n2 , sg m 1 n1  1, n2 
 g f m n1 , n2 
32
CONCLUSION
Summary
multi-class base-stock controlled systems
numerical investigation of the structure of the optimal policy
for maximizing the weighted average of fill rates
optimal policy for
I
min  ci Si
 i 1

w
FR
(
S
,...,
S
)




i
i
1
I
i 1

I
smaller (S1 ,..., S I ) than LQ (symmetric case),  (asymmetric case), FCFS policies give
smaller expected inventory when r is not too small
accurate heuristics adapted for extensions: asymmetric case, more than two types of items
disadvantage: not that easy to implement compared to LQ,  and FCFS policies
Future Work
optimizing base-stock levels
set-up time
type-dependent processing time
I

min  ci Si FRi ( S1 ,..., S I )   i  i  instead of working with aggregate fill rate
 i 1

how to determine the values of wi?
I
 w FR (S ,..., S )
i 1
i
i
33
1
I