YA/EM'2004 - Yöneylem Araştırması/Endüstri Mühendisliği - XXIV Ulusal Kongresi, 15-18 Haziran 2004, Gaziantep - Adana
İKİ SINIFLI, BAZ-STOK DENETİMİNDEKİ BİR SİSTEMİN DİNAMİK
ÇİZELGELEMESİ İÇİN SEZGİSEL YÖNTEMLERLE GELİŞTİRİLEN
ALTERNATİF POLİTİKALAR
Bora Kat, Zeynep Müge Avşar
Orta Doğu Teknik Üniversitesi, Endüstri Mühendisliği Bölümü, 06531, Ankara
Özet: Bu çalışmada baz-stok denetimindeki iki tip parçanın birer birer işlendiği bir sistem için en iyi
çizelgeleme politikaları araştırılmaktadır. Performans ölçütü gelen talebi anında karşılama oranıdır, bu oran
parça tiplerinin farklılıkları gözönüne alınarak ağırlıklı bir ortalama olarak hesaplanmaktadır.Parçaların talep
süreçleri Poisson, tesisin parça başına işleme süresinin üstel olduğu durum çalışılmaktadır. En iyi politikanın
performansı sıklıkla kullanılan parçaları gelilş sıralarında işleme ve işlenmek üzere sırada bekleyen parça
tiplerinden sayısı fazla olanı öncelikle işleme politikalarıyla karşılaştırılmaktadır. En iyi politikanın yaklaşık
olarak belirlenebilmesi için sezgisel yaklaşımlar önerilmektedir.
Anahtar Kelimeler: Kuyruk sistemleri, Dinamik çizelgeleme, Baz-stok, Karşılama oranı
ALTERNATIVE POLICIES BY HEURISTICS FOR DYNAMIC SCHEDULING OF
A TWO-CLASS BASE-STOCK CONTROLLED SYSTEM
Abstract: Dynamic scheduling of a single facility processing items one by one is studied. Inventories of two
types of items are managed by base-stock control policies for the case of Poisson demand and exponential
processing times. The decision criterion under which structure of the optimal scheduling policy is investigated
is a weighted average of the fill rates of the items. Performance of the optimal policy is compared to those of
two well-known policies, LQ (Longest Queue) and FCFS (First-Come-First-Served), and alternative polices
are generated by heuristic approaches to approximate the optimal policy.
Keywords: Queuing systems, Dynamic scheduling, Base-stock, Fill rate
Consider a production facility capable of processing I types of items. There are no set-up costs or
change-over times. Inventories of item i is managed according to well-known base-stock policy with basestock level Si , i=1, ..., I. Demands of each item are met from the respective stock, and those that are not
satisfied immediately are backordered. Figure 1 shows how the system is operated satisfying the following
balance equations
ni  ni  Si  ki
i  1,..., I
implied by the base-stock policies. ni , ni and ki denote the number of items of type i to be processed,
number in stock and number backordered, respectively.


Figure1. I-item system
It is possible to consider the system outlined above as a manufacturing system or a repair shop. In the
case of a repair shop, demands represent failures of repairable items. Every time an item fails, it joins its own
queue in the repair shop in order to be repaired. One has to decide how many spares to be kept in the repair
shop. Many spares bring extra cost, on the other hand fewer spares would increase the probability of having
interruptions at sites where the repairable items are in use as in the case of many military applications.
The closely related work in the literature is summarized next. Zheng and Zipkin (1997) proposed a
simple recursive scheme for the exact calculation of the joint distribution of the queue lengths and the
marginal distributions for the same problem as ours under the LQ policy. Ha (1997) also studies the same
problem with holding and backorder costs to minimize the expected discounted cost over an infinite horizon.
Van Houtum et al. (1997) investigate the performance of the symmetric LQ system with two variants;
threshold rejection and threshold addition, which provide lower and upper bounds for the original LQ system.
In this study we consider the simple case where there are two types of items, I=2. n=(n1,n2) fully
describes state of the two-item system under consideration. Demands for the items occur according to
independent Poisson Processes with rates i . As for the case of single-server facility, which is studied in the
first place as a building block of a further generalized investigation, the processing time for item i is
exponentially distributed with rate  i . The cost formulation considered is
c(n)=  1.1n1  S1   2 .1n2  S 2  


1  2


Note that minimization of long-run average cost is then equivalent to maximization of weighted average of
fill rates. Then, the recursive (multi-period) formulation for dynamically scheduling the items is



f m n1 , n2   cn1 , n2   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



f0 n1, n2   0, f1 n1, n2   cn1, n2  
for all n1, n2 where fm(n) is the total cost when the current state is n and there are m periods to go until the
end of the planning horizon and   1  2   . Since it is always possible to redefine the time scale,   1
can be assumed without loss of generality. As the solution algorithm, value iteration is used without
truncating the infinite state space but narrowing it as the number of remaining periods decreases.
Figure 2 shows results of value iteration, i.e., the optimal decisions at each state, for two different
values of     with   i and S  S i for i=1, 2. According to these sketches in Figure 2, the optimal
policy is characterized by two switching curves; one is a decreasing curve, Bn1  , and the other is an
increasing curve, An1  intersecting at state S1 , S 2 . The optimal policy is LQ (Longest Queue) when n1  S1
and n2  S 2 namely Region-I, and SQ (Shortest Queue) when n1  S1 and n2  S 2 , namely Region-II. In the
remaining regions, the optimal policy is determined by the switching curve Bn1  .
(a) S1=S2=9
(b) S1=S2=9
Figure 2. The optimal scheduling policy
Analyzing the structure of the optimal policy, five heuristic approaches are proposed to approximate the
optimal policy. What is observed by the numerical experiments as for the optimal policy being LQ in RegionI and SQ in Region-II is dictated in all heuristics. For a state in one of the remaining regions, heuristic 1 and 2
are proposed to approximate the switching curve Bn1  in two different ways. For heuristic 3(4), we follow
SQ(LQ) policy in Region III(I) and LQ(SQ) in the remaining regions. Finally for heuristic 5, LQ policy is
followed if ni  n2  2S and SQ otherwise. The comparison of the performances of these heuristics with the
optimal policy and the two well-known policies, LQ and FCFS, can be seen in Table 1.
Table 1. Performances (fill rate percentages) of the Policies
S


Optimal
LQ
FCFS
Heuristic 1
0.7
Heuristic 2
Heuristic 3
Heuristic 4
Heuristic 5
Optimal
LQ
FCFS
Heuristic 1
0.9
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
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
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
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
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
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
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
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
97.19
95.48
95.07
97.01
97.17
96.48
94.72
96.92
We have seen that it is possible to generate good heuristic approaches for the symmetric case. We
are now studying the asymmetric case where the demand rates are different across the items and investigating
whether the heuristic approaches giving good results for the symmetric problem can be modified for the
asymmetric case. The next step will be to analyze the problem for unequal processing rates.
References
Zheng, Y.S. and Zipkin, P. A queueing model to analyze the value of centralized inventory information.
Opns. Res., 38, 296-307, 1990.
Ha, A., Optimal dynamic scheduling policy for a make-to-stock production system. Opns. Res., 45, 42-53,
1997.
Houtum, G.J. Van, Adan, I.J.B.F and Wal, J. Van Der, The symmetric longest queue system.
Communications in Statistics: Stochastic Models, 13, 105-120, 1997.