A periodic tabular policy for scheduling of a single stage production-inventory system

Computers & Industrial Engineering 62 (2012) 21–28

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A periodic tabular policy for scheduling of a single stage production-inventory system q Shashank Garg a,⇑, D. Krishna Sundar b, K. Ravikumar c a

Handheld Solutions & Research Labs Pvt. Ltd., 36, 20th Main 1st Cross, 1st Stage BTM Layout, Bangalore 560 068, India Indian Institute of Management Bangalore, Bannerghatta Road, Bangalore 560 076, India c Adarsh Palm Retreat D-103, Devarabisenahalli, Varthur Road, Outer Ring Road, Bangalore 560 103, India b

a r t i c l e

i n f o

Article history: Received 31 March 2011 Received in revised form 19 August 2011 Accepted 21 August 2011 Available online 26 August 2011 Keywords: Inventory control Base stock policies M/G/1 queue Server vacations The newsboy problem

a b s t r a c t In this paper, we consider scheduling of a multi-item single stage production-inventory system in the presence of uncertainty regarding demand patterns, production times and switchover times. For a given specification of base-stock levels of individual items and under (S  1, S) requests for replenishment policy, a mathematical program to minimize long-run average system wide costs is formulated. We derive approximations for the first two moments of demand over lead time using residual service analysis of vacation queue models. Subsequently, we develop an approximate convex program for the original cost model and determine optimal production frequencies for individual types. Based on these relative frequencies, we determine a table size and devise an efficient heuristic to construct a tabular sequence in which individual items appear according to their respective absolute frequencies and items are positioned such that variance of their inter-visit times is minimized. A numerical study that demonstrates effectiveness of the proposed policy against cyclic policies is given. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction In a typical manufacturing setting, multiple items compete for the availability of a common production facility and production of units of individual items is initiated on the basis of outstanding orders for the item. In this paper, the focus is on a setting, where uncertainty prevails regarding the demand patterns and production times. Switching between different items incurs constant (random) sequence independent setup times and an explicit cost for setup as well. A variety of strategy classes have been proposed to govern these systems effectively with a view to minimize system-wide costs. These can be classified as (i) static and (ii) dynamic strategies. Dynamic strategies determine at any point in time which type of the items and the corresponding number of items, if any, is to be produced in the facility on the basis of complete state of the system which may include inventory levels of all items and the most recent assignment of the production facility. On the other hand, static policies use only the state information that pertains to the item currently being produced. Optimal policies for many common performance measures are often dynamic and can be obtained through solution of a dynamic program in multi-dimensional space. In many practical cases, such dynamic programs suffer from curse of q

The manuscript was processed by Area Editor T.C. Edwin Cheng.

⇑ Corresponding author.

E-mail addresses: [email protected] (S. Garg), [email protected] (D. Krishna Sundar), [email protected] (K. Ravikumar). 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.08.013

dimensionality. Also, even in the cases, where the optimal strategies could be computed in a reasonable amount of time, the policies have complex structure and can hardly be implemented in practice.

1.1. Literature survey In contrast with the set of dynamic policies, the class of static policies offers a better co-ordination of various other inter-related activities such as raw material procurement; and external setup and hence have a good practical appeal in manufacturing framework. Also, recent works stand as evidence to the fact that restriction to static policies in the context of scheduling of production-inventory systems comes with only a moderate loss of optimality. See for instance, Markowitz, Reiman, and Wein (2000) for a comparative performance study of a dynamic cyclic base stock rules and a minor variant of a static cyclic base stock policy. Federgruen and Katalan (1998) demonstrate that significant improvement in performance is achievable by replacing the static cyclic rules in the above case by general periodic strategies which produce different items with different relative frequencies. In Federgruen and Katalan (1998), a loose lower bound on second moment for shortfall distribution was derived in terms of individual item frequencies and a convex program was formulated using these bounds to derive approximate values for these frequencies. To obviate difficulties arising out of fully dynamic policies, a class of semi-dynamic policies has been proposed in Browne and

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S. Garg et al. / Computers & Industrial Engineering 62 (2012) 21–28

Yechiali (1989), Duenyas and Van Oyen (1995), Duenyas and Van Oyen (1996), Zipkin (1986), Leachman, Xiong, et al. (1991), Bourland and Yano (1991) and Bourland and Yano (1994). A detailed review of the policies is discussed in the literature by Federgruen and Katalan (1994), Federgruen and Katalan (1996) and Federgruen and Katalan (1998). There are no accurate analytical techniques that can evaluate this class of policies if the number of classes is more than two. The only class of policies which admits evaluation and optimization is the class of static base stock policies. The works of Federgruen and Katalan (1994) and Federgruen and Katalan (1996) suggest an efficient methodology to evaluate any given base stock policy and to derive optimal base stock policies for a given periodic sequence of items, with a particular reference to the cyclic policy case. In their subsequent work, Federgruen and Katalan (1998) provide a methodology to arrive at an effective periodic sequence that outperforms cyclic base stock policy. Another paper by Anupindi and Tayur (1998) suggests a simulation based method to compute optimal base stock levels for a given cyclic sequence. In communication networks scenario, several authors derive polling table policies for various performance measures such as weighted average of steady state waiting times and workloads. See Boxma, Levy, and Weststrate (1990), Boxma, Levy, and Weststrate (1991a), Boxma, Hanoch, and Weststrate (1991b) and Bertsimas and Xu (1993) for this topic. The effect of product variety on inventory costs has been examined in a capacitated production-inventory system to show that inventory costs increase almost linearly in the number of products and that, the rate of increase is sensitive to demand variability, demand and capacity levels, and setup times by Benjaafar, Kim, and Viswanadham (2004). Baker and Keller (2010) proposed solving the tardiness problem in single-machine sequencing by using integer programming formulations. Ravikumar and Bassamboo (2000) have described scheduling problems in a multi-item production-inventory system in the presence of random setup and production times. Traditional inventory models such as the economic production quantity model (EPQ) and the economic order quantity model (EOQ) have been formulated to minimize the holding cost or the ordering cost, components of the inventory-related costs, under ideal conditions of no defective products. However, the occurrence of defective products will cause rework on the same production line. A batch production system may produce a few defective products which require rework and determining the optimal batch quantity under rework conditions poses additional challenges. Several studies have also been made to understand optimal lot sizing of EPQ and EOQ inventory models that include rework due to defective products on the same production line. Jamal, Sarker, and Mondal (2004) proposed two different operational policies to determine the optimal batch size and minimize the total system cost in a single-stage production system on which rework is done. Cárdenas-Barrón (2007) pointed out numerical errors in the operational policies developed in Jamal et al. (2004). Subsequently, Cárdenas-Barrón (2008) proposed a simple improved algebraic derivation of the policies to determine the optimal batch size. As an extension of the single-stage production policies developed in Jamal et al. (2004), two different policies for an optimal batch quantity in a multi-stage production system have been proposed in Sarker, Jamal, and Mondal (2008). The first policy proposed rework of defective products within the same production cycle, whereas the second policy proposed rework of defective products after N cycles, at the end of the last regular cycle. The choice of rework policy is to be determined by the workin-process inventory costs. Cárdenas-Barrón (2009a) proposed a numerical correction to these policies. Further, Cárdenas-Barrón (2009b) has provided an EPQ inventory model of a single-stage production system in which all defective products are reworked in the same production cycle using planned backorders.

Cheng, Gupta, and Wang (2000) have provided a detailed review of the literature on flow-shop scheduling involving machine setup times. The study of single-machine scheduling with learning to reduce processing times has been studied by Wang and Wang (2010) and Cheng, Cheng, et al. (2011). The probabilistic safety stock n-item inventory system having varying ordering cost and zero lead-time that obtains the optimal maximum inventory levels using a geometric programming approach has been studied by Fergany (2005). A comprehensive survey of research in production-inventory systems under stochastic economic lot scheduling is also provided by Winands, Adan, and van Houtum (2005).

2. Description of the scheduling problem Consider a production system with N distinct items, demands for which are generated by independent Poisson processes with ki as the rate at which demand arises for item i, i = 1, 2, . . ., N. Let P k ¼ Ni¼1 ki . With appropriate modifications, analysis to be presented here can be carried over to compound Poisson demand streams as well. The N items are produced at common processing facility that can produce a single unit of any item at a time. Production times for individual units of an item are assumed to be i.i.d random variables with c.d.f Fi(.) and mean l1 for i = 1, 2, . . ., N. A posi sibly random setup time with cdf Gi and mean si is incurred when setting up the facility to produce item i. Consecutive setup times are independent. The utilization rate for item i is qi ¼ lki and the i P utilization of the system equals q ¼ Ni¼1 qi . Assume that the system is stable i.e., q < 1. A demand which finds zero inventory is backlogged. The following costs are incurred:  C hi = the inventory carrying cost rate for item i per unit of time (i = 1, 2, . . . ,N);  C bi = the backorder cost rate for item i per unit of time (i = 1, 2, . . . , N);  Ki = Sequence independent setup cost for item i, i = 1, 2, . . . , N The objective is to find a production sequence to minimize the long-run average total cost. Let Bi denote the base stock level for item i, i = 1, 2, . . . , N. An arriving demand which finds non-zero inventory will deplete the inventory level by a unit and each such depletion initiates a production order request at the facility. Also, a demand arriving into the system when the inventory level of the corresponding item is zero, will initiate a back order. We assume that no priority is assigned to the back logged demands. In such a scenario, one can model the production facility as a multi-class M/G/1 queue. Further, it is assumed that once production is initiated on a particular item, the facility continues to serve it exhaustively, that is, until its inventory level reaches its base stock level. A periodic tabular policy is specified by  base-stock level vector B: = (B1, B2, . . .BN)  a table T: = (T(1), T(2), . . . , T(M)), where T(j) e 1, 2, . . . , N and M denotes the size of the table T.  the idling policy specified by the vector of idle time d: = (d1, d2, . . . , dN). (B, T, d) is used to denote any such periodic tabular policy. Execution of the foregoing policy results in a cycle in which items are replenished according to the sequence specified by the table and production of units of a given item, i, is continued until the inventory level of the item hits its base-stock level Bi. Define shortfall for item i at any time t as the amount by which inventory level of item i at time t, Ii(t) falls below its base stock

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S. Garg et al. / Computers & Industrial Engineering 62 (2012) 21–28

level Bi. Note that the shortfall is not equal to the number of backorders. For stable policies such as periodic policies, using regenerative arguments, one can show that steady state values of Li(t) and Ii(t), denoted by Li and Ii respectively, exist and values are related by:

Li ¼ Bi  Ii

ð2:1Þ

Further, let the steady state cycle length random variable be denoted by C. Its mean value is given by M P

E½C ¼ i¼1

sðTðiÞ þ di ð2:2Þ

1q

Now one can represent the long run expected average cost for a given periodic policy (B, T, d) as follows: M P

"Z # t X 1 þ TC ¼ C bi ðLi ðtÞ  Bi Þ þ limsup þ limsup E t E½C t!1 t!1 0 i "Z # t X 1  E C hi ðBi  Li ðtÞÞþ t 0 i K TðiÞ

i¼1

ð2:4Þ

It is interesting to note that {Li, i = 1, 2, . . . , N} is independent of B but does depend on the table T and d, the vector of idle times. For a given table T and a given idle time vector d, the optimal base stock levels, Bi, i = 1, 2, . . . , N are obtained from the solution of a newsboy problem with Li as the demand distribution which is given by

C bi C bi þ C hi

 ð2:5Þ

Note that Bi in Eq. (2.5) can be computed only when complete distribution of Li is known. More often than not, it is difficult to find the distribution function of Li even in a simplified framework. In our problem, Li depends on the policy (B, T, d) and for a given policy, regenerative arguments can be used to arrive at expressions for moments of Li. This knowledge of moments will help us to approximate the objective value in (2.4) by the solution of the distribution free newsboy problem. A detailed account of the distribution free newsboy problem is provided in the literature by Gallego and Moon (1993). To make the discussion self-contained a brief account of the distribution free newsboy problem is given. Consider the following component terms in (2.4):

nbi :¼ C bi E½Li  Bi þ þ C hi E½Bi  Li þ 8i

ð2:6Þ

(2.6) is the steady state average holding cost and back order cost incurred for item i and is equal to total cost incurred in the newsboy problem when the demand follows the law of Li above. Scarf (1958) provided an upper-bound for (2.6) when only the mean and variance of the distribution of the demand Li, rather than the complete distribution, are known. It is easy to show that

nbi 6

1 ðC bi þ C hi Þ 2

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  r2Li þ ðBi  E½Li Þ2  ðBi  E½Li Þ

þ C hi ðBi  E½Li Þ

Hence if an approximation for r2Li can be obtained, then (2.8) provides a way to establish relationship between the policy (B, T, d) and the total cost given in (2.4). We formalize these in the following sections. As mentioned earlier, under (S  1, S) replenishment for requests policy, the dynamics of the system can be analyzed using a multi-class M/G/1 queue model. Since demands for different items join different queues and server switches for service from one queue to another as specified by (B, T, d), then it is possible to relate the time elapsed between two successive visits to a given queue with server’s vacation. Since a periodic tabular policy ensures stability, vacation periods have finite moments. However it is difficult to arrive at exact expressions for the moments of vacation even in a special case of periodic policies, the cyclic base stock policy. In view of this, we derive variance of steady state queue length in an M/G/1 queue with server vacations and use this knowledge in deriving approximation to (2.8), and hence to (2.4). 3. Determination of production frequencies

M P

 Bi ¼ min k : P½Li 6 k P

ð2:8Þ

Bi

ð2:3Þ

Since by construction the policy (B, T, d) imposes regenerative dynamics, the long run average terms in the Eq. (2.3) above converge to their respective steady state expected values. See Wolff (1988) for further discussion. Thus, Eq. (2.3) can be re-written as

K TðiÞ X X TC ¼ i¼1 C bi EðLi  Bi Þþ þ C hi EðBi  Li Þþ þ E½C i i

qffiffiffiffiffiffiffiffiffiffiffiffi min nbi ¼ C bi C hi rLi

The problem of determining production frequencies is analyzed in two phases. Firstly, the variance of numbers of customers is derived in an M/G/1 queue with server vacation under steady state conditions. The assumption is that if server goes on vacation at the end of each busy period and after return from a vacation, if the system is empty, then the server goes on a new vacation. Further, assume that the vacation periods are random and form an i.i.d sequence. 3.1. Variance of the number of customers in the system in an M/G/1 queue with server vacation Consider an M/G/1 queue with Poisson arrival rate k and i.i.d service times. Lemma 1. Let L be the steady state number of customers in the system, and W the steady state waiting time of a customer in the system. Then

VarðLÞ ¼ k2 VarðWÞ þ E½L

ð3:1Þ

Proof. Consider the moment generating function of L:

GL ðzÞ ¼ ¼

1 X n¼0 1 X

h

zn PðL ¼ nÞ zn

Z 0

n¼0

1

ekt ðktÞn f ðtÞdt n!

where f(t) is the p.d.f of the waiting time, W

¼

Z

1

ektþkzt f ðtÞdt

0

)GL ðzÞ ¼ GW ðk  kzÞ where GW(z) is the moment generation function of the waiting time. Using this relation, the variance of L can be obtained as:

VarðLÞ ¼ GL ðzÞjz¼1  ðGL ðzÞjz¼1 Þ2 þ GL ðzÞjz¼1 ð2Þ

ð1Þ

ð1Þ

) VarðLÞ ¼ k2 E½W 2   k2 ðE½WÞ2 þ kE½W ð2:7Þ

where r2Li denotes the variance of Li. Since (2.7) is convex in b i :¼ Bi  E½Li , its minimum value can be obtained from the first orB der conditions. The objective at the minimum value of Bi is given by

Using Little’s formula and the definition of variance, we obtain:

VarðLÞ ¼ k2 VarðWÞ þ E½L Hence proved.

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S. Garg et al. / Computers & Industrial Engineering 62 (2012) 21–28

Now consider an M/G/1 queue when the server goes on vacations at the end of each busy period. A new arrival to an idle system, rather than going into service immediately, waits for the end of the ongoing vacation period. Also, if the system is empty at the end of a vacation, the server takes a new vacation. Assume the sequence of vacations, fV n gnP1 is i.i.d. A new arrival into the system has to wait in the queue for the completion of the ongoing service or vacation and then for the service of all the customers ahead of it. Assume that the i.i.d vacation sequence {Vn} has finite first and second moments E[V] and E[V2] respectively. Further, let k denote the Poisson arrival rate, and fX n gnP1 denote the i.i.d sequence of service times with finite first, second and third moments. The residual time seen by ith arrival can be either remaining time for completion of customer in service or remaining time in vacation in process when the arrival occurs. Let r(s) be the residual time at time s. Let N(t) be the number of services completed by time t and Nv(t) the number of vacations completed by time t. At any time t when a service or vacation is just completed, it is easy to see that the following holds:

1 t

Z 0

t

E½R ¼

ð3:2Þ



Xe

w:p:q

Ve

w:p:ð1  qÞ

Qþ1 X

"

! Xj

¼ E Var

j¼1

Q þ1 X

!# X j jQ

þ Var E

" Qþ1 X

j¼1

#! X j jQ

j¼1

¼ VarðXÞ þ E½Q VarðXÞ þ E½X2 VarðQ Þ

ð3:8Þ

Note that, if L is the number of customers in the system in the steady state, then Q = L  1 during a service (that is, if a typical arrival observes that the server is busy) and Q = L, when the server is on vacation (that is, if a typical arrival notices that the server is on vacation). Thus

 Q¼

L  1 w:p:q w:p:ð1  qÞ

L

kE½R þq 1q

ð3:10Þ

VarðQ Þ ¼ qE½ðL  1Þ2  þ ð1  qÞE½L2   E½Q2 ¼ VarðLÞ þ qð1  qÞ

ð3:3Þ

where Xe and Ve are the equilibrium excess distributions of the service time and the vacation period respectively. Thus

ð3:9Þ

Thus

E½Q  ¼ E½L  q

Observe that (1  q) is the fraction of server’s time occupied in vacations. From (3.3), it can be re-interpreted that R has the following distribution

R¼

Var

2

kE½X  ð1  qÞE½V  þ 2 2E½V

ð3:7Þ

where W is the steady state waiting time in the system. Also

E½L ¼

Thus it follows that the steady state residual time R has its mean as: 2

Xj

j¼1

NP v ðtÞ

1 2 1 2 X V NðtÞ i¼1 2 i Nv ðtÞ i¼1 2 i ¼ þ t t NðtÞ N v ðtÞ

!

Also note that

NðtÞ Nv ðtÞ 1X1 2 1 X 1 2 Xi þ V rðsÞds ¼ t i¼1 2 t i¼1 2 i NðtÞ P

VarðWÞ ¼ VarðRÞ þ Var

Q þ1 X

ð3:11Þ

Using Lemma (3.1) and equations (3.7), (3.8), (3.10) and (3.11) to obtain:

VarðLÞ ¼

k2 VarðXÞðE½L  q þ 1Þ þ q3 ð1  qÞ þ k2 VarðRÞ þ E½L 1  q2 ð3:12Þ

where Var(R) and E[L] are as given in (3.5) and (3.9) respectively. Eq. (3.12) is used to derive an approximation for the system wide costs for any given (B, T, d). 3.2. An Approximate mathematical program

E½R ¼ qE½X e  þ ð1  qÞE½V e 

ð3:4Þ

Now variance of R can be written as

VarðRÞ ¼ qE½X 2e  þ ð1  qÞE½V 2e   E½R2 ¼

q E½X 3  3 E½X

þ

1  q E½V 3   E½R2 3 E½V

ð3:5Þ

The waiting time of the ith customer in the system can be expressed as:

W i ¼ Ri þ

i1 X

Xj

ð3:6Þ

j¼iNi

where Ri and Ni are the residual time seen and the number of customers found in queue, respectively, by the ith customer upon his arrival. Let Q be the steady state number of customers in the queue at a typical arrival epoch. Since occupation distribution upon arrival is typical, Q has the same distribution as the steady state number of customers in queue. We use Q to denote both and can be interpreted appropriately based on its context. It is easy to see that, Ri is independent of the sum appearing in (3.6). Also, since {Xj}’s are i.i.d, we can rewrite the steady state version of (3.6) as

Consider a policy (B, T, d). At time t = 0, the facility starts production of item T(1) and continues production until the inventory level hits BT(1). Then the facility switches to item T(2), after possibly idling for a duration ta1, incurring a random or non-random setup time, sT(2). The procedure is continued until item T(M) is processed and then the policy is repeated. For any policy (B, T, d), the expected long run average cost is given by (2.4). As mentioned in Section 2, for a given table, optimal base stock levels can be determined from (2.5) provided complete distribution of the shortfall for each item is known. Since such distribution is difficult to arrive at, we invoke the distribution free newsboy solution which requires only the knowledge of moments of shortfall. Let as assume that variance of shortfall for item i, Var(Li) is known. Then from (2.8) it follows that the total cost can be approximated by

TC ¼

n hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i X C bi C hi VarðLi Þ þ K i fi

ð3:13Þ

i¼1

where fi is the long run average switching frequency. Let mi, i = 1, 2, . . ., N be the absolute frequencies with which items appear in the production sequence specified by the table T. PN Note that i¼1 mi ¼ M. Also, note that, from the regenerative arguments, it can be shown that

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S. Garg et al. / Computers & Industrial Engineering 62 (2012) 21–28

mi E½C

fi ¼

ð3:14Þ

cases, one can arrive at a simplified program which can be easily solved. Case 1 Assume that ai  ci s2i and bi si  ci s2i . Then

ð3:15Þ

ci s2i k qffiffiffiffiffiffiffi ffi¼ iþk si 2 2 g i ci si

ð3:16Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ffi ci ; and ) g i ¼ si ki þ ksi

Also, under stable conditions, the following holds true:

E½Cð1  qÞ ¼

n X

mi si þ

M X

i¼1

di

i¼1

mi ð1  qÞ n M P P mi si þ di

) fi ¼

i¼1

i¼1

V ji

For, j = 1, 2, . . ., mi, let be the inter-visit time between the j  1 and j th visit to queue i for service. That is, V ji is the time between the end of busy period on the server’s j  1 st visit to the start of jth visit to item i. Federgruen and Katalan (1996) prove that the steady state inter-visit time between two consecutive visits to i will converge to Vi, which has a mixed distribution of equilibrium excess distribution of individual inter-visit times V ji ; j ¼ 1; 2; . . . ; mi . Observe that mi X

E½V i ðjÞ ¼ ð1  qi ÞE½C

ð3:17Þ

If we use Vi instead of

in (3.17), we get

1  qi E½V i  ¼ fi

n P

i¼1

mi si þ

i¼1

M P

ð3:19Þ di

i¼1

If we assume that the variance of the vacations are negligible, then (3.13) reduces to the following approximate mathematical program:

Min

n X i¼1

"sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # bi si ci s2i þ 2 þ ki g i ai þ gi gi

subject to

n X

gi ¼ 1



3.4. Generation of the Item sequence for production

K i ð1  qÞ si

Now, using the Lagrangean multiplier approach it is possible to solve (3.20). That is, we need to choose an appropriate k such that the following hold:

bi si g i þ 2ci s2i k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  i ¼ k8i si 2 2 2 2g i ai g i þ bi si g i þ ci si and;

n X

gi ¼ 1

Based on the results presented in the foregoing, we summarize the procedure to compute production frequencies as follows:

In the next section, we provide a method to construct an appropriate production sequence for items, which conforms to the underlying assumptions in deriving the mathematical program (3.13).

ki qi E½X 2i  þ þ qi 2ð1  qi Þ E½X i 

E½X i  4 " # 3 ki VarðX i Þ ki k2i qi ð1  qi Þ E½X 2i  ð1  qi Þ þ  bi ¼ 2 2 E½X i  ð1  qÞ 2 " # 2 k2 ð1  qi Þ k2i ð1  qi Þ ð1  qi Þ2 ci ¼ i  3 4 ð1  qÞ2 ki ¼

In both the above cases, the problem reduces to choosing an appropriate k that satisfies (3.25) or (3.27) as appropriate and finding the corresponding gi ’s and thus, the relative frequencies, fi’s for i = 1, 2, . . ., N.

1. Compute ai, bi, ci and ki in (3.20) 2. Check if any of the cases presented in the previous are satisfied. P 3. Compute gi s from (3.24), (3.26) varying k such that igi = 1.

qi k3i VarðX i Þ E½X 2i  2 k2 q E½X 3i  þ ki VarðX i Þ þ ð1  qi Þq3i þ i i 2ð1  qi Þ E½X i  3 E½X i  !2

ð3:27Þ

ð3:21Þ

where

q2i k2i E½X 2i 

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bi pffiffiffiffi si ¼1 2 ai ðki þ ksi Þ

ð3:26Þ

3.3. Computation of production frequencies

ð3:20Þ

i¼1

ai ¼

bi si g i ki pffiffiffiffi ¼ þ k g 2i ai g i si

n X

mi si

ð3:25Þ

Case 2 Consider the case when ai g 2i  bi si g i  ci s2i . It follows that

ð3:18Þ

Denote by

g i :¼

i¼1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi ffi ci ¼1 si ki þ ksi

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bi pffiffiffiffi ) g i ¼ si ; and 2 ai ðki þ ksi Þ

j¼1

V ji

n X

ð3:24Þ

ð3:22Þ

ð3:23Þ

i¼1

(3.22) is a sixth degree equation and any numerical technique can be used to find a solution. However, by restricting to the following

Our procedure for sequence construction is along the lines of Federgruen and Katalan (1998). For a table size M and the corresponding absolute frequencies of items mi, the relative positioning of items can be treated as a scheduling problem with M jobs, with mi as the P number of items of type i, i = 1, 2, . . ., N and Ni¼1 mi ¼ M. Different heuristics have been proposed for this problem in various related contexts. The Golden ratio heuristic and Dobson’s makespan heuristic are some examples and are widely used in the scheduling literature. In the Golden ratio rule, the items are positioned such that the number of entries between two consecutive appearances of an item in the table is equalized. This construction is based on the Fibonacci sequence. The Dobson’s makespan rule finds a sequence, which equalizes the inter-visit times between consecutive appearances and is based on the power-of-two method which assumes that the frequencies mi, i = 1, 2, . . ., N can be rounded off to integers which are powers of two. Note that we derived the approximate mathematical program under the assumption that variance in inter-visit

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S. Garg et al. / Computers & Industrial Engineering 62 (2012) 21–28

time are negligible. In view of this assumption, Dobson’s rule is a possible choice for construction of item sequence. However, representation of frequencies in powers of two can be a serious limitation in many cases. To surmount this, we suggest a heuristic procedure which attempts to find a sequence that equalizes the inter-visit times. To this end, we start with an initial sequence generated by Golden ratio rule as described below. Since absolute frequencies can be determined only when the size of the table T is fixed, we determine the table size based on the relative frequencies fi, i = 1, 2, . . ., N by finding the smallest integer M such that the maximum error incurred by rounding off the absolute frequencies mi = Mfi is within a tolerance limit. Fix up such an M as the table size. With a given value of M, generate an initial sequence based on the Golden ratio rule. For this, create mi copies of item i, i = 1, 2, . . ., N and assign indices to these jobs from 1 to M. Associate an index I(k) to a job k, k = 1, 2, . . ., M as follows:

IðkÞ ¼ ku1 ðmodulo1Þ

ð4:1Þ pffiffi 51 2

where the Fibonacci number, u1 = and position the jobs in the table according to the increasing order of the indices, I(k). We will rearrange the terms in the above sequence such that for all items, the mean inter visit time between any two consecutive visits is the same. To achieve this, observe that if inter-visit times, V ji ; j ¼ 1; 2; . . . ; mi for the item i have identical mean values, then

E½V i  ¼

ð1  qi ÞE½C þ si mi

ð4:2Þ

We suggest the following recursive procedure for rearrangement of terms in the initial sequence so that the sequence generated conforms to our approximation.  Select the first item, say k, in the initial sequence and remove all mk copies of item k appearing in the sequence retaining the order of the remaining elements of the sequence.  Now concatenate the remaining sequence into mk  1 subsequences such that maximum of deviations from the target value (4.2) is minimum.  Repeat the procedure over all i = 1, 2, . . ., N until no improvement is observed over N consecutive repetitions.  Replace the existing entries for item k by the newly obtained sequence if it incurs a lower cost. The above heuristic can be improved further by incorporating efficient heuristics if we concentrate on sum of absolute deviations of individual inter-visit times of any item from its mean value. In this case, the problem can be formulated as a shortest path problem. However, in our methodology we do not attempt to invoke any such heuristics existing in the literature. 4. Experimental study In order to check the performance, and also the robustness, of the periodic policy derived from the above approximate mathematical program against cyclic policies, we have conducted extensive experimental study varying all relevant parameters of the model described in the foregoing. In our design of our experimental scenarios we have ignored the conditions cited in Case 1 and Case 2 in our selection of system parameters. In all the scenarios, we have considered four queues (i = 1, . . . , 4) assuming Poisson arrivals and exponential service times. Further, we have assumed that switching times between any pair of queues are sequence independent and are equal with value five time units. Similarly, in all the scenarios switching costs are assumed to be, again, sequence independent with value 800. The overall system utilization, P P viz., q ¼ 4i¼1 qi ¼ 4i¼1 lki is varied from 0.3 to 0.90, where i

ki ; i ¼ 1; . . . ; 4 denotes the arrival rate at i and rac1li, i = 1, . . ., 4 denotes the mean service time at queue i. The values of initial set of input parameters in different scenarios are listed in Table A.1 below. For the given set of parameters, the tabular policy corresponding to each scenario is found as detailed in the previous section. For Scenario 1, Scenario 2 the table size, M, turned out to be 32, whereas for other scenarios the table size is 48. The tabular sequence in each of the scenarios has been determined using the algorithm in Section 3.4. In our performance study, we use exhaustive cyclic policies for comparisons against tabular policies. An exhaustive cyclic policy follows a pre-determined polling order for serving the queue. Each queue once selected is served exhaustively. We evaluate the tabular policy of each scenario against all possible exhaustive cyclic policies of that scenario. In other words, the total cost is evaluated against each possible enumeration of cyclic policies and the best cyclic policy is identified. Further, each scenario is repeated for five simulation runs and the total cost for policies are averaged. Later, experiments are repeated by varying the ratio between backorder and holding cost for the queue with the highest utilization. Table A.2 below gives details of total cost comparisons averaged over all simulation runs against each scenario. It is interesting to note that in the scenarios corresponding to q = 0.3, 0.4, the total cost resulting from the best cyclic policy is almost close to that of the tabular policy of the scenario, sometimes outperforming the tabular policy. This can be mainly attributed to the approximation involved in our derivation of tabular policy. Also note that at higher system utilization, the gap between the best cyclic policy and the tabular policy is pronounced. At higher utilization rates, to balance the impact of bias due to holding costs, we tried to vary the ratio of backorder cost to holding cost of the queue(s) with higher values of utilization. Again, the same trend in performance can be seen from the results. These results clearly demonstrate relative effectiveness of tabular policies with respect to cyclic policies.

5. Conclusions and future work In production-inventory systems, static policies are of great value compared against their dynamic counter parts. In this paper the focus was on periodic tabular policies which may be viewed as generalized version of cyclic policies, and to derive a tabular policy by formulating an approximate mathematical program for total cost minimization over tabular policies. Later, a comparative performance study of the tabular policy with respect to cyclic policies was provided which demonstrated that though cyclic policies offer comparable performance with respect to tabular policies at low utilization rates, the performance gap widens significantly as system utilization tends to increase. The experiments have only been done on the class of exhaustive policies, which are not necessarily optimal from the perspective of system wide costs. In such scenarios, it would be valuable to experiment on the class of threshold policies. Below we list some directions for future research. In the present paper, we considered the production of multiple standardized items on a single machine with limited capacity and random set-up times under random demand and random production times: a problem that falls under the genre of stochastic economic lot-sizing problems (SELSP). The primary focus of the paper has been optimization of total cost, that is, the sum of set-up, holding, and back-logging costs, specifically under local lot-sizing policies—that is, base-stock policy derived based on the inventory level of the product that is currently set up. As a result of such localized policies, one single product, for which high demand arrives for a certain duration, may dominate the machine for a while

S. Garg et al. / Computers & Industrial Engineering 62 (2012) 21–28

leading to stock-outs, high costs, and high variability in cycle lengths of other products. Since the problem addressed herein is generally intended to model bottleneck machines in any production environment, it is important to have these lot-sizing decisions depend on global information, that is, on the stock levels of all individual product and on the state of the machine – a complex but practically relevant problem worth addressing. Another important question of practical relevance is to study impact of process variations on machine performance; a compre-

27

hensive sensitivity analysis with respect to input distributions can be carried out to that effect. In fact, in many situations, responsiveness to variations is of more practical relevance than the total operational costs. A feature of our practical problem that could not be modeled in the current version of our paper is the perishability nature of the finished product and of the raw material. This feature warrants development of production plans incorporating bounds on cycle lengths and safety-stocks – a theoretically challenging constrained optimization problem. We would wish to take up this part in our future work.

Table A.1 Input parameters. Scenario

Queue

ki

li

Holding cost

Backorder cost

Acknowledgements

q = 0.3

1 2 3 4

1.0 1.0 1.0 1.0

10 10 20 20

150 150 10 150

10 100 150 10

The authors would like to thank the two anonymous referees for their constructive comments and suggestions that helped to improve this paper.

q = 0.4

1 2 3 4

1.0 1.0 1.0 1.0

5 10 20 20

150 150 10 150

10 100 150 10

q = 0.5

1 2 3 4

1.0 1.0 1.0 1.0

5 5 20 20

150 150 10 150

10 100 150 10

q = 0.6

1 2 3 4

1.0 1.0 1.0 1.0

5 5 10 10

150 150 10 150

10 100 150 10

q = 0.7

1 2 3 4

1.0 1.0 1.0 1.0

5 5 5 10

150 150 10 150

10 100 150 10

q = 0.8

1 2 3 4

1.0 1.0 1.0 1.0

5 5 5 5

150 150 10 150

10 100 150 10

q = 0.9

1 2 3 4

1.0 1.0 1.0 1.0

3 5 5 5

150 150 10 150

10 100 150 10

Table A.2 Performance comparison with cyclic policies. Scenario

Cb Ch

q = 0.3

1 5 20

6263 3387 1690

6279 3190 1511

q = 0.4

1 5 20

8879 4800 2781

9424 4321 2933

q = 0.5

1 5 20

13,721 9875 8311

16,324 12,386 8765

q = 0.6

1 5 20

23,118 17,424 16,525

32,175 23,516 18,113

q = 0.7

1 5 20

57,823 44,525 38,331

62,111 58,124 49,110

q = 0.8

1 5 20

93,167 88,183 73,456

112,345 103,117 99,324

q = 0.9

1 5 20

162,348 142,879 139,312

198,006 173,238 157,437

Cost of tabular policy (Tp)

Cost of best cyclic policy (Tc)

Appendix A See Tables A.1 and A.2.

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