A disruption recovery model for a single stage production-inventory system

European Journal of Operational Research 222 (2012) 464–473

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

A disruption recovery model for a single stage production-inventory system H. Hishamuddin a,b, R.A. Sarker a,⇑, D. Essam a a b

School of Engineering and Information Technology, University of New South Wales, ADFA Campus, Northcott Drive, Canberra 2600, Australia Department of Mechanical and Materials Engineering, Faculty of Engineering and Built Environment, National University of Malaysia (UKM), 43600 Bangi, Selangor, Malaysia

a r t i c l e

i n f o

Article history: Received 21 June 2011 Accepted 15 May 2012 Available online 28 May 2012 Keywords: Disruption Economic production quantity Production- inventory system

a b s t r a c t This paper presents a newly developed disruption recovery model for a single stage production and inventory system, where the production is disrupted for a given period of time during the production up time. The model is categorized as a constrained non-linear optimization program which we have solved using an efficient heuristic developed in this paper. The model was also solved using an evolutionary algorithm and a comparison of the results from both methods was performed. The heuristic was able to accurately solve the model with significantly less time compared to the evolutionary algorithm. It can be shown that the optimal recovery schedule is dependent on the shortage cost parameters, as well as the extent of the disruption. The proposed model offers a potentially useful tool to help manufacturers decide on the optimal recovery plan in real time whenever the production system experiences a sudden disruption. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction A supply chain is a system that consists of facilities or entities whose activities involve the transformation of raw materials into finished products that are later delivered from the supplier to the end customer. More often than not, conventional supply chains are designed to operate smoothly in a problem-free environment. In reality, however, unexpected events such as machine breakdowns, transportation failures, labor strikes, and natural disasters are bound to happen and are often inevitable. As a result, this may cause disruptions at different levels of the supply chain, from the upstream to the downstream stages, and the effects are often times destructive. For instance, the 1995 earthquake that hit Kobe left vast damage to all of the transportation links in Kobe, and nearly destroyed the world’s sixth-largest shipping port. The 7.2 scale Richter quake substantially affected Toyota, where an estimated production of 20,000 cars, equivalent to $200 million worth of revenue was lost due to parts shortages (Sheffi, 2005). Without a proper response to such events, a manufacturer would have to incur considerably higher additional costs to recover from the negative impacts of disruption. Realizing the potential losses from such events, enterprises have recently shown a growing interest in incorporating risk management into their operations. Two common strategies to manage the risk of disruptions include mitigation and contin-

⇑ Corresponding author. Tel.: +61 4 25886611; fax: +61 2 62688276. E-mail addresses: [email protected] (H. Hishamuddin), r.sarker@adfa. edu.au (R.A. Sarker), [email protected] (D. Essam). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.05.033

gency tactics (Tomlin, 2006). The former strategy requires a firm to act in advance of a disruption, while the latter is taking action only during the occurrence of a disruption. Implementing mitigation and contingency tactics is not free; rather it involves a cost that influences the attractiveness of an optimal strategy for a given firm. In the literature on supply-disruption where the supplier is not always available, numerous studies have been performed for inventory models under the continuous review framework with deterministic demand, where supplier availability is modeled as an alternating renewal process (Berk and Arreola-Risa, 1994; Parlar and Berkin, 1991; Parlar and Perry, 1995, 1996). Under the periodic review framework, Parlar et al. (1995), Song and Zipkin (1996), and Ozekici and Parlar (1999) have analyzed an inventory model with backorders in a random supply environment modeled as a Markov chain. There also exist works that study both supply and demand disruption in their model (Weiss and Rosenthal, 1992; Xiao and Yu, 2006; Ross et al., 2008). Tomlin (2006) examines the optimal strategy for a single product system with two suppliers: one that is unreliable and another that is reliable but expensive. Schmitt et al. (2010) and Chen et al. (2012) extend the work of Tomlin (2006) to study the system with stochastic demand. Furthermore, Schmitt and Snyder (2012) conducted a study on the comparison between single period and multiple period settings for an inventory system subject to yield uncertainty and supply disruption. To do this, they extended the paper by Chopra et al. (2007) which only considered the single period case. Other variations of supply disruptions in stochastic inventory models include the works by Arreola-Risa and DeCroix (1998), Li et al. (2004), Mohebbi (2003, 2004), Mohebbi and Hao (2006),

H. Hishamuddin et al. / European Journal of Operational Research 222 (2012) 464–473

Mohebbi and Hao (2008), Moinzadeh and Aggarwal (1997), and Qi et al. (2009). Most of the papers cited above consider inventory mitigation as the disruption-management strategy, in which additional inventory is held in the system for the entire period to protect against disruptions. These studies design their inventory models such that supply uncertainty occurrences are included in the original objective function by modifying the original non-disruption models. The majority of the studies are likely to result in stationary higher ordering quantities or bigger stock levels for the entire planning horizon, which may cause a firm to incur unnecessarily high holding costs over the long run. Thus, inventory mitigation tactics may not be of interest for firms that prefer a more cost effective solution to managing disruptions. An alternative solution to this problem would be the use of contingency or recovery tactics. Studies on optimal recovery strategies for disruptions, on the other hand, are rather scarce. In the production and inventory literature with regards to the Economic Lot Scheduling Problem (ELSP), Gallego (1994) considered how to schedule production after a single schedule disruption by proposing a base stock policy. His work was extended by Eisenstein (2005) who introduced the Dynamic Produce-Up-To (Dynamic PUT) policies. Tang and Lee (2005) proposed rules for recovering from a machine breakdown or other forms of interruption using relaxation and heuristic methods. Rather than avoiding stock-out costs, their focus was to minimize inventory holding cost in front of the production facilities. Recovery strategies to demand disruptions have also been explored in the works by Qi et al. (2004) and Yang et al. (2005). Xia et al. (2004) developed a recovery strategy for an Economic Production Quantity (EPQ) system subject to disruption in the form of parameter changes. The purpose of their study was to recover from the disruption within a short time window, spanning two to three production cycles, at minimum disruption costs (incorporating penalty costs for deviations in the original schedule). These studies propose various methods of schedule recovery, but none examines the optimal recovery duration, while minimizing the recovery costs, given a production-inventory system with partial backlogging options. To the best of our knowledge, we are the first to consider this as it has never been reported in the previous literature. This paper is an extension of the work by Xia et al. (2004). It presents a newly developed real-time recovery model for a single stage production-inventory system subject to disruption. We consider a production facility that manufactures a single product in batches at a constant time interval following the EPQ model. However, it is assumed that a random disruption occurs during a cycle that disables the production from running as scheduled. This may be due to parts shortages, a machine breakdown or any other form of interruption. After the disruption occurs, a specified duration, known as the recovery time window, is allocated for recovery. During this window, changes are made to the original production schedule in order to satisfy customer orders. Similar to other disruption management models, the original production schedule is restored by the end of the recovery time window, focusing on the attempt to preserve, as much as possible, the original operational plan. The production facility we consider faces four types of costs: setup cost, inventory holding cost, backorder cost, and lost sales cost. The objective of the model is to determine the optimal manufacturing batch sizes and optimal recovery duration for the production run in the recovery time window (recovery schedule), so that the expected total cost is minimized. The model can be classified as a constrained non-linear programming model and is solved using a set of heuristics that has been developed as part of this study. Considering the complexity of this model, we have also implemented a class of evolutionary algorithms, known as the evolution strategy (Runarsson and Yao, 2000), to also solve the model. To judge the quality of solutions, a comparison of the

465

results from the two methods was performed. The results showed that the optimal recovery schedule is highly dependent on the relationship between the backorder and lost sale costs, as well as the length of the disruption. Given the various risks and difficulties that firms face nowadays, the proposed model could prove to be a very useful tool for manufacturers who wish to make quick decisions on an optimal recovery plan after the occurrence of a disruption. The main contributions of the paper can be summarized as follows: 1. The development of a recovery model for a production-inventory system with disruption in the form of schedule interruptions that is not known a priori. Additionally, the model considers stock-out costs consisting of both backorder and lost sale costs, as opposed to the penalty costs or complete backlogging/lost sales considered in previous works. 2. The introduction of an efficient heuristic approach that determines the optimal recovery plan for disruption, subject to the system’s costs and constraints. The heuristic is able to decide the new recovery plan in a short amount of time. For better understanding of the problem, we provide the definition of the different terms used in this paper as below. Disruption: Any form of interruption in production. Disruption duration: The duration of production interruption. Recovery window: The duration allocated for recovery, where changes are made to the original schedule. The original schedule is recovered by the end of this duration. Recovery plan: The new schedule which includes the optimal quantity of backorders and lost sales in the recovery time window and the optimal number of production cycles required for recovery. Back order: The portion of an order that cannot be delivered at the scheduled time, but will be delivered at a later date when available Lost sale: When a demand occurs and the item is out of stock, the customer will not wait for the stock to be replenished so the demand is lost. The remainder of this paper is structured as follows. In Section 2, the model formulation is presented. The solution approach is proposed in Section 3. Section 4 discusses the related computational results and analysis of the quality of the solution. Section 5 extends the study to the multiple disruption case. Finally, Section 6 summarizes the paper and provides directions for future research.

2. Model development In the following sub-sections, we derive a mathematical model accounting for disruption in a single stage production-inventory system. The first subsection will discuss the system description of the model, which will then be followed by development of the cost model.

2.1. System description In this research, we assume that the current non-disruption production-inventory system is based on the EPQ system, of lotfor-lot production, similar to Sarker and Khan (2001). A typical inventory system of this type is shown in Fig. 1. The following notations are used in developing the model:

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Fig. 1. Original production schedule under ideal conditions.

A D H P Q Td q T0 u te tf T

q B L Xi Ti St d

setup cost for a cycle ($/setup) demand rate for a product (units/year) annual inventory holding cost ($/unit/year) production rate (units/year) production lot size in the original schedule (units), as in (1) disruption period pre-disruption production quantity in a cycle production time for q production down time for a normal cycle (setup time + idle time), ðT  Q =PÞ start of recovery time window end of recovery time window production cycle time for a normal cycle (Q/D) production up time for a normal cycle (Q/P) unit back order cost per unit time ($/unit/time) unit lost sales cost ($/unit) production quantity for cycle i in the recovery window (units) production up time for cycle i in the recovery window ðX i =PÞ setup time for a cycle idle time for a cycle

The fundamental cycle time, T, is comprised of the production up time, q, and down time,u. There exists production downtime in the system due to the assumption that demand is less than the available production capacity ðD < PÞ. Each cycle has a setup time; therefore, a down time for a production cycle is the sum of setup time ðSt Þ and idle time ðdÞ. The optimal production lot size (Q) for the above ideal system is formulated as follows:

Q¼

rffiffiffiffiffiffiffiffiffi 2AP H

ð1Þ

In our model, however, we assume that a disruption occurs in the above system and interrupts the production from running as scheduled. Following the disruption, a recovery time window is allocated to allow for schedule recovery. Recovery takes place by utilizing the production idle times in the original schedule. Like other disruption management models, the term recovery is defined as restoring the original production schedule within a short time period, while minimizing the relevant costs. In the presence of disruption, production can only resume when the problem is rectified. Hence, there may be some shortages in the system due to the production delay created by the disruption. Unsatisfied customer demand during this stock out period may become backorders that are produced during the recovery time window, and the remainder will become lost sales. Additionally, lost sales may occur during any of the cycles in the recovery time window. The concept of the recovery time window was adopted from Xia et al. (2004), although our model assumes that the disruption occurs randomly and is not known in advance. In other words, the firm does not have the opportunity to prepare in advance for a disruption by increasing inventory or safety stock before the actual disruption occurs. We define the recovery time window as n normal production cycle times from the start of a disruption. During the recovery period, the production schedule is modified such that the length of n cycles in the recovery schedule is equal to n cycles in the original schedule. Moreover, the recovery schedule includes the disruption length, T d . The backorder costs, B, will be a function of the time delayed in units of $/unit/unit time. One of the advan-

Fig. 2. Production inventory curve with disruption.

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   X1 þ q Q ¼ B ðX 1 þ qÞ T d þ St þ  P P

tages of our model is the ability to determine the optimal quantity of backorders and lost sales in each recovery cycle. The problem described above is illustrated in Fig. 2. The dotted line represents the inventory curve for the original schedule in the absence of disruption, while the solid line represents the inventory curve for the new recovery plan with the presence of disruption. The disruption duration ðT d Þ is indicated by the shaded area under the graph. Additionally, the recovery time window is shown as the interval between t e and t f . One point is worth mentioning, goods that are produced during the recovery time window are considered as backorders. If otherwise (the goods could be shipped on time), this would mean that the original schedule has been restored, thus t f should be positioned earlier and n should be smaller. This condition is consistent with our assumption that the schedule is restored at tf .

The sum of all the cost components above gives the total relevant costs of the recovery plan. Therefore, the mathematical model is presented below:

2.2. Mathematical representation

MinTCðX i ; nÞ ¼ðA  nÞ þ

In this subsection, a cost model is developed for the productioninventory system described earlier. Let T d be the disruption period (see Fig. 2). Without loss of generality, we assume that T d is less than the normal production cycle time, T. We define the decision variable X i as the production quantity for cycle i in the recovery time window and T i as its respective production time, where i ¼ 1; 2; . . . ; n. Furthermore, we define q as the pre-disruption production quantity in the disrupted cycle and T 0 as the production time for q. Note that the disruption may occur anywhere in a cycle, either at the start or in the middle of a cycle, where a certain quantity of goods has already been produced. For the latter case, q is a positive integer. For our particular model, the time horizon is finite, such that only the costs in the recovery window are considered. The setup cost equation is rather straight forward and can be obtained by:

ð2Þ

The inventory holding cost is derived as the unit inventory holding cost, H, multiplied by the total inventory during the recovery time, which is equivalent to the area under the curve. This is calculated as: 

 1 1 1 1 qT 0 þ qðT d þ St Þ þ qT 1 þ ðX 1  qÞT 1 þ X 2 T 2 þ    þ X n T n 2 2 2 2   H q X1 X1 X2 Xn q  þ 2qðT d þ St Þ þ 2q þ ðX 1  qÞ  þ X2  þ    þ Xn  ¼ 2 P P P P P ! 2 2 2 2 H q X1 X1 X1 X2 X ¼ þ 2qðT d þ St Þ þ 2q  q þ þ þ  þ n 2 P P P P P P !

¼

n H q2 X1 1 X X2 þ 2qðT d þ St Þ þ q  þ 2 P P i¼1 i P

¼

n X H q2 þ 2qP  ðT d þ St Þ þ q  X 1 þ X 2i 2P i¼1

! ð3Þ

Next, the backorder cost formulation can be derived by multiplying the unit backorder costs, B, with the backorder units of each cycle i and it’s time delay, given that the delay is a positive value:

¼ B ðX 1 þ qÞ  Delay1 þ

ð6Þ

Finally, the lost sales cost is obtained as:

¼ LðnQ  ðX 1 þ qÞ  X 2      X n Þ ! n X Xi ¼ L nQ  ðX 1 þ qÞ 

ð7Þ

i¼2

n X H q2 þ 2qP  ðT d þ St Þ þ q  X 1 þ X 2i 2P i¼1

!!

    X1 þ q Q  þ B ðX 1 þ qÞ T d þ St þ P P   n i X X Xj q Q þ X i T d þ i  St þ  uði  1Þ þ i P P P i¼2 j¼1 þ L nQ  ðX 1 þ qÞ 

n X Xi

!!!

!!

ð8Þ

i¼2

Subject to the following constraints:

X1 6 Q  q Xi 6 Q

ð9Þ for i ¼ 2; 3; . . . ; n

ð10Þ

n X X i 6 PðnT  nSt  T d Þ  q

ð11Þ

i¼1

¼ A  ðNumber of setupsÞ ¼ A  ðnÞ

¼H

!!   n i X X Xj q Q  uði  1Þ þ X i T d þ i  St þ þ i P P P i¼2 j¼1

n X

!

X i  Delayi

X1 þ q þ

n n X X X i P nTD  nQ  Xi  q i¼2

! ð12Þ

i¼1

i X X j P i  Q þ ði  1ÞP  u  P  T d  P  iSt  q

ð13Þ

j¼1

All decision variables are non-negative and integers. The objective function (8) comprises of the four cost components mentioned earlier, each separated in parenthesis. Eqs. (9) and (10) ensure that the production lot sizes in the recovery schedule are less than that of the original schedule due to the delivery and transportation requirements. Eq. (11) represents the production capacity constraint, Eq. (12) ensures that all demand during the recovery period is accounted for and Eq. (13) guarantees a non-negative delay for backorders. By solving the above model for X i and n subject to the constraints (9)–(13), one can obtain the optimal recovery plan for the production system under disruption. Without disruption, this model will reduce to the original EPQ model as in (1) that was presented earlier. 2.3. Special cases

ð4Þ

Two special cases of the above are briefly discussed below.

ð5Þ

2.3.1. Disruption at the start of a cycle A special case of the model is considered here in which the disruption occurs at the beginning of a production cycle. For this case, goods have not been produced before the disruption, or similarly, q ¼ 0. The mathematical model for this particular case can be derived as:

i¼2

The delay for backorders in each cycle i, is calculated below:

Delayi ¼ T d þ i  St þ

i X Xj j¼1

P

þ

  q Q  uði  1Þ i P P

Thus, the backorder cost can be derived as:

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MinTCðX i ; nÞ ¼ ðA  nÞ þ

n HX X2 2P i¼1 i

!

in which case, there will be some amount of lost sales. However, for the case when backorders are more attractive than lost sales, it would be desirable for X 1 ¼ Q  q and X i ¼ Q , 8i; i – 1, where

!!!   n i X X Xj Q  uði  1Þ X i T d þ i  St þ i P P i¼1 j¼1 !! n X Xi þ L nQ 

þ B

n¼

i¼1

ð14Þ

P  Td PðT  St Þ  Q

ð25Þ

is the minimum number of cycles to fulfill the demand without any lost sales. Since n must be an integer, we use dne, which completes the proof. h

Subject to the following constraints:

Xi 6 Q n X X i 6 PðnT  nSt  T d Þ i¼1 n n X X X i P nTD  nQ  Xi i¼1

ð15Þ ð16Þ !

The next corollary is an opposite consequence of Theorem 1. Corollary 1. For a given A, H, B and L, the number of recovery cycles PT d when there exists lost sales in the recovery will be n < PðTS t ÞQ schedule.

ð17Þ

i¼1

i X X j P i  Q þ ði  1ÞP  u  P  T d  P  iSt

ð18Þ

j¼1

Proof. Clearly, lost sales will be encouraged for quick recovery when Bis significantly larger than L. Therefore, some amount of backorders will become lost sales resulting in X 1 < Q  q or X i < Q , 8i; i – 1 implying that

All decision variables are non-negative and integer.

proves that n < 2.3.2. Equal lot sizes In addition to the case in which disruption occurs at the beginning of a cycle, we have also considered the scenario where the revised lot sizes are equal in each cycle in the recovery plan. In mathematical terms, this can be noted as X 1 ¼ X 2 ¼    ¼ X, thus T 1 ¼ T 2 ¼    ¼ T x . The mathematical model for this type of scenario is formulated as follows:   H  n  X2 MinTCðX; nÞ ¼ðA  nÞ þ 2P   n n X ðn þ 1Þ  St þ ðn þ 1Þ  þ BX 2 2 P   n Q n  ðn þ 1Þ   ðn þ 1Þ  u þ BXnðT d þ uÞ þ ðLnðQ  XÞÞ 2 P 2

Subject to : X 6 Q

ð19Þ

ð20Þ

nX 6 PðnT  nSt  T d Þ

ð21Þ

nX P nTD  nðQ  XÞ

ð22Þ

2.4. Analysis of the model 2.4.1. Bounds for n The following theorem gives bounds on the number of recovery cycles (n) that is used in developing the solution procedure for the main model. Theorem 1. For a given A, H, B and L, when backorders are more attractive than lost sales, the number of recovery cycles will be dne, PT d where n ¼ PðTS . t ÞQ Proof. First, using Eq. (11) and solving for n, one will obtain

Pn nP

i¼1 X i

þ P  Td þ q PðT  St Þ

PT d PðTSt ÞX i

PT d < PðTS from (25). This t ÞQ

when there is lost sales in the optimal

solution. h 2.4.2. Effects of d and Td on recovery performance The approach of using idle time to hedge against lost time due to disruption has been used in several works in the literature (Gallego, 1994; Tang and Lee, 2005; Eisenstein, 2005) and this approach has been utilized in our model as well. Production systems are commonly designed to incorporate idle times (d) in their schedules to compensate for various uncertainties that arise (Bourland and Yano, 1994). These idle times, or slack capacity as the term used by some authors, can be utilized to help absorb uncertainties due to variances in manpower, equipment, and demand. Moreover, slack capacity in the system is essential for schedule recovery, otherwise, recovery from a disruption would be impossible (Eisenstein, 2005). This agrees with the claim made by Yang et al. (2005) that Disruption Management is valuable to firms that have a certain amount of spare capacity in their production systems. Having spare capacity is one issue, but the size of a firm’s spare capacity is another issue worth highlighting. According to Tomlin (2006)city has an influence on the firm’s ability to recover from a disruption. Production schedules with smaller spare capacity or percentage idle time would generally take longer time to recover. This observation would also be valid for our model. In the paper by Tang and Lee (2005), a condition must hold where the sum of the idle times is required to be larger than the disruption duration in order for the original schedule to be restored (we refer the reader to the first theorem in Tang and Lee (2005)). This condition would also be applicable to our model, but only for the case when all shortages are backordered. If some lost sales are desirable in the optimal solution, this condition would no longer be necessary, as demonstrated in Corollary 2 below.

ð23Þ

From Eqs. (9) and (10), X 1 6 Q  q and X i 6 Q , 8i; i – 1 indicates the respective upper bounds. Substituting these upper bounds in (23), we get

nP

PT d PðTSt ÞQ

nQ þ P  T d PðT  St Þ

ð24Þ

PT d Rearranging (24) will reduce to n P PðTS . For values of t ÞQ X1 < Q  q or Xi < Q , 8i; i – 1, it is clear that PT d PT d PðT  St Þ  X i P PðT  St Þ  Q . This implies that PðTS < , PðTSt ÞQ t ÞX i

Corollary 2. When lost sales exist in the optimal solution, the sum of the idle times (d) in the recovery window is less than the disruption duration, T d . Proof. Recall that Eq. (25) from Theorem 1, which was a result when all shortages are backordered. Rearranging Eq. (25), one will obtain

n¼

Td Td or similarly n ¼ d T  St  QP

ð26Þ

H. Hishamuddin et al. / European Journal of Operational Research 222 (2012) 464–473

However, with the presence of lost sales, the recovery period n will be shorter. Let us denote this new n as n_ls. It is clear that n ls < Tdd . Rearranging this expression, we obtain n ls  d < T d , which implies that the sum of idle times in the recovery time window for the lost sales scenario is less than the disruption duration, thus proving the above corollary. Another point worth noting is, regardless of the relationship between B and L, recovery requires longer time when the disruption length ðT d Þ increases, given that the other parameters are fixed. The next corollary proves this observation. h

Corollary 3. For a given A, H, B, and L, the longer the disruption duration, T d , the longer it will take to recover. Proof. It is clear that after rearranging (26) one will obtain T d ¼ n  d. Taking into account the lost sales case simultaneously, one may replace it with this term:

Td P n  d

ð27Þ

Notice that the parameter d is fixed for a particular production system, whereas T d and n may change with different disruption occurrences. Referring to (27), an increase in T d will consequently result in an increase in n. This shows that number of cycles taken to recover is larger when the disruption length is longer and completes the proof. h

obtained as multiples of the original schedule’s optimal lot size (Q) and are then allocated into the new recovery schedule. If a fraction of Q exists (/required), the optimal solution would be to assign this fraction of the lot size to the first cycle. This minimizes the delay for back orders, allowing minimum backorder costs to be incurred. Any unmet demand will become lost sales. The third strategy follows the steps of Strategy 2; however, it allocates the production lot sizes according to the required minimum amount of backorders such that all delays are positive. The idea behind the last two strategies were developed from analysis of the model’s hard constraints, which are Eqs. (11) and (13). The heuristic begins by evaluating the relationship between the backorder cost (B) and the lost sales cost (L). If B is less than or equal to L, the problem would be solved using Strategy I. If L is less than B, a search procedure is performed by solving the model using all the above strategies and the strategy with the lowest TC value is chosen as the optimal solution. Alternatively, we also list out the heuristic steps used to solve the model when n is a user input. The detailed procedure for the heuristic is presented below: Algorithm 1. n is a decision variable Step 1: Step 2: Step 3: Strategy I:

3. Solution approach 3.1. An overview One unique feature of our model is the ability to determine the optimal recovery duration for a particular disruption. In this paper, the number of recovery cycles, n, must be an integer. Notice that if one can fix the value of n, solving the model will be easier. However, our model also considers n to be a decision variable, thus increasing the complexity of the model. The solution approach, which will be presented in the following subsection, is capable of solving the model for both of these cases (that is n as either a decision variable or a given value). This allows a manager of a firm to choose between fixing the recovery period to a desired duration, and obtaining the optimal recovery duration as decided by the model.

Step 4: Strategy II:

Initialize the parameters and variables Calculate the maximum number of cycles to recover from a disruption, n_max, from Eq. (25) If B 6 L, set n ¼ dn maxe (a) Set the production quantity for the first cycle equal to Q minus q, i.e. X 1 ¼ Q  q (b) Set the production quantity for all the remaining cycles equal to Q, i.e. X i ¼ Q (c) Calculate TC (d) Record the solution and go to Step 5 If B > L, set K ¼ 1 and n ¼ dn maxe þ 1  K (a) Calculate the available production capacity, as in Eq. (28):

Av C ¼ PðnT  nSt  T d Þ  q

ð28Þ

(b) Find the lot size allocation as a multiple of Q, as in Eq. (29)

MQ ¼

3.2. A proposed heuristic In this study, an efficient heuristic approach has been developed as a solution approach to determine the optimal values of X i and n for the recovery model. The recovery model is a complex constrained mixed integer program, thus needs a specialized software to solve the model. Having a heuristic that consists of simple steps will eliminate the hassle of acquiring such a software and will facilitate firms to obtain a recovery schedule in real-time. Furthermore, as we want to study SC problems with multiple nodes in the future, such a heuristic is imperative to ensure the ease of its application to a larger problem scope. The heuristic consists of three main strategies. Strategy 1: the total back order plan, Strategy 2: the available capacity allocation and Strategy 3: the minimum backorder requirement. The first strategy considers that all shortages become backorders and the recovery time window is extended to the maximum value of n. In the second strategy, the feasible total production quantity is determined based on the total available time or similarly, available capacity, during the recovery period. The production lot sizes are

469

Strategy III:

Av C Q

ð29Þ

MQ denotes the number of lot sizes in the recovery schedule with the quantity Q. If MQ is noninteger, the fraction is multiplied by Q to obtain the production quantity for cycle 1 ðX 1 Þ. If MQ þ q=Q > n, ignore Strategy II and go to Step 4(e). (c) Set the production quantity for the remaining cycles as Q, i.e. X 2 ; . . . ; X n ¼ Q (d) Calculate TCðKÞ using Eq. (8) (e) Calculate the minimum backorders required using Eq. (30)

BRq ¼ n  Q þ ðn  1ÞP  u  P  T d  P  nSt  q ð30Þ

Step 5:

(f) Repeat steps (b)–(d) but replacing AvC with BRq (g) If K ¼ 1 or similarly n ¼ dn maxe execute Strategy I (Step 3 (a)–(c)) and choose the solution from Strategy I, II or III with the lowest TC value. Otherwise, choose the solution from Strategy II or III with the lower TC value. (h) If TCðKÞ < TCðK  1Þ, set K ¼ K þ 1, and repeat Step 4 (a)–(h) Otherwise, record the solution and go to Step 5 Stop

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H. Hishamuddin et al. / European Journal of Operational Research 222 (2012) 464–473 Table 2 Parameters for five test instances and the results for the optimal n value.

Algorithm 2. n is a user input Step 1: Step 2: Step 3:

Step 4:

Step 5:

Initialize the parameters and enter the user input (n) Calculate n_max from Eq. (25) If B 6 L, If n ¼ dn maxe Solve using Strategy I Else, if n < dn maxe Solve using Strategy II If L < B, If n 6 dn maxe Solve using Step 4 (a)–(g) in Algorithm 1 Stop

4. Computational experience In this section, several experiments are presented to demonstrate the applicability and performance of the proposed model and method. The mathematical model presented in this paper was solved using the heuristic developed in Section 3 for 500 random test problems. The test problems were generated by randomly changing the cost parameters, the pre-disruption quantity and the disruption duration for the given intervals. Table 1 summarizes the range of data of the problems that were tested. In addition, we have also tested the heuristic for the bounds of each parameter by fixing either its upper or lower bound value, and then randomly generating data for the rest of the parameters based on the given ranges. We formulated 600 test problems, where 12 data sets with 50 problems in each set were randomly generated. Notice that the upper and lower limits for parameters T d and q are shown in terms of other parameters, which were determined based on the assumptions made for this model. In order to judge the quality of the heuristic solutions, the same test problems as above were solved using a population based evolutionary algorithm known as evolution strategy ðl; kÞ-ES with stochastic raking. ES is a population based stochastic search algorithm widely used for optimization problem solving. We have chosen ðl; kÞ-ES with stochastic ranking as an alternative method to compare our system’s solutions as the algorithm is suitable for constrained optimization. Moreover, comparing solutions is necessary to evaluate the effectiveness of our heuristic. If an error exists, this will indicate that the heuristic is incorrect. Due to space limitations, we refer interested readers to Runarsson and Yao (2000) for stochastic ranking and to Runarsson and Yao (2000) and Schwefel (1995) for constraint handling in optimization. Ten independent runs were performed based on a (10, 200)-ES with a total of 1750 generations before termination. The value of Pf used was 0.45. As used in Runarsson and Yao (2000), P f is the probability of using only the objective function for comparisons in ranking. The solution procedure for both methods was coded in MATLAB and executed on an Intel Core Duo processor with 1.99 GB RAM and a 2.66 GHz CPU. The results of the experiments are ideal. There are no errors or differences between the heuristic and ES solutions for the totally random experiment. For the ran-

Table 1 Range of data for randomly generated test problems. Parameters

Range of data

Setup cost, A Inventory holding cost, H Back order cost, B Lost sales cost, L Disruption duration, T d Pre-disruption quantity, q

[20, 500] [1, 10] [10, 100] [1, 100] [u  St, T] [0, Q]

Test instance

A

H

B

L

Td

St

q

n

1 2 3 4 5

20 20 50 20 50

1.2 1.2 2.4 1.2 2.4

10 10 10 100 100

15 15 15 1 1

0.003 0.003 0.03 0.003 0.003

0.000057 0.000057 0.000057 0.000057 0.000057

0 100 100 100 100

6 6 5 3 3

dom experiment with fixed bounds, the average error was 0.0001%, which can be considered negligible. 4.1. Numerical examples For illustrative purposes, five different test instances were generated by arbitrarily changing the cost parameters and these were later solved by the proposed heuristic and ES. The parameters for the five test instances are shown in Table 2. Table 3 summarizes the results of the experiment, which shows the best objective value and the running time found by the two approaches. Apart from the ability to produce the quality solutions, the heuristic has a significantly lower running time compared to the ES method. Each of the test instances can be solved by the proposed heuristic in less than one second. The total cost function, TC, with respect to n was plotted using the data of example 4 and is shown in Fig. 3. The plot demonstrates a convex function curve, which proves that the model procedure developed in this paper provides good solutions. An analysis of the results shows that the solution highly depends on the relationship between the backorder cost and the lost sales cost parameters. The optimal quantity of backorders and lost sales would be a tradeoff between the backorder and lost sales cost. For a certain range of backorder cost that is lower than the lost sales cost, it is found that backorders are more attractive. Thus the optimal X 1 and X i values are found to be equal to their upper limits, which is Q-q and Q respectively. On the other hand, when the backorder cost is significantly higher than the lost sales cost, the solution will have some amount of lost sales. In this case, there will be a cycle where the production quantity, X i is less than Q, which will constantly be the first cycle in the recovery schedule so that the backorder cost is minimized. Interestingly, the optimal recovery duration, n, will be shorter for the latter scenario than the former. An explanation for this finding is as follows: As the backorder cost becomes larger, it is more cost effective to have some lost sales in the recovery schedule. Thus, when a portion of the demand becomes lost sales, the quantity to be produced (the sum of X i Þ is reduced and the time for recovery will be shorter. One point worth noting is that the optimal recovery duration is independent of the q values, given that the other parameters remain the same (see test instances 1 and 2). To further illustrate the above findings, an analysis was performed to show the effect of increased backorder cost, B, on n, backorder quantity and lost sales quantity, when the lost sales cost, L, is significantly low (see Fig. 4). The base parameters were set at P ¼ 5  106 ; D ¼ 4  106 ; A ¼ 20; H ¼ 1:2; St ¼ 0:000057; T d ¼ 0:003 and q ¼ 0. With L fixed at $1/unit and B increasing from $10 to $5000, it can be observed that the lost sales quantity is found to increase, while the backorder quantity eventually reduces to zero. Notice also that the optimal n value gradually decreases as B increases. 4.2. Sensitivity analysis A sensitivity analysis was carried out to show the effects of the various parameters on the total cost for the proposed model. We

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H. Hishamuddin et al. / European Journal of Operational Research 222 (2012) 464–473 Table 3 Comparison of the results between the heuristic and ES. Test instance

Total cost

1 2 3 4 5

Heuristic

ES

1468.322 1747.195 1468.535 9932.575 10272.981

1468.322 1747.195 1468.535 9932.575 10272.981

Running time

0.0 0.0 0.0 0.0 0.0

ES

0.3 0.2 0.2 0.2 0.2

1 minute 8 seconds 1 minute 25 seconds 1 minute 40 seconds 50.6 seconds 47 seconds

Inventory Holding Cost (H) vs Total Cost (TC)

12000

2500.00

11000

2000.00

TC ($)

10000 9000 8000

1500.00 1000.00 500.00

7000

n

10 9 8 7 6 5 4 3 2 1 0

9.5

95

85

5000

75

1000

0.00 65

500

2000.00 55

0.00

4000.00

45

10000.00

6000.00

35

20000.00

8000.00

25

n

30000.00

200

10000.00

15

LSQ (units)

40000.00

12000.00

5

50000.00

150

8.5

Back Order Cost (B) vs Total Cost (TC) 14000.00

TC ($)

BOQ (untis)

n (cycles)

Quantity (units)

60000.00

100

7.5

Fig. 6. The change of TC with respect to H.

Effect of B on n , back order quantity (BOQ ) and lost sale quantity ( LSQ )

50

6.5

H ($/unit/unit time)

Fig. 3. TC vs. n using the data from Example 4.

10

5.5

5

4.5

4

3.5

3

2.5

2

1.5

0.00 1

0.5

6000

Heuristic (seconds)

TC versus n

13000

TC

% Error

B ($/unit back order)

Backorder Cost, B ($/unit/unit time)

Fig. 7. The change of TC with respect to B.

Fig. 4. Effect of B, backorder quantity, lost sales quantity and TC when L is fixed at $1/unit.

Lost Sales (L) vs Total Cost (TC)

500

440

380

320

260

200

140

80

20

A ($/setup)

100.5

0.00

90.5

0.5

500.00

80.5

1000.00

70.5

1500.00

60.5

2000.00

50.5

2500.00

40.5

TC ($)

3000.00

30.5

TC ($)

3500.00

20.5

4000.00

1600.00 1400.00 1200.00 1000.00 800.00 600.00 400.00 200.00 0.00 10.5

Setup Cost (A) vs Total Cost (TC)

L ($/unit lost) Fig. 8. The change of TC with respect to L.

Fig. 5. The change of TC with respect to A.

assume that P ¼ 5  106 and D ¼ 4  106 for all cases. Other parameters, if not varied, are fixed as A ¼ 20; H ¼ 1:2; B ¼ 10; L ¼ 15; St ¼ 0:000057, and T d ¼ 0:003. Fig. 5 shows the behavior of the total cost for different values of setup costs varying from

$20 to $500. It can be observed that the total cost for the system increases with an increase in setup cost when the other parameters are fixed. A similar behavior of the total cost is observed as holding costs increases from $0.5 to $10, as indicated in Fig. 6. Fig. 7 depicts that the backorder cost has an increasing effect on total costs as

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schedule, and no be the cycle where the new disruption occurs, measured from the start of first recovery schedule. Note that the new recovery time window will start from cycle no . For the new schedule, T d is used as the new T d value in the heuristic. All of the parameters introduced earlier are needed to compute T d , which is formulated as follows:

Disruption Duration (Td) vs Total Cost (TC) 2500.00

TC ($)

2000.00 1500.00 1000.00

"

500.00

T d ¼ 0.00347

0.00307

0.00267

0.00227

0.00187

0.00147

0.00067

0.00107

0.00

ð31Þ

The procedure to obtain the new recovery schedule is as follows:

Td (time)

1. Compute the value T d . 2. Initialize all parameters of the heuristic that were presented in Section 4.1. 3. Input T d and q ¼ qr along with the other relevant parameters into the heuristic. 4. Solve to obtain the new recovery schedule.

Fig. 9. The change of TC with respect to T d .

Initial Production Quantity (q) vs Total Cost (TC) 1472.00 1471.00

TC ($)

# j 1 r r1 n r h X X X qj X ji X qi  ðn0  1Þ  T þ r T dj þ þ P P P j¼1 j¼1 i¼1 j¼1

To demonstrate this further, we provide the following numerical example: Consider a disruption that occurred as depicted in Test Instance 2 (see Section 4). Moreover, consider that a second disruption occurs at the start of the 3rd cycle of the recovery schedule with duration of 0.0025. The parameter values for this example are as follows:

1470.00 1469.00 1468.00 1467.00 1400

1200

1000

800

600

400

200

0

1466.00

q (units) Fig. 10. The change of TC with respect to q.

well, as it is seen to increase linearly as backorder cost increases from $5 to $100. A similar effect on the total cost is found as the pre- disruption quantity increases from 0 to 1400 units, as seen in Fig. 10. As the lost sales cost increases, the total cost does not change and is constant over the entire range of the lost sales cost, as shown in Fig. 8. This is due to the result that all shortages tend to be backordered when the lost sales cost is larger than the backorder cost. Thus no lost sales exist in the system. Finally, when the disruption duration is increased from 0.00065 to 0.003665, the total cost is seen to increase with a parabolic curve, as shown in Fig. 9. 5. Recovery from multiple disruptions It is the nature of disruptions that one may occur after another and also that most occur unexpectedly. Thus, when facing disruptions in real time, we tackle the first and seek its recovery schedule. Subsequently, when another disruption occurs, we look at the inventory on hand, define a new recovery window, and then tackle it as another disruption problem. We find it vital to emphasize how this real time mechanism works in our model. In this section, we discuss a special case in which one or more disruptions occur before the recovery of a previous disruption has been achieved. In other words, we will propose a method of how to handle the case of multiple disruptions. The procedure is similar to the single disruption case, except with the introduction of some new parameters. First of all, let us define the new parameters. Let T dj be the disth ruption duration for the j disruption, r be the total number of disruptions that occurred, X ji be the production quantities in the th recovery schedule for the j disruption, qj be the initial production th quantity before the j disruption, nTj be the cycle where disruption j occurred as measured from the start of the previous recovery

A ¼ 20;

H ¼ 1:2;

B ¼ 10;

L ¼ 15;

P ¼ 5000000;

D ¼ 4000000; St ¼ 0:000057; q1 ¼ 100; q2 ¼ 0; T d1 ¼ 0:003; T d2 ¼ 0:0025; nT2 ¼ 3; and n0 ¼ 3: Using Eq. (31), we obtain the value of T d as 0.004209. Initializing all parameters in the heuristic and solving gives us the new schedule. For the new optimal plan, we have:

n ¼ 8;

TC ¼ 2598:6

The above method can be used to determine the recovery schedule, even if more than one disruption occurs during the recovery stage. Once recovery is achieved, we treat any new disruption just like the single disruption case. 6. Conclusion In this study, a recovery model for a single stage inventory system subject to disruption has been developed, which is able to determine the optimal production quantities and the number of cycles for recovery, in order to minimize the total recovery cost. The model is a non-linear constrained integer program and is solved using a new heuristic that we have developed. In addition to the heuristic, the model was also solved using an evolution algorithm. A summary of the results was presented for 500 randomly generated problems and the solutions from the two techniques were compared. The heuristic approach is capable of accurately solving the problems with significantly less time compared to the ES method. From the analysis of results, it is shown that the optimal recovery schedule is highly dependent on the relationship between the backorder and the lost sales cost, as well as the extent of the disruption. The particular model developed in this paper would be useful for the pharmaceutical industry (Sarker and Khan, 2001) or others of a similar type, that faces a disruption in their production system. Due to their high quality control requirements, the goods from a single batch may not be separated and must be shipped only after the whole lot completes quality inspection. Furthermore, the disruptions that occur may be due to a machine breakdown, raw material supply shortage, or any other type of interruption that

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prevent the production from running normally during a cycle. The model is suitable for manufacturers who want to seek an optimal recovery schedule without affecting the entire schedule already on hand, with the consideration of opting for lost sales. In summary, the proposed model offers a potentially very useful tool to help manufacturers make prompt and accurate decisions on the optimal recovery plan, whenever a sudden disruption occurs in the production system. There are several directions that this research could continue. Future research is currently ongoing for extensions of the model to a supply chain system with multiple stages. One may also consider extending the study to multiple nodes in each stage. Another area worth exploring is considering the different shipment policies that exist in today’s supply chains.

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