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A recovery model for a two-echelon serial supply chain with consideration of transportation disruption
Computers & Industrial Engineering 64 (2013) 552–561
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A recovery model for a two-echelon serial supply chain with consideration of transportation disruption q 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 Architecture, National University of Malaysia (UKM), 43600 Bangi, Selangor, Malaysia
a r t i c l e
i n f o
Article history: Received 22 December 2011 Received in revised form 16 November 2012 Accepted 21 November 2012 Available online 8 December 2012 Keywords: Transportation disruption Lot sizing Disruption management Two stage inventory–production system Supply chain
a b s t r a c t Supply chains are becoming increasingly competitive and complex in order to effectively meet customer demands. These characteristics make supply chains vulnerable to various risks, including disruptions. In this study, a recovery model is explored for a two-stage production and inventory system with the possibility of transportation disruption. This model is capable of determining the optimal ordering and production quantities during the recovery window, and ensuring that the total relevant costs are minimized, while seeking to recover the original schedule. An efficient heuristic was developed to solve the model. The results showed that the optimal recovery schedule is highly dependent on the relationship between the backorder cost and the lost sales cost parameters. In addition, the heuristic was able to give quality solutions for the model, with very small deviations of the heuristic solutions from the optimal value. Such tools are useful in assisting managers towards effective decision making, particularly in determining the optimal recovery strategy for the longevity and sustainability of their firms undergoing disruptions. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The nature and complexity of today’s supply chains (SCs) make them vulnerable to various risks. These risks may fall into different terms, such as disruptions, uncertainties, and disturbances. SC disruption, particularly, is defined as an event that interrupts the material flows in the SC, resulting in an abrupt cessation of the movement of goods. SC disruptions can be caused by internal or external sources to the SC, including natural disasters, transportation failure, labor dispute, terrorism, war, and political instability. In recent years, we have come to see many disruption occurrences that have severely affected SCs. 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 $200 million worth of revenue, was lost due to parts shortages (Sheffi, 2005). SC disruptions are costly and it is crucial that managers take appropriate measures of response to reduce its negative effects.
q
This manuscript was processed by Area Editor Mohamad Y. Jaber.
⇑ Corresponding author at: School of Engineering and Information Technology, University of New South Wales, ADFA Campus, Northcott Drive, Canberra 2600, Australia. Tel.: +61 2 62688989; fax: +61 2 62688581. E-mail addresses: [email protected] (H. Hishamuddin), r.sarker@adfa. edu.au (R.A. Sarker), [email protected] (D. Essam). 0360-8352/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2012.11.012
Disruption Management (DM) is a line of study that has recently gained the interest of researchers. One of the goals of DM is to implement the correct strategies that will enable the SC to quickly return to its original state, while minimizing the relevant costs associated with recovery of the disruption (Qi, Bard, & Yu, 2004). Two common strategies to manage the risk of disruptions include mitigation and contingency (or recovery) 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 recovery tactics is not free; rather it involves a cost that influences the attractiveness of an optimal strategy for a given firm. Transportation disruption, in particular, is slightly different from other forms of SC disruptions, in that it only stops the flow of goods, whereas other disruptions may stop the production of goods as well. It is distinctive in that the goods in transit halt, even though the other operations of the SC are intact (Wilson, 2007). Giunipero and Eltantawy (2004) noted that a potential transportation disruption is a source of risk and that it could quickly cripple the entire SC. Transportation disruption can cause late deliveries, which may lead to production stoppages costs, lost sales and lost of customer’s goodwill (Guiffrida & Jaber, 2008). Furthermore, a transportation disruption may affect the condition of the valuable goods in transit. Due to the rise of organized crime and terrorist activities, the cost of goods lost during transportation is estimated at billions of dollars per year, with manufacturers suffering losses amounting to approximately five times the value of those goods
H. Hishamuddin et al. / Computers & Industrial Engineering 64 (2013) 552–561
damaged or stolen. The floods that hit Bangkok in 2011 caused vast damage to inventories in sugar mills and firms faced increased raw material cost and shortages, partly due to transportation disruption (Fernquest, 2011). Managers are forced to seek cost effective ways to react to these unexpected occurrences, mainly to alleviate the damaging impact it could bring to the firm. The model that we have developed in this research thus addresses this vital aspect of transportation disruption. This paper proposes a newly developed real-time rescheduling mechanism for an economic lot sizing problem of a two stage supply chain system subject to transportation disruption. A recovery duration known as the recovery time window (Hishamuddin, Sarker, & Essam, 2010, 2012; Xia, Yang, Golany, Gilbert, & Yu, 2004) is allocated after the disruption to allow changes in the production and ordering schedule. The objective is to determine the new optimal recovery schedule that minimizes the overall recovery costs of the system. Similar to other DM models, the original production schedule is restored by the end of the recovery time window, focusing on the attempt to preserve the original operational plan as much as possible. The contents of the paper are organized as follows. Section 2 presents the related literature review. Section 3 discusses the model development and analysis. This section includes derivation of the cost functions. Section 4 deals with the solution approach for the model. Section 5 addresses the related computational results and analysis. Lastly, Section 6 summarizes our research findings and offers potential directions for future research.
2. Literature review 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 & Arreola-Risa, 1994; Li, Xu, & Hayya, 2004; Parlar & Berkin, 1991; Parlar & Perry, 1995). Under the periodic review framework, Parlar, Wang, and Gerchak (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 & Rosenthal, 1992; Xiao & Yu, 2006). 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, Snyder, and Shen (2010) and Chen, Zhao, and Zhou (2012) extend the work of Tomlin (2006) to study the system with stochastic demand. Furthermore, Schmitt and Snyder (in-press) 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, Reinhardt, and Mohan (2007) which only considered the single period case. Other variations of supply disruptions in stochastic inventory models are also available in literatures (Arreola-Risa & DeCroix, 1998; Li et al., 2004; Mohebbi, 2003; Moinzadeh & Aggarwal, 1997). Snyder et al. (2012) provides an extensive review of supply chain models with disruption. Most of the works cited above consider inventory mitigation as a DM strategy, in which additional inventory is held in the system for the entire period to protect against disruptions. The majority of the studies are likely to result in stationary higher ordering quantities or bigger stock levels for the entire planning horizon. Carrying additional inventory can be very costly, unless if the disruption is predictable, the items have low holding costs or the products
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will not be obsolete (Wilson, 2007). Therefore, inventory mitigation tactics may not be of interest for firms that prefer a more lean and cost effective solution to managing disruptions. This in turn justifies the need for more recovery strategies in the presence of SC disruptions. Studies on optimal recovery strategies for disruptions exist in the literature, but 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. Xiao-Feng and Ming (2012) explored the optimal recovery strategies of an assemble-to-order SC subject to supply disruption. Recovery strategies to demand disruptions have also been explored in the work by Qi et al. (2004). In the work by Xia et al. (2004), a recovery strategy was developed for a twostage production and inventory 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. Hishamuddin et al. (2012) studied a recovery mechanism for a single stage production–inventory system subject to supply disruption, in which a heuristic was developed to obtain the new recovery schedule within the recovery time window. The objective was to seek the optimal production and ordering lot sizes, as well as the optimal back ordering and lost sales quantity, while minimizing the overall recovery costs. The study of transportation disruption in particular has received much less attention, despite the many harmful effects that it may impose on the SC, as mentioned in the earlier section. Giunipero and Eltantawy (2004) in their study discussed about transportation disruption in general, but did not specify the strategies on how to face it. Wilson (2007) investigated the effect of transportation disruption on SC performance using system dynamics. The work concluded that the most severe impact is experienced when transportation disruption exists between the tier 1 supplier and the warehouse. This disruption location is considered in our study. Zhang and Figliozzi (2010) conducted a survey on the effects of delay and disruption on international freight transport chains. Unnikrishnan and Figliozzi (2011) proposed an online freight network assignment model for network flows experiencing disruptions. Although some mitigation strategies were suggested in most of these papers in general, there is still a lack of computational methods to face transportation disruption in the SC and production– inventory context. The recovery model proposed in this paper is an extension of the work by Hishamuddin et al. (2012). While the former study only considered the single stage, our work explores a two echelon supply chain, where disruption is in the form of a transportation disruption that is not known in advance. In other words, there is no pre-disruption period in our model. Hence, the firm does not have the opportunity to take mitigation measures before the occurrence of the disruption, which reflects many real world problems that occur without warning. The main contributions of the paper can be summarized as follows: 1. The development of a recovery model for a two stage serial SC system with transportation disruption. 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.
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2. The introduction of an efficient heuristic approach that determines the optimal recovery plan for the supplier and the retailer, subject to the system’s costs and constraints. The heuristic is able to give quality solutions for the model. The proposed model could prove to be a very useful tool for manufacturers to make quick decisions on an optimal recovery plan after experiencing a transportation disruption.
T B1, B2 L1, L2 qt CT W T1i
3. Model formulation T2i In the following subsections, we address the system’s description of the model. This is followed by a mathematical representation of the model, in which the problem for the damaged lot case is formulated.
n m z
3.1. System description
Ii
In this study, we consider a two stage production and inventory system consisting of a manufacturer and a retailer. The manufacturer has production and inventory, and thus follows the economic production quantity model, while the retailer only has inventory and follows the economic order quantity model. Our model, however, assumes that the transport that delivers the goods from the supplier experiences a disruption that interrupts the timely delivery of goods to the retailer. The disruption may be caused by an accident or a natural disaster, such as a flood, earthquake, or snow blizzard, which disrupts the truck from operating normally. In addition, the goods in transit may or may not be damaged during the disruption. For the damaged lot case, the manufacturer needs to reproduce the damaged lot in order to satisfy the demand. Therefore, there will be changes in the original production schedule at the manufacturer’s side. The retailer only receives the goods after the production of the damaged lot is completed. This delay will result in shortages to the retailer. The notations used in developing the cost function are as follows:
f1
Decision variables Xi production lot size of cycle i in the recovery schedule for stage 1 (units) Si order lot size of cycle i in the recovery schedule for stage 2 (units) n number of cycles in the recovery window z number of optimal production lots in the recovery window Other parameters and notation A1 setup cost for the first stage ($/setup) A2 ordering cost for the second stage ($/order) D demand rate for the system (units/year) H1, H2 annual inventory cost for stages 1 and 2 ($/unit/year) P production rate (units/year) Q1 production lot size for stage 1 in the original schedule (units) Q2 ordering lot size for stage 2 in the original schedule (units) Bq back order quantity for stage 2 Lq lost sales quantity for stage 2 Td disruption period q production up time for a normal cycle (Q/P) u production down time for a normal cycle te start of recovery time window tf end of recovery time window
f2
f3
production cycle time for a normal cycle (Q/D) unit back order cost per unit time for stages 1 and 2 ($/unit/time) unit lost sales cost for stages 1 and 2 ($/unit) truck capacity unit transportation cost for each delivery ($/ shipment) warehouse capacity for stage 2 (units) production time for cycle i in the recovery window for stage 1 production time for cycle j in the recovery window for stage 2 number of cycles in the recovery window number of lots in the recovery window number of optimal production lots in the recovery window inventory level at the end of cycle i in the recovery window the penalty function for delay in recovering the original schedule in the first stage the penalty function for delay in recovering the original schedule of the second stage handled by the first stage the penalty function for delay in recovering the original schedule in the second stage
As a preliminary study, we have chosen the lot-for-lot policy to be applied to the model. For this particular type of shipment policy, the manufacturing lot size for the first stage is equal to the ordering lot size of the second stage (Q1 = Q2 = Q) under ideal conditions, due to the coordination of the two stage system. The current production–inventory system is a modified version of the non-disrupted model proposed by Banerjee (1986), where the optimal production lot size (Q) is:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2PDðA1 þ A2 Þ Q¼ H1 D þ H 2 P
ð1Þ
The proposed model is able to determine the new recovery plan, consisting of the supplier’s production quantities and the retailer’s ordering quantities. Like other DM models, the term recovery is defined as restoring the original production schedule within a short time period, while minimizing the relevant costs (Qi et al., 2004). Note that recovery is achieved when both stages are back to their original schedule. Extra costs are incurred in order to recover the system from the disruption, including backorder costs (B1, B2), lost sales costs (L1, L2) and penalty costs (f1, f2, f3) for both the manufacturer and the retailer. For this particular study, we have made the assumption that the lost sales costs are always higher than the backorder costs. We believe that this assumption is crucial for any lot sizing model, as losing sales is always more expensive than having backorders. The time horizon is finite, such that only costs in the recovery window are considered. Fig. 1 depicts the inventory lines for stage 1 (manufacturer) and stage 2 (retailer). The dotted line represents the original non-disrupted schedule; whereas the solid line represents the new recovery schedule with the presence of disruption. The zebra shaded triangle shows the amount of shortages composed of backorders (Bq) and lost sales (Lq) incurred by the retailer during the disruption period, Td. In this figure, we have n = 3 recovery cycles and z = 4 production lots in the recovery time window (te –tf). We define the decision variable Xi as the production quantity for cycle i in the recovery time window for the first stage
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Fig. 1. Production inventory curve for a two stage supply chain for the damaged lot case.
(manufacturer) and T1i as its respective production time, where i = 1, 2, . . . , n. Another decision variable Si is the ordering quantity for cycle i in the recovery window for the second stage (retailer) and T2i is its respective consumption time. Without loss of generality, we assume that Td is less than the normal production cycle time, T. After a disruption of time Td occurs, recovery takes place by utilizing the production idle times, d, in the original schedule. The following are additional assumptions of the proposed model: 1. 2. 3. 4.
The demand rate is less than the production rate, i.e. D < P. The transportation has limited capacity (qt). The retailer owns limited warehouse capacity (W). No shortages are allowed during the subsequent cycles following the disruption. 5. The second stage follows the zero-order inventory policy, where an order is made only when the on-hand inventory reaches zero. 6. The first stage is allowed to have left over inventory at the end of the cycles in the recovery window.
Let z be the optimal number of production lots in the recovery window, n be the number of cycles in the recovery window beginning from the start of disruption, m be the number of lots, i.e. the demand to be satisfied and y be a binary parameter to represent the state of goods. The relationship between n, m and y can be stated as follows:
1 if lot is damaged 0
if lot is undamaged
3.2. Damaged goods For this particular case, we assume that the goods being transported are damaged during the disruption (y = 1). The total costs for the second stage will first be formulated as this will ease the later determination of the costs for the first stage, since the production schedule for the first stage is dependent on the order schedule of the second stage. The ordering cost equation for the second stage is rather straight forward and can be obtained by:
¼ A2 ðNumber of ordersÞ ¼ A2 ðz 1Þ
ð3Þ
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 H2 ððS1 BqÞ T 21 þ S2 T 22 þ S3 T 23 þ Þ 2 1 ðS1 BqÞ S2 S3 ¼ H2 ðS1 BqÞ þ S2 þ S3 þ 2 D D D H2 ðS1 BqÞ2 þ S22 þ S23 þ ¼ 2D ! z1 X H2 2 2 Si ðS1 BqÞ þ ¼ 2D i¼2 ¼
3.2. Mathematical representation
m ¼ n þ y where y ¼
that it is better to include these costs as a total and not as a time average. In the next subsection, the cost function for the damaged lot case is presented.
ð2Þ
The two-stage model we consider faces six types of cost: setup cost, inventory holding cost, back order cost, lost sales cost, transportation cost, and penalty cost. The average costs are calculated for all the above costs except for the back order and lost sale cost. Given that shortages only occur during the first cycle, we believe
ð4Þ
The amount of shortages occurring during the disruption period can be derived as Td D. The lost sales quantity for the second stage P is calculated as Lq ¼ nQ z1 i¼1 Si . The backorder quantity is the amount of shortages minus the lost sales quantity given by Bq = Td D Lq. The retailer incurs a backorder penalty cost of B2 per unit backordered. Therefore, the backorder cost formulation for the second stage can be derived as:
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¼
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B2 T d ððT d DÞ LqÞ 2
ð5Þ
Finally, the lost sales cost is obtained as:
¼ L2 nQ
z1 X
! Si
ð6Þ
i¼1
The penalty function derived in this model is based on the assumption that the longer it takes to recover the original schedule, the higher the associated penalties. These penalties represent the extra costs incurred by the system when there are changes in the original plan. Here we have derived it as a function of the number of cycles for recovery, as below: 2
¼ f3 ðn Þ The sum of all the cost components above gives the total relevant costs of the recovery plan for the second stage, as presented below:
2 3 P 2 þ f 3ðn2 Þ ðA2 ðz 1ÞÞ þ H2D2 ðS1 BqÞ2 þ z1 i¼2 Si 5 TC 2 ðSi ; zÞ ¼ 4 nT !! z1 X B2 T d þ Si ððT d DÞ LqÞ þ L2 nQ 2 i¼1 ð7Þ Next, the total relevant costs for recovery for the first stage will be calculated. The setup cost is given by:
¼ A1 ðzÞ
the overall effects of the difference between backorder cost and lost sales cost on the system’s new schedule, rather than the effect of these parameters between individual stages. The backorder cost and lost sales cost for the manufacturer follows the concept by Cachon and Zipkin (1999) and is given by
B1 T d ððT d DÞ LqÞ 2 and z1 X Q ðyÞ þ nQ Si
L1
ð9Þ
The inventory cost for the first stage is:
1 1 1 ¼ H1 I0 T 11 þ X 1 T 11 þ I1 T 12 þ X 2 T 12 þ I2 T 13 þ X 3 T 13 2 2 2 1 þI3 T 14 þ X 4 T 14 þ 2 1 1 1 ¼ H1 I0 þ X 1 T 11 þ I1 þ X 2 T 12 þ I2 þ X 3 T 13 2 2 2 1 þ I3 þ X 4 T 14 þ 2 1 X1 1 X2 1 X3 þ I1 þ X 2 þ I2 þ X 3 ¼ H1 I0 þ X 1 2 2 2 P P P 1 X4 þ I3 þ X 4 þ 2 P ! z X 1 Xi ð10Þ Ii1 þ X i ¼ H1 2 P i¼1 For this model, it is assumed that the manufacturer incurs a penalty for backorders and lost sales at the retailer’s. In other words, the manufacturer incurs a cost whenever a customer is unable to purchase the manufacturer’s product from the retailer because it is out of stock. The two firms are assumed to have the same unit backorder and lost sales cost i:e: BB21 ¼ 1 and LL21 ¼ 1 . This indicates that both the supplier and the retailer have an equal level of concern for customer service and ensures that the supply chain is treated as a centralized system (Cachon & Zipkin, 1999). The solution will thus be optimal when the total cost is minimized for the entire SC rather than for the individual firms as in the decentralized system. Moreover, our main focus is to investigate
ð12Þ
respectively. Note that instead of having a parameter that constitutes the fraction of shortages that are backordered or lost like other models do (Park, 1982), our model instead determines this by way of optimization to ensure the overall cost of the system is minimized. Notice that for the lost sales cost formulation, we have considered the damaged lot as lost sales, which is given by Q(y). The transportation cost for each delivery can be formulated as:
CT
z1 X Si i¼1
ð13Þ
qt
Lastly, the penalty for delay in recovery is given as f1 ðn2 Þ þ f2 ðn2 Þ. Thus, the first stage’s average total cost for the recovery plan is represented as follows: 3 z1 P X X z Si 1 i þ CT þ f1 ðn2 Þ þ f2 ðn2 Þ7 6ðA1 ðzÞÞ þ H1 i¼1 Ii1 þ 2 X i P qt 7 6 i¼1 7 TC 1 ðX i ; zÞ ¼ 6 7 6 nT 5 4 2
ð8Þ
for i ¼ 1; 2; . . . ; z
!
i¼1
Let us define Ii as the inventory level at the end of cycle i in the recovery window, where
Ii ¼ Ii1 þ X i Si
ð11Þ
þ
!! z1 X B1 T d ððT d DÞ LqÞ þ L1 QðyÞ þ nQ Si 2 i¼1 ð14Þ
The optimal recovery plan for the damaged lot case is obtained by solving the following mathematical problem, which minimizes the total cost of recovery for the two-stage system:
Min ½TC 1 ðX i ; i ¼ 1; . . . ; zÞ þ TC 2 ðSi ; i ¼ 1; . . . ; zÞ
ð15Þ
subject to the following constraints (16)–(22):
Si 6 W
for i ¼ 1; 2; . . . ; z 1
z X X i 6 nPT
ð16Þ ð17Þ
i¼1 z X X i P mTD Lq
ð18Þ
i¼1 i i1 1X 1X Bq Xj 6 Sj P j¼2 D j¼1 D
for i ¼ 2; . . . ; z
ð19Þ
Io ¼ Iz ¼ 0
ð20Þ
Sz ¼ Q
ð21Þ
Xi 6 Q
ð22Þ
The objective function (15) comprises of the two total cost components of the first stage (7) and the second stage (14). Eq. (16) ensures that the inventory storage at the retailer’s side does not exceed its warehouse capacity. Eq. (17) represents the production capacity constraint; whereas Eq. (18) ensures that all demand during the recovery period is accounted for. Eq. (19) ensures that the retailer receives its shipments on time and never runs out of stock.
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Eq. (20) states that there is zero inventory at the start and end of the recovery window and Eq. (21) guarantees recovery of the original schedule after m cycles. Lastly, Eq. (22) ensures that the production quantities do not exceed Q due to shipping limitations. The above model can be categorized as a constrained integer nonlinear programming model. By solving the above model (16) for Xi, Si, n and z subject to the constraints (16)–(22), one can obtain the optimal recovery plan for the two stage supply chain system under disruption. Without disruption, this model will reduce to the original model as in (1) that was presented earlier. The model, as well as the solution method that is presented in the following section, could also be applied to the problem of a firm facing an increase in demand, along with goods that are damaged or lost in a disruption. Moreover, it is not difficult for one to obtain the optimal recovery plan for the case of an undamaged lot or a partially damaged lot (y = 0 or y is a fraction, respectively). This can be easily done by solving the same mathematical problem as the above, with slight modifications.
Heuristic Algorithm (1) Initialize the parameters and variables (2) Calculate the initial number of cycles to recover, n_start, using Eq. (23)
n start ¼ (3) (4) (5) (6)
(7)
Set K = 0, J = 0 Set n = n_start + J, z = n_start + K, X 1 ¼ Q ; X z ¼ Q and Lq ¼ 0 Calculate the production capacity, PnT and the demand to fulfill, mTD Check the following conditions: If T d > T 11 and PnT < mTD, set Lq ¼ ðT d T 11 ÞP Otherwise, retain Lq ¼ 0 Calculate the Xi, values
(8)
DD ¼ mTD
Corollary 2. The optimal policy for the model is for X i ¼ Si for 8i. Proof. From Eq. (9), we have Ii = Ii1 + Xi Si for 8i; i ¼ 1; . . . ; z. Substituting Ii ¼ 0 and Ii1 ¼ 0 into (9), will reduce to X i ¼ Si , which completes the proof for the corollary. h The detailed procedure for the heuristic is presented below. To generate a feasible solution for the problem under consideration, the following algorithm is used.
ð24Þ
z X X i Lq
ð25Þ
i¼1
(9)
Check the condition for DD: If DD > 0 (a) Calculate the new Lq value, Lq⁄, using Eq. (26)
Lq ¼ Lq þ
DD Lf þ 1
where Lf ¼
z1 i1 X P i¼2
D
ð26Þ
(b) Calculate Bq⁄, Bq ¼ T d D Lq (c) Calculate the new Xi, values using Eq. (24) by replacing Bq with Bq⁄ (d) Calculate TC using Eq. (15) and record the solution as TC(K) If DD < 0 (a) Set t = 2 (b) Calculate and replace the Xi values obtained from step (7) with the following
Corollary 1. The optimal policy for the model is for Ii ¼ 0 for 8i. Proof. Since the proof is rather lengthy, we refer the interested reader to the original work by Xia et al. (2004), proof of Theorem 2(a). h The following corollary is a consequence of Corollary 1.
for i ¼ 2; . . . ; z 1
Using Eq. (25), calculate DD, which is the difference between the demand to be fulfilled and the total of all Xi when Lq = 0.
4.1. A proposed heuristic In this research, a computationally efficient heuristic method has been developed to find the optimal recovery schedule for the model presented in the earlier section. 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. The heuristic we developed can be used as a module for re-scheduling of the production–inventory sub-system, due to disruption, within the company’s planning process. The module can be run immediately after the disruption occurs and the customized output will provide the decisions without further processing of outputs and interpretations. 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 two results that follow are derived for the model and are used for developing the steps in the heuristic algorithm. The next corollary is a result from the original work by Xia et al. (2004), which has been applied to our problem.
ð23Þ
! i i1 X P X Xi ¼ X j Bq Xj D j¼1 j¼2
4. Solution approach Here, we propose a method that was specially designed to solve the model in hand. A numerical example is then presented to demonstrate the applicability of the proposed method.
Q Pu
mTD Xi ¼
zt X Xi þ Q i¼1
t1 ¼ z t þ 1; . . . ; z 1
! Lq for i ð27Þ
given that Xi > 0 and constraint (19) is not violated. (c) Calculate TC (d) Set t = t + 1 and repeat step (b). If constraint (19) is violated, stop. (e) Record the solution with the lowest TC value as TC(K). (10) If K = 0, set K = K + 1 and repeat steps (4)–(9). Else, if TC(K) < TC(K 1), set K = K + 1 and repeat steps (4)–(9). Otherwise, record the solution. The solution is the minimum TC for a specified n value, where the optimal z (continued on next page)
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is determined. Record it as TC(J). (11) If J = 0, set J = J + 1, reset K = 0, and repeat steps (4)–(10) Else, if TC(J) < TC(J 1), set J = J + 1, reset K = 0, and repeat steps (4)–(10). Otherwise, record the solution. This optimal solution corresponds to the overall minimum total cost of the optimal solutions found for all possible n values. (12) Stop When solving the model, it is worth noting that Td may also depend on the variable T11. For instance, it may be more optimal for the time of the first production lot to be longer than the actual disruption duration being experienced (T11 > Td). In this case, Td must be replaced by T11 in the optimization solving process. Otherwise, Td remains as the user input. The following theorem is an important finding that led to the development of step 9 of algorithm 1 in the heuristic. Theorem DD > 0, P 1. If i1 it is optimal to have 1 unit of lost sales for z1 P each þ 1 units of DD. i¼2 D
Proof. When lost sales exist, one can derive the formula for the new Xi, X 0i from Eq. (24), as
! i i1 X P X Xi ¼ X j Bq þ Lq Xj D j¼1 j¼2
for i ¼ 2; . . . ; z 1:
ð28Þ
i1 The difference X 0i X i is calculated as DP Lq for i = 2, . . . , z 1. P For any particular value of z, the effect of 1 unit Lq on zi¼1 X i is given Pz1 P i1 by i¼2 D Lq. As defined earlier, DD is the amount of unfulfilled demand given by Eq. (25). Naturally, we want DD to be zero. We assume that mTD is fixed, thus one can only reduce DD to zero Pz by increasing Lq, which in turn will increase i¼1 X i . Therefore, we Pz1 P i1 have the expression DD ¼ Lq þ i¼2 D Lq. Solving this for Lq will yield the optimal Lq value from the unmet demand, formulated as Lq ¼ Pz1 DPD i1 . This completes the theorem proof. h ð Þ þ1 i¼2 D
4.2. Numerical example We will solve a numerical example to show the applicability of the model. The problem data are as follows:
A1 ¼ 200; L1 ¼ 15; y ¼ 1;
A2 ¼ 20; L2 ¼ 15;
H1 ¼ 1:2; T d ¼ 0:003;
H2 ¼ 1:8;
B1 ¼ 1;
P ¼ 5000000;
B2 ¼ 1;
D ¼ 4000000;
qt ¼ 10000
Parameter
Range
A1 A2 H1 H2 B1,2 L1,2 Td
[20, 500] [20, 500] [1 ,10] [1 ,10] [5, 80] [20, 100] [u, T]
the optimal solution occurs at n = 4 and z = 9, where TC( n, z) = 586878.8. 5. Numerical analysis This section presents several experiments to demonstrate the performance of the proposed model and method developed in this research. The mathematical model was solved using the developed heuristic for 500 random test problems. From this total, 50 test problems were generated by randomly changing all the parameters for the given intervals. The remaining 450 test problems on the other hand, tested the heuristic for the upper and lower bounds of each parameter while the other parameters were random. 9 data sets with 50 test problems in each set were randomly generated. Table 2 summarizes the range of data of the problems that were tested. Notice that the upper and lower limits for parameter Td are shown in terms of other parameters based on the assumptions made for this model. To evaluate the performance of the proposed heuristic, the results of the heuristic are compared to the solutions obtained by the LINGO 10.0 optimization software. The percentage of error, which is the deviation of the objective function value of the heuristic (TC_H) from the optimal value of TC obtained by LINGO (TC_L) was used as a quality criterion. The percentage of error was calculated using the following equation:
%error ¼
TC H TC L 100 TC L
The experiment shows fairly good results with solution errors that vary from 0.00% to 0.000026% with an average of 0.0000005%. The deviation of the heuristic solutions from the optimal value is significantly small and can be considered negligible, thus proving that the heuristic algorithm is reliable in generating optimal solutions for the recovery model. The total cost function, TC, with respect to z for different n values was plotted using the data from the numerical example presented earlier. In Fig. 2, the plot shows convex curves for each n
The solution procedure described in the previous subsection was coded in MATLAB and executed on an Intel Core Duo processor with 1.99 GB RAM and a 2.66 GHz CPU. The computational results are presented in Table 1. We can see from the results that, based on the stopping criterion in step (10), for n = 4 and n = 5 the algorithm stops at z = 9 and z = 11 respectively. Comparing the two TC values,
Table 1 Computational results of numerical example. n
z
TC(n, z)
Remark
4 5
9 11
586878.8 603265.3
Lowest TC
ð29Þ
Fig. 2. TC versus z for different n values.
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559
Fig. 3. Effect of increasing L1 and L2 on n, Bq and Lq.
Fig. 4. The change of TC with respect to A1 and A2.
value, which proves that the model and solution procedure developed in this paper provides good solutions. The global minimum corresponds to the lowest point of all the n curves, in which case is at n = 4 and z = 9 for Fig. 2. Analysis of the results shows that the solution is highly dependent on the relationship between the backorder cost and the lost sales cost parameters. The optimal number of backorders and lost sales is a tradeoff between the back order and lost sales cost. This finding can be observed in Fig. 3, which shows the effect of increased lost sales cost for both stages, L1 and L2 on n, Bq, and Lq, when the back order cost is significantly low. With back order cost fixed at $1, and lost sales cost increasing from $2 to $145, it can be seen that the backorder quantity increases whereas the lost sales quantity decreases and eventually reduces to zero. Another important finding is that the optimal recovery duration, n, gradually increases. This observation can be explained as follows: For a low lost sales cost, it is found that lost sales are more attractive. Thus, when a portion of demand becomes lost sales, the quantity to be produced is reduced and the duration for recovery will be shorter. However, as the lost sales cost becomes larger, it is more cost effective to have backorders. Therefore, the amount of lost sales will reduce and the recovery duration will become longer for the latter case. This important finding is beneficial as it facilitates suppliers and retailers in predicting the amount of time needed to recover from a disruption before the firm can continue its normal operations. Moreover, recovery information is vital for firms to assist
Fig. 5. The change of TC with respect to H1 and H2.
Fig. 6. The change of TC with respect to L1 and L2.
in proper management of resources, such as labor, raw material and machines. In addition to the above experiment, a sensitivity analysis to explore the effects of various key model parameters with respect to total costs was conducted as well. It can be seen in Figs. 4, 6 and 7 that the setup cost, lost sales cost and back order cost have a linear increasing effect on total cost, which is an intuitive observation. A different trend is observed with the inventory holding
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Fig. 7. The change of TC with respect to B1 and B2.
the solutions, the results from the heuristic were compared with LINGO. A numerical example was also provided to demonstrate the applicability of the model to real life problems and a sensitivity analysis was conducted with respect to the key model parameters. Analysis of the results showed that the optimal recovery schedule is highly dependent on the relationship between the backorder cost and the lost sales cost parameters. In addition, the heuristic was able to give quality solutions for the model, with very small deviations of the heuristic solutions from the optimal value. This proves that the use of the proposed heuristic is acceptable and reliable. The presented model is useful to assist decision makers who can take a pro-active approach in maintaining business continuity in the event of a transportation disruption in the supply chain system. Managers of manufacturing companies may easily utilize the model and heuristic to solve daily or occasional transportation disruptions that affect their production system. Efficiently achieving a cost-effective recovery schedule in real time is crucial as it enhances productivity and overall business operations. There are several directions in which this research could continue. An important extension of the paper may involve extending the model to a multi-echelon supply chain with more than two stages and of also having multiple retailers. In addition, incorporating different shipment policies in the study is another potential and interesting area for future work.
References
Fig. 8. The change of TC with respect to Td.
cost, where total cost decreases following a parabolic curve as the parameter increases, as shown in Fig. 5. The total cost behavior as disruption duration increases was also examined. As illustrated in Fig. 8, the total cost remains constant until a certain point where it starts to increase linearly. This TC behavior can be explained by the condition that has been set for Td as described in the previous section, in which case that if Td is smaller than T11, then Td is replaced by T11. Looking at the curve, Td is found to be smaller than T11 initially, and so it is replaced by T11, which explains the static TC pattern. However, the Td value becomes less than T11 as the curve progresses, which consequently results in the increasing TC trend. Hence, the larger the Td value, the higher the stock out quantity, and thus the total cost curve reflects the increasing cost of the total stock outs that exist in the system due to this change in Td. 6. Conclusion In this study, a disruption recovery model for a two stage production and inventory system subject to transportation disruption was analyzed. The objective of the study was to determine the optimal production and ordering quantities for the supplier and retailer, as well as the duration for recovery, which yields the minimum relevant costs of the system. The cost structure for the above model was developed for the case where goods in transit were damaged during the disruption. As for the solution procedure, a self developed heuristic was used to obtain optimal solutions for the proposed model, where a total of 500 randomly generated test problems were solved using the heuristic. To judge the quality of
Arreola-Risa, A., & DeCroix, G. A. (1998). Inventory management under random supply disruptions and partial backorders. Naval Research Logistics, 45, 687–703. Banerjee, A. (1986). A joint economic-lot-size model for purchaser and vendor. Decision Sciences, 17, 292–311. Berk, E., & Arreola-Risa, A. (1994). Note on ‘‘Future supply uncertainty in EOQ models’’. Naval Research Logistics, 41, 129–132. Cachon, G. P., & Zipkin, P. H. (1999). Competitive and cooperative inventory policies in a two-stage supply chain. Management Science, 45, 936–953. Chen, J., Zhao, X., & Zhou, Y. (2012). A periodic-review inventory system with a capacitated backup supplier for mitigating supply disruptions. European Journal of Operational Research, 219, 312–323. Chopra, S., Reinhardt, G., & Mohan, U. (2007). The importance of decoupling recurrent and disruption risks in a supply chain. Naval Research Logistics, 54, 544–555. Eisenstein, D. D. (2005). Recovering cyclic schedules using dynamic produce-up-to policies. Operations Research, 53, 675–688. Fernquest, J. (2011). Floods hit auto and electronics exports. Bangkok Post. Gallego, G. (1994). When is a base stock policy optimal in recovering disrupted cyclic schedules? Naval Research Logistics, 41, 317–333. Giunipero, L. C., & Eltantawy, R. A. (2004). Securing the upstream supply chain: A risk management approach. International Journal of Physical Distribution & Logistics Management, 34, 698–713. Guiffrida, A. L., & Jaber, M. Y. (2008). Managerial and economic impacts of reducing delivery variance in the supply chain. Applied Mathematical Modelling, 32(10), 2149–2161. Hishamuddin, H., Sarker, R. A., Essam, D. (2010). A recovery model for an economic production quantity problem with disruption. In: Paper presented at the international conference of industrial engineering and engineering management, Macau. Hishamuddin, H., Sarker, R. A., & Essam, D. (2012). A disruption recovery model for a single stage production–inventory system. European Journal of Operational Research, 222(3), 464–473. Li, Z., Xu, S. H., & Hayya, J. (2004). A periodic-review inventory system with supply interruptions. Probability in the Engineering and Informational Sciences, 18, 33–53. Mohebbi, E. (2003). Supply interruptions in a lost-sales inventory system with random lead time: [doi: 10.1016/S0305-0548(01)00108-3]. Computers & Operations Research, 30, 411–426. Moinzadeh, K., & Aggarwal, P. (1997). Analysis of a production/inventory system subject to random disruptions. Management Science, 43, 1577–1588. Ozekici, S., & Parlar, M. (1999). Inventory models with unreliable suppliers in a random environment. Annals of Operations Research, 91, 123–136. Park, K. S. (1982). Inventory model with partial backorders. International Journal of Systems Science, 13, 1313–1317. Parlar, M., & Berkin, D. (1991). Future supply uncertainty in EOQ models. Naval Research Logistics, 38, 107–121. Parlar, M., & Perry, D. (1995). Analysis of a (Q, r, T) inventory policy with deterministic and random yields when future supply is uncertain. European Journal of Operational Research, 84, 431–443.
H. Hishamuddin et al. / Computers & Industrial Engineering 64 (2013) 552–561 Parlar, M., Wang, Y., & Gerchak, Y. (1995). A periodic review inventory model with Markovian supply availability: [doi: 10.1016/0925-5273(95)00115-8]. International Journal of Production Economics, 42, 131–136. Qi, X., Bard, J. F., & Yu, G. (2004). Supply chain coordination with demand disruptions: [doi: 10.1016/j.omega.2003.12.002]. Omega, 32, 301–312. Schmitt, A. J., Snyder, L. V., & Shen, Z.-J. M. (2010). Inventory systems with stochastic demand and supply: Properties and approximations. European Journal of Operational Research, 206, 313–328. Sheffi, Y. (2005). The resilient enterprise: Overcoming vulnerability for competitive advantage. Massachusetts: Cambridge. Snyder, L. V., Atan, Z., Peng, P., Rong, Y., Schmitt, A. J., & Sinsoysal, B. (2012). OR/MS Models for Supply Chain Disruptions: A Review. Available from http://ssrn.com/ abstract=1689882. Song, J.-S., & Zipkin, P. H. (1996). Inventory control with information about supply conditions. Management Science, 42, 1409–1419. Tang, L. C., & Lee, L. H. (2005). A simple recovery strategy for economic lot scheduling problem: A two-product case. International Journal of Production Economics, 98, 97–107. Tomlin, B. (2006). On the value of mitigation and contingency strategies for managing supply chain disruption risks. Management Science, 52, 639–657.
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Contents lists available at SciVerse ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
A recovery model for a two-echelon serial supply chain with consideration of transportation disruption q 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 Architecture, National University of Malaysia (UKM), 43600 Bangi, Selangor, Malaysia
a r t i c l e
i n f o
Article history: Received 22 December 2011 Received in revised form 16 November 2012 Accepted 21 November 2012 Available online 8 December 2012 Keywords: Transportation disruption Lot sizing Disruption management Two stage inventory–production system Supply chain
a b s t r a c t Supply chains are becoming increasingly competitive and complex in order to effectively meet customer demands. These characteristics make supply chains vulnerable to various risks, including disruptions. In this study, a recovery model is explored for a two-stage production and inventory system with the possibility of transportation disruption. This model is capable of determining the optimal ordering and production quantities during the recovery window, and ensuring that the total relevant costs are minimized, while seeking to recover the original schedule. An efficient heuristic was developed to solve the model. The results showed that the optimal recovery schedule is highly dependent on the relationship between the backorder cost and the lost sales cost parameters. In addition, the heuristic was able to give quality solutions for the model, with very small deviations of the heuristic solutions from the optimal value. Such tools are useful in assisting managers towards effective decision making, particularly in determining the optimal recovery strategy for the longevity and sustainability of their firms undergoing disruptions. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The nature and complexity of today’s supply chains (SCs) make them vulnerable to various risks. These risks may fall into different terms, such as disruptions, uncertainties, and disturbances. SC disruption, particularly, is defined as an event that interrupts the material flows in the SC, resulting in an abrupt cessation of the movement of goods. SC disruptions can be caused by internal or external sources to the SC, including natural disasters, transportation failure, labor dispute, terrorism, war, and political instability. In recent years, we have come to see many disruption occurrences that have severely affected SCs. 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 $200 million worth of revenue, was lost due to parts shortages (Sheffi, 2005). SC disruptions are costly and it is crucial that managers take appropriate measures of response to reduce its negative effects.
q
This manuscript was processed by Area Editor Mohamad Y. Jaber.
⇑ Corresponding author at: School of Engineering and Information Technology, University of New South Wales, ADFA Campus, Northcott Drive, Canberra 2600, Australia. Tel.: +61 2 62688989; fax: +61 2 62688581. E-mail addresses: [email protected] (H. Hishamuddin), r.sarker@adfa. edu.au (R.A. Sarker), [email protected] (D. Essam). 0360-8352/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cie.2012.11.012
Disruption Management (DM) is a line of study that has recently gained the interest of researchers. One of the goals of DM is to implement the correct strategies that will enable the SC to quickly return to its original state, while minimizing the relevant costs associated with recovery of the disruption (Qi, Bard, & Yu, 2004). Two common strategies to manage the risk of disruptions include mitigation and contingency (or recovery) 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 recovery tactics is not free; rather it involves a cost that influences the attractiveness of an optimal strategy for a given firm. Transportation disruption, in particular, is slightly different from other forms of SC disruptions, in that it only stops the flow of goods, whereas other disruptions may stop the production of goods as well. It is distinctive in that the goods in transit halt, even though the other operations of the SC are intact (Wilson, 2007). Giunipero and Eltantawy (2004) noted that a potential transportation disruption is a source of risk and that it could quickly cripple the entire SC. Transportation disruption can cause late deliveries, which may lead to production stoppages costs, lost sales and lost of customer’s goodwill (Guiffrida & Jaber, 2008). Furthermore, a transportation disruption may affect the condition of the valuable goods in transit. Due to the rise of organized crime and terrorist activities, the cost of goods lost during transportation is estimated at billions of dollars per year, with manufacturers suffering losses amounting to approximately five times the value of those goods
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damaged or stolen. The floods that hit Bangkok in 2011 caused vast damage to inventories in sugar mills and firms faced increased raw material cost and shortages, partly due to transportation disruption (Fernquest, 2011). Managers are forced to seek cost effective ways to react to these unexpected occurrences, mainly to alleviate the damaging impact it could bring to the firm. The model that we have developed in this research thus addresses this vital aspect of transportation disruption. This paper proposes a newly developed real-time rescheduling mechanism for an economic lot sizing problem of a two stage supply chain system subject to transportation disruption. A recovery duration known as the recovery time window (Hishamuddin, Sarker, & Essam, 2010, 2012; Xia, Yang, Golany, Gilbert, & Yu, 2004) is allocated after the disruption to allow changes in the production and ordering schedule. The objective is to determine the new optimal recovery schedule that minimizes the overall recovery costs of the system. Similar to other DM models, the original production schedule is restored by the end of the recovery time window, focusing on the attempt to preserve the original operational plan as much as possible. The contents of the paper are organized as follows. Section 2 presents the related literature review. Section 3 discusses the model development and analysis. This section includes derivation of the cost functions. Section 4 deals with the solution approach for the model. Section 5 addresses the related computational results and analysis. Lastly, Section 6 summarizes our research findings and offers potential directions for future research.
2. Literature review 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 & Arreola-Risa, 1994; Li, Xu, & Hayya, 2004; Parlar & Berkin, 1991; Parlar & Perry, 1995). Under the periodic review framework, Parlar, Wang, and Gerchak (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 & Rosenthal, 1992; Xiao & Yu, 2006). 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, Snyder, and Shen (2010) and Chen, Zhao, and Zhou (2012) extend the work of Tomlin (2006) to study the system with stochastic demand. Furthermore, Schmitt and Snyder (in-press) 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, Reinhardt, and Mohan (2007) which only considered the single period case. Other variations of supply disruptions in stochastic inventory models are also available in literatures (Arreola-Risa & DeCroix, 1998; Li et al., 2004; Mohebbi, 2003; Moinzadeh & Aggarwal, 1997). Snyder et al. (2012) provides an extensive review of supply chain models with disruption. Most of the works cited above consider inventory mitigation as a DM strategy, in which additional inventory is held in the system for the entire period to protect against disruptions. The majority of the studies are likely to result in stationary higher ordering quantities or bigger stock levels for the entire planning horizon. Carrying additional inventory can be very costly, unless if the disruption is predictable, the items have low holding costs or the products
553
will not be obsolete (Wilson, 2007). Therefore, inventory mitigation tactics may not be of interest for firms that prefer a more lean and cost effective solution to managing disruptions. This in turn justifies the need for more recovery strategies in the presence of SC disruptions. Studies on optimal recovery strategies for disruptions exist in the literature, but 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. Xiao-Feng and Ming (2012) explored the optimal recovery strategies of an assemble-to-order SC subject to supply disruption. Recovery strategies to demand disruptions have also been explored in the work by Qi et al. (2004). In the work by Xia et al. (2004), a recovery strategy was developed for a twostage production and inventory 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. Hishamuddin et al. (2012) studied a recovery mechanism for a single stage production–inventory system subject to supply disruption, in which a heuristic was developed to obtain the new recovery schedule within the recovery time window. The objective was to seek the optimal production and ordering lot sizes, as well as the optimal back ordering and lost sales quantity, while minimizing the overall recovery costs. The study of transportation disruption in particular has received much less attention, despite the many harmful effects that it may impose on the SC, as mentioned in the earlier section. Giunipero and Eltantawy (2004) in their study discussed about transportation disruption in general, but did not specify the strategies on how to face it. Wilson (2007) investigated the effect of transportation disruption on SC performance using system dynamics. The work concluded that the most severe impact is experienced when transportation disruption exists between the tier 1 supplier and the warehouse. This disruption location is considered in our study. Zhang and Figliozzi (2010) conducted a survey on the effects of delay and disruption on international freight transport chains. Unnikrishnan and Figliozzi (2011) proposed an online freight network assignment model for network flows experiencing disruptions. Although some mitigation strategies were suggested in most of these papers in general, there is still a lack of computational methods to face transportation disruption in the SC and production– inventory context. The recovery model proposed in this paper is an extension of the work by Hishamuddin et al. (2012). While the former study only considered the single stage, our work explores a two echelon supply chain, where disruption is in the form of a transportation disruption that is not known in advance. In other words, there is no pre-disruption period in our model. Hence, the firm does not have the opportunity to take mitigation measures before the occurrence of the disruption, which reflects many real world problems that occur without warning. The main contributions of the paper can be summarized as follows: 1. The development of a recovery model for a two stage serial SC system with transportation disruption. 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.
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2. The introduction of an efficient heuristic approach that determines the optimal recovery plan for the supplier and the retailer, subject to the system’s costs and constraints. The heuristic is able to give quality solutions for the model. The proposed model could prove to be a very useful tool for manufacturers to make quick decisions on an optimal recovery plan after experiencing a transportation disruption.
T B1, B2 L1, L2 qt CT W T1i
3. Model formulation T2i In the following subsections, we address the system’s description of the model. This is followed by a mathematical representation of the model, in which the problem for the damaged lot case is formulated.
n m z
3.1. System description
Ii
In this study, we consider a two stage production and inventory system consisting of a manufacturer and a retailer. The manufacturer has production and inventory, and thus follows the economic production quantity model, while the retailer only has inventory and follows the economic order quantity model. Our model, however, assumes that the transport that delivers the goods from the supplier experiences a disruption that interrupts the timely delivery of goods to the retailer. The disruption may be caused by an accident or a natural disaster, such as a flood, earthquake, or snow blizzard, which disrupts the truck from operating normally. In addition, the goods in transit may or may not be damaged during the disruption. For the damaged lot case, the manufacturer needs to reproduce the damaged lot in order to satisfy the demand. Therefore, there will be changes in the original production schedule at the manufacturer’s side. The retailer only receives the goods after the production of the damaged lot is completed. This delay will result in shortages to the retailer. The notations used in developing the cost function are as follows:
f1
Decision variables Xi production lot size of cycle i in the recovery schedule for stage 1 (units) Si order lot size of cycle i in the recovery schedule for stage 2 (units) n number of cycles in the recovery window z number of optimal production lots in the recovery window Other parameters and notation A1 setup cost for the first stage ($/setup) A2 ordering cost for the second stage ($/order) D demand rate for the system (units/year) H1, H2 annual inventory cost for stages 1 and 2 ($/unit/year) P production rate (units/year) Q1 production lot size for stage 1 in the original schedule (units) Q2 ordering lot size for stage 2 in the original schedule (units) Bq back order quantity for stage 2 Lq lost sales quantity for stage 2 Td disruption period q production up time for a normal cycle (Q/P) u production down time for a normal cycle te start of recovery time window tf end of recovery time window
f2
f3
production cycle time for a normal cycle (Q/D) unit back order cost per unit time for stages 1 and 2 ($/unit/time) unit lost sales cost for stages 1 and 2 ($/unit) truck capacity unit transportation cost for each delivery ($/ shipment) warehouse capacity for stage 2 (units) production time for cycle i in the recovery window for stage 1 production time for cycle j in the recovery window for stage 2 number of cycles in the recovery window number of lots in the recovery window number of optimal production lots in the recovery window inventory level at the end of cycle i in the recovery window the penalty function for delay in recovering the original schedule in the first stage the penalty function for delay in recovering the original schedule of the second stage handled by the first stage the penalty function for delay in recovering the original schedule in the second stage
As a preliminary study, we have chosen the lot-for-lot policy to be applied to the model. For this particular type of shipment policy, the manufacturing lot size for the first stage is equal to the ordering lot size of the second stage (Q1 = Q2 = Q) under ideal conditions, due to the coordination of the two stage system. The current production–inventory system is a modified version of the non-disrupted model proposed by Banerjee (1986), where the optimal production lot size (Q) is:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2PDðA1 þ A2 Þ Q¼ H1 D þ H 2 P
ð1Þ
The proposed model is able to determine the new recovery plan, consisting of the supplier’s production quantities and the retailer’s ordering quantities. Like other DM models, the term recovery is defined as restoring the original production schedule within a short time period, while minimizing the relevant costs (Qi et al., 2004). Note that recovery is achieved when both stages are back to their original schedule. Extra costs are incurred in order to recover the system from the disruption, including backorder costs (B1, B2), lost sales costs (L1, L2) and penalty costs (f1, f2, f3) for both the manufacturer and the retailer. For this particular study, we have made the assumption that the lost sales costs are always higher than the backorder costs. We believe that this assumption is crucial for any lot sizing model, as losing sales is always more expensive than having backorders. The time horizon is finite, such that only costs in the recovery window are considered. Fig. 1 depicts the inventory lines for stage 1 (manufacturer) and stage 2 (retailer). The dotted line represents the original non-disrupted schedule; whereas the solid line represents the new recovery schedule with the presence of disruption. The zebra shaded triangle shows the amount of shortages composed of backorders (Bq) and lost sales (Lq) incurred by the retailer during the disruption period, Td. In this figure, we have n = 3 recovery cycles and z = 4 production lots in the recovery time window (te –tf). We define the decision variable Xi as the production quantity for cycle i in the recovery time window for the first stage
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Fig. 1. Production inventory curve for a two stage supply chain for the damaged lot case.
(manufacturer) and T1i as its respective production time, where i = 1, 2, . . . , n. Another decision variable Si is the ordering quantity for cycle i in the recovery window for the second stage (retailer) and T2i is its respective consumption time. Without loss of generality, we assume that Td is less than the normal production cycle time, T. After a disruption of time Td occurs, recovery takes place by utilizing the production idle times, d, in the original schedule. The following are additional assumptions of the proposed model: 1. 2. 3. 4.
The demand rate is less than the production rate, i.e. D < P. The transportation has limited capacity (qt). The retailer owns limited warehouse capacity (W). No shortages are allowed during the subsequent cycles following the disruption. 5. The second stage follows the zero-order inventory policy, where an order is made only when the on-hand inventory reaches zero. 6. The first stage is allowed to have left over inventory at the end of the cycles in the recovery window.
Let z be the optimal number of production lots in the recovery window, n be the number of cycles in the recovery window beginning from the start of disruption, m be the number of lots, i.e. the demand to be satisfied and y be a binary parameter to represent the state of goods. The relationship between n, m and y can be stated as follows:
1 if lot is damaged 0
if lot is undamaged
3.2. Damaged goods For this particular case, we assume that the goods being transported are damaged during the disruption (y = 1). The total costs for the second stage will first be formulated as this will ease the later determination of the costs for the first stage, since the production schedule for the first stage is dependent on the order schedule of the second stage. The ordering cost equation for the second stage is rather straight forward and can be obtained by:
¼ A2 ðNumber of ordersÞ ¼ A2 ðz 1Þ
ð3Þ
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 H2 ððS1 BqÞ T 21 þ S2 T 22 þ S3 T 23 þ Þ 2 1 ðS1 BqÞ S2 S3 ¼ H2 ðS1 BqÞ þ S2 þ S3 þ 2 D D D H2 ðS1 BqÞ2 þ S22 þ S23 þ ¼ 2D ! z1 X H2 2 2 Si ðS1 BqÞ þ ¼ 2D i¼2 ¼
3.2. Mathematical representation
m ¼ n þ y where y ¼
that it is better to include these costs as a total and not as a time average. In the next subsection, the cost function for the damaged lot case is presented.
ð2Þ
The two-stage model we consider faces six types of cost: setup cost, inventory holding cost, back order cost, lost sales cost, transportation cost, and penalty cost. The average costs are calculated for all the above costs except for the back order and lost sale cost. Given that shortages only occur during the first cycle, we believe
ð4Þ
The amount of shortages occurring during the disruption period can be derived as Td D. The lost sales quantity for the second stage P is calculated as Lq ¼ nQ z1 i¼1 Si . The backorder quantity is the amount of shortages minus the lost sales quantity given by Bq = Td D Lq. The retailer incurs a backorder penalty cost of B2 per unit backordered. Therefore, the backorder cost formulation for the second stage can be derived as:
556
¼
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B2 T d ððT d DÞ LqÞ 2
ð5Þ
Finally, the lost sales cost is obtained as:
¼ L2 nQ
z1 X
! Si
ð6Þ
i¼1
The penalty function derived in this model is based on the assumption that the longer it takes to recover the original schedule, the higher the associated penalties. These penalties represent the extra costs incurred by the system when there are changes in the original plan. Here we have derived it as a function of the number of cycles for recovery, as below: 2
¼ f3 ðn Þ The sum of all the cost components above gives the total relevant costs of the recovery plan for the second stage, as presented below:
2 3 P 2 þ f 3ðn2 Þ ðA2 ðz 1ÞÞ þ H2D2 ðS1 BqÞ2 þ z1 i¼2 Si 5 TC 2 ðSi ; zÞ ¼ 4 nT !! z1 X B2 T d þ Si ððT d DÞ LqÞ þ L2 nQ 2 i¼1 ð7Þ Next, the total relevant costs for recovery for the first stage will be calculated. The setup cost is given by:
¼ A1 ðzÞ
the overall effects of the difference between backorder cost and lost sales cost on the system’s new schedule, rather than the effect of these parameters between individual stages. The backorder cost and lost sales cost for the manufacturer follows the concept by Cachon and Zipkin (1999) and is given by
B1 T d ððT d DÞ LqÞ 2 and z1 X Q ðyÞ þ nQ Si
L1
ð9Þ
The inventory cost for the first stage is:
1 1 1 ¼ H1 I0 T 11 þ X 1 T 11 þ I1 T 12 þ X 2 T 12 þ I2 T 13 þ X 3 T 13 2 2 2 1 þI3 T 14 þ X 4 T 14 þ 2 1 1 1 ¼ H1 I0 þ X 1 T 11 þ I1 þ X 2 T 12 þ I2 þ X 3 T 13 2 2 2 1 þ I3 þ X 4 T 14 þ 2 1 X1 1 X2 1 X3 þ I1 þ X 2 þ I2 þ X 3 ¼ H1 I0 þ X 1 2 2 2 P P P 1 X4 þ I3 þ X 4 þ 2 P ! z X 1 Xi ð10Þ Ii1 þ X i ¼ H1 2 P i¼1 For this model, it is assumed that the manufacturer incurs a penalty for backorders and lost sales at the retailer’s. In other words, the manufacturer incurs a cost whenever a customer is unable to purchase the manufacturer’s product from the retailer because it is out of stock. The two firms are assumed to have the same unit backorder and lost sales cost i:e: BB21 ¼ 1 and LL21 ¼ 1 . This indicates that both the supplier and the retailer have an equal level of concern for customer service and ensures that the supply chain is treated as a centralized system (Cachon & Zipkin, 1999). The solution will thus be optimal when the total cost is minimized for the entire SC rather than for the individual firms as in the decentralized system. Moreover, our main focus is to investigate
ð12Þ
respectively. Note that instead of having a parameter that constitutes the fraction of shortages that are backordered or lost like other models do (Park, 1982), our model instead determines this by way of optimization to ensure the overall cost of the system is minimized. Notice that for the lost sales cost formulation, we have considered the damaged lot as lost sales, which is given by Q(y). The transportation cost for each delivery can be formulated as:
CT
z1 X Si i¼1
ð13Þ
qt
Lastly, the penalty for delay in recovery is given as f1 ðn2 Þ þ f2 ðn2 Þ. Thus, the first stage’s average total cost for the recovery plan is represented as follows: 3 z1 P X X z Si 1 i þ CT þ f1 ðn2 Þ þ f2 ðn2 Þ7 6ðA1 ðzÞÞ þ H1 i¼1 Ii1 þ 2 X i P qt 7 6 i¼1 7 TC 1 ðX i ; zÞ ¼ 6 7 6 nT 5 4 2
ð8Þ
for i ¼ 1; 2; . . . ; z
!
i¼1
Let us define Ii as the inventory level at the end of cycle i in the recovery window, where
Ii ¼ Ii1 þ X i Si
ð11Þ
þ
!! z1 X B1 T d ððT d DÞ LqÞ þ L1 QðyÞ þ nQ Si 2 i¼1 ð14Þ
The optimal recovery plan for the damaged lot case is obtained by solving the following mathematical problem, which minimizes the total cost of recovery for the two-stage system:
Min ½TC 1 ðX i ; i ¼ 1; . . . ; zÞ þ TC 2 ðSi ; i ¼ 1; . . . ; zÞ
ð15Þ
subject to the following constraints (16)–(22):
Si 6 W
for i ¼ 1; 2; . . . ; z 1
z X X i 6 nPT
ð16Þ ð17Þ
i¼1 z X X i P mTD Lq
ð18Þ
i¼1 i i1 1X 1X Bq Xj 6 Sj P j¼2 D j¼1 D
for i ¼ 2; . . . ; z
ð19Þ
Io ¼ Iz ¼ 0
ð20Þ
Sz ¼ Q
ð21Þ
Xi 6 Q
ð22Þ
The objective function (15) comprises of the two total cost components of the first stage (7) and the second stage (14). Eq. (16) ensures that the inventory storage at the retailer’s side does not exceed its warehouse capacity. Eq. (17) represents the production capacity constraint; whereas Eq. (18) ensures that all demand during the recovery period is accounted for. Eq. (19) ensures that the retailer receives its shipments on time and never runs out of stock.
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Eq. (20) states that there is zero inventory at the start and end of the recovery window and Eq. (21) guarantees recovery of the original schedule after m cycles. Lastly, Eq. (22) ensures that the production quantities do not exceed Q due to shipping limitations. The above model can be categorized as a constrained integer nonlinear programming model. By solving the above model (16) for Xi, Si, n and z subject to the constraints (16)–(22), one can obtain the optimal recovery plan for the two stage supply chain system under disruption. Without disruption, this model will reduce to the original model as in (1) that was presented earlier. The model, as well as the solution method that is presented in the following section, could also be applied to the problem of a firm facing an increase in demand, along with goods that are damaged or lost in a disruption. Moreover, it is not difficult for one to obtain the optimal recovery plan for the case of an undamaged lot or a partially damaged lot (y = 0 or y is a fraction, respectively). This can be easily done by solving the same mathematical problem as the above, with slight modifications.
Heuristic Algorithm (1) Initialize the parameters and variables (2) Calculate the initial number of cycles to recover, n_start, using Eq. (23)
n start ¼ (3) (4) (5) (6)
(7)
Set K = 0, J = 0 Set n = n_start + J, z = n_start + K, X 1 ¼ Q ; X z ¼ Q and Lq ¼ 0 Calculate the production capacity, PnT and the demand to fulfill, mTD Check the following conditions: If T d > T 11 and PnT < mTD, set Lq ¼ ðT d T 11 ÞP Otherwise, retain Lq ¼ 0 Calculate the Xi, values
(8)
DD ¼ mTD
Corollary 2. The optimal policy for the model is for X i ¼ Si for 8i. Proof. From Eq. (9), we have Ii = Ii1 + Xi Si for 8i; i ¼ 1; . . . ; z. Substituting Ii ¼ 0 and Ii1 ¼ 0 into (9), will reduce to X i ¼ Si , which completes the proof for the corollary. h The detailed procedure for the heuristic is presented below. To generate a feasible solution for the problem under consideration, the following algorithm is used.
ð24Þ
z X X i Lq
ð25Þ
i¼1
(9)
Check the condition for DD: If DD > 0 (a) Calculate the new Lq value, Lq⁄, using Eq. (26)
Lq ¼ Lq þ
DD Lf þ 1
where Lf ¼
z1 i1 X P i¼2
D
ð26Þ
(b) Calculate Bq⁄, Bq ¼ T d D Lq (c) Calculate the new Xi, values using Eq. (24) by replacing Bq with Bq⁄ (d) Calculate TC using Eq. (15) and record the solution as TC(K) If DD < 0 (a) Set t = 2 (b) Calculate and replace the Xi values obtained from step (7) with the following
Corollary 1. The optimal policy for the model is for Ii ¼ 0 for 8i. Proof. Since the proof is rather lengthy, we refer the interested reader to the original work by Xia et al. (2004), proof of Theorem 2(a). h The following corollary is a consequence of Corollary 1.
for i ¼ 2; . . . ; z 1
Using Eq. (25), calculate DD, which is the difference between the demand to be fulfilled and the total of all Xi when Lq = 0.
4.1. A proposed heuristic In this research, a computationally efficient heuristic method has been developed to find the optimal recovery schedule for the model presented in the earlier section. 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. The heuristic we developed can be used as a module for re-scheduling of the production–inventory sub-system, due to disruption, within the company’s planning process. The module can be run immediately after the disruption occurs and the customized output will provide the decisions without further processing of outputs and interpretations. 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 two results that follow are derived for the model and are used for developing the steps in the heuristic algorithm. The next corollary is a result from the original work by Xia et al. (2004), which has been applied to our problem.
ð23Þ
! i i1 X P X Xi ¼ X j Bq Xj D j¼1 j¼2
4. Solution approach Here, we propose a method that was specially designed to solve the model in hand. A numerical example is then presented to demonstrate the applicability of the proposed method.
Q Pu
mTD Xi ¼
zt X Xi þ Q i¼1
t1 ¼ z t þ 1; . . . ; z 1
! Lq for i ð27Þ
given that Xi > 0 and constraint (19) is not violated. (c) Calculate TC (d) Set t = t + 1 and repeat step (b). If constraint (19) is violated, stop. (e) Record the solution with the lowest TC value as TC(K). (10) If K = 0, set K = K + 1 and repeat steps (4)–(9). Else, if TC(K) < TC(K 1), set K = K + 1 and repeat steps (4)–(9). Otherwise, record the solution. The solution is the minimum TC for a specified n value, where the optimal z (continued on next page)
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is determined. Record it as TC(J). (11) If J = 0, set J = J + 1, reset K = 0, and repeat steps (4)–(10) Else, if TC(J) < TC(J 1), set J = J + 1, reset K = 0, and repeat steps (4)–(10). Otherwise, record the solution. This optimal solution corresponds to the overall minimum total cost of the optimal solutions found for all possible n values. (12) Stop When solving the model, it is worth noting that Td may also depend on the variable T11. For instance, it may be more optimal for the time of the first production lot to be longer than the actual disruption duration being experienced (T11 > Td). In this case, Td must be replaced by T11 in the optimization solving process. Otherwise, Td remains as the user input. The following theorem is an important finding that led to the development of step 9 of algorithm 1 in the heuristic. Theorem DD > 0, P 1. If i1 it is optimal to have 1 unit of lost sales for z1 P each þ 1 units of DD. i¼2 D
Proof. When lost sales exist, one can derive the formula for the new Xi, X 0i from Eq. (24), as
! i i1 X P X Xi ¼ X j Bq þ Lq Xj D j¼1 j¼2
for i ¼ 2; . . . ; z 1:
ð28Þ
i1 The difference X 0i X i is calculated as DP Lq for i = 2, . . . , z 1. P For any particular value of z, the effect of 1 unit Lq on zi¼1 X i is given Pz1 P i1 by i¼2 D Lq. As defined earlier, DD is the amount of unfulfilled demand given by Eq. (25). Naturally, we want DD to be zero. We assume that mTD is fixed, thus one can only reduce DD to zero Pz by increasing Lq, which in turn will increase i¼1 X i . Therefore, we Pz1 P i1 have the expression DD ¼ Lq þ i¼2 D Lq. Solving this for Lq will yield the optimal Lq value from the unmet demand, formulated as Lq ¼ Pz1 DPD i1 . This completes the theorem proof. h ð Þ þ1 i¼2 D
4.2. Numerical example We will solve a numerical example to show the applicability of the model. The problem data are as follows:
A1 ¼ 200; L1 ¼ 15; y ¼ 1;
A2 ¼ 20; L2 ¼ 15;
H1 ¼ 1:2; T d ¼ 0:003;
H2 ¼ 1:8;
B1 ¼ 1;
P ¼ 5000000;
B2 ¼ 1;
D ¼ 4000000;
qt ¼ 10000
Parameter
Range
A1 A2 H1 H2 B1,2 L1,2 Td
[20, 500] [20, 500] [1 ,10] [1 ,10] [5, 80] [20, 100] [u, T]
the optimal solution occurs at n = 4 and z = 9, where TC( n, z) = 586878.8. 5. Numerical analysis This section presents several experiments to demonstrate the performance of the proposed model and method developed in this research. The mathematical model was solved using the developed heuristic for 500 random test problems. From this total, 50 test problems were generated by randomly changing all the parameters for the given intervals. The remaining 450 test problems on the other hand, tested the heuristic for the upper and lower bounds of each parameter while the other parameters were random. 9 data sets with 50 test problems in each set were randomly generated. Table 2 summarizes the range of data of the problems that were tested. Notice that the upper and lower limits for parameter Td are shown in terms of other parameters based on the assumptions made for this model. To evaluate the performance of the proposed heuristic, the results of the heuristic are compared to the solutions obtained by the LINGO 10.0 optimization software. The percentage of error, which is the deviation of the objective function value of the heuristic (TC_H) from the optimal value of TC obtained by LINGO (TC_L) was used as a quality criterion. The percentage of error was calculated using the following equation:
%error ¼
TC H TC L 100 TC L
The experiment shows fairly good results with solution errors that vary from 0.00% to 0.000026% with an average of 0.0000005%. The deviation of the heuristic solutions from the optimal value is significantly small and can be considered negligible, thus proving that the heuristic algorithm is reliable in generating optimal solutions for the recovery model. The total cost function, TC, with respect to z for different n values was plotted using the data from the numerical example presented earlier. In Fig. 2, the plot shows convex curves for each n
The solution procedure described in the previous subsection was coded in MATLAB and executed on an Intel Core Duo processor with 1.99 GB RAM and a 2.66 GHz CPU. The computational results are presented in Table 1. We can see from the results that, based on the stopping criterion in step (10), for n = 4 and n = 5 the algorithm stops at z = 9 and z = 11 respectively. Comparing the two TC values,
Table 1 Computational results of numerical example. n
z
TC(n, z)
Remark
4 5
9 11
586878.8 603265.3
Lowest TC
ð29Þ
Fig. 2. TC versus z for different n values.
H. Hishamuddin et al. / Computers & Industrial Engineering 64 (2013) 552–561
559
Fig. 3. Effect of increasing L1 and L2 on n, Bq and Lq.
Fig. 4. The change of TC with respect to A1 and A2.
value, which proves that the model and solution procedure developed in this paper provides good solutions. The global minimum corresponds to the lowest point of all the n curves, in which case is at n = 4 and z = 9 for Fig. 2. Analysis of the results shows that the solution is highly dependent on the relationship between the backorder cost and the lost sales cost parameters. The optimal number of backorders and lost sales is a tradeoff between the back order and lost sales cost. This finding can be observed in Fig. 3, which shows the effect of increased lost sales cost for both stages, L1 and L2 on n, Bq, and Lq, when the back order cost is significantly low. With back order cost fixed at $1, and lost sales cost increasing from $2 to $145, it can be seen that the backorder quantity increases whereas the lost sales quantity decreases and eventually reduces to zero. Another important finding is that the optimal recovery duration, n, gradually increases. This observation can be explained as follows: For a low lost sales cost, it is found that lost sales are more attractive. Thus, when a portion of demand becomes lost sales, the quantity to be produced is reduced and the duration for recovery will be shorter. However, as the lost sales cost becomes larger, it is more cost effective to have backorders. Therefore, the amount of lost sales will reduce and the recovery duration will become longer for the latter case. This important finding is beneficial as it facilitates suppliers and retailers in predicting the amount of time needed to recover from a disruption before the firm can continue its normal operations. Moreover, recovery information is vital for firms to assist
Fig. 5. The change of TC with respect to H1 and H2.
Fig. 6. The change of TC with respect to L1 and L2.
in proper management of resources, such as labor, raw material and machines. In addition to the above experiment, a sensitivity analysis to explore the effects of various key model parameters with respect to total costs was conducted as well. It can be seen in Figs. 4, 6 and 7 that the setup cost, lost sales cost and back order cost have a linear increasing effect on total cost, which is an intuitive observation. A different trend is observed with the inventory holding
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Fig. 7. The change of TC with respect to B1 and B2.
the solutions, the results from the heuristic were compared with LINGO. A numerical example was also provided to demonstrate the applicability of the model to real life problems and a sensitivity analysis was conducted with respect to the key model parameters. Analysis of the results showed that the optimal recovery schedule is highly dependent on the relationship between the backorder cost and the lost sales cost parameters. In addition, the heuristic was able to give quality solutions for the model, with very small deviations of the heuristic solutions from the optimal value. This proves that the use of the proposed heuristic is acceptable and reliable. The presented model is useful to assist decision makers who can take a pro-active approach in maintaining business continuity in the event of a transportation disruption in the supply chain system. Managers of manufacturing companies may easily utilize the model and heuristic to solve daily or occasional transportation disruptions that affect their production system. Efficiently achieving a cost-effective recovery schedule in real time is crucial as it enhances productivity and overall business operations. There are several directions in which this research could continue. An important extension of the paper may involve extending the model to a multi-echelon supply chain with more than two stages and of also having multiple retailers. In addition, incorporating different shipment policies in the study is another potential and interesting area for future work.
References
Fig. 8. The change of TC with respect to Td.
cost, where total cost decreases following a parabolic curve as the parameter increases, as shown in Fig. 5. The total cost behavior as disruption duration increases was also examined. As illustrated in Fig. 8, the total cost remains constant until a certain point where it starts to increase linearly. This TC behavior can be explained by the condition that has been set for Td as described in the previous section, in which case that if Td is smaller than T11, then Td is replaced by T11. Looking at the curve, Td is found to be smaller than T11 initially, and so it is replaced by T11, which explains the static TC pattern. However, the Td value becomes less than T11 as the curve progresses, which consequently results in the increasing TC trend. Hence, the larger the Td value, the higher the stock out quantity, and thus the total cost curve reflects the increasing cost of the total stock outs that exist in the system due to this change in Td. 6. Conclusion In this study, a disruption recovery model for a two stage production and inventory system subject to transportation disruption was analyzed. The objective of the study was to determine the optimal production and ordering quantities for the supplier and retailer, as well as the duration for recovery, which yields the minimum relevant costs of the system. The cost structure for the above model was developed for the case where goods in transit were damaged during the disruption. As for the solution procedure, a self developed heuristic was used to obtain optimal solutions for the proposed model, where a total of 500 randomly generated test problems were solved using the heuristic. To judge the quality of
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