A disruption recovery plan in a three-stage production-inventory system

Computers & Operations Research 57 (2015) 60–72

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A disruption recovery plan in a three-stage production-inventory system Sanjoy Kumar Paul a,b,n, Ruhul Sarker a, Daryl Essam a a b

School of Engineering and Information Technology, The University of New South Wales, Canberra, Australia Department of Industrial and Production Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh

art ic l e i nf o

a b s t r a c t

Available online 17 December 2014

This paper proposes a recovery plan for managing disruptions in a three-stage production-inventory system under a mixed production environment. First, a mathematical model is developed to deal with a disruption at any stage while maximizing total profit during the recovery-time window. The model is solved after the occurrence of a disruption event, with changed data used to generate a revised plan. We also propose a new and efficient heuristic for solving the developed mathematical model. Second, multiple disruptions are considered, where a new disruption may or may not affect the recovery plans of earlier disruptions. The heuristic, developed for a single disruption, is extended to deal with a series of disruptions so that it can be implemented for disruption recovery on a real-time basis. We compare the heuristic solutions with those obtained by a standard search algorithm for a set of randomly generated disruption test problems, and that show the consistent performance of our developed heuristic with lower computational times. Finally, some numerical examples and a real-world case study are presented to demonstrate the benefits and usefulness of our proposed approach. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Production-inventory system Disruption recovery Mixed-production environment Heuristic Pattern search

1. Introduction Batch production is a very common and popular technique in manufacturing systems in which products are produced in batches to minimize the overall production cost while maximizing utilization of the available capacity. Sometimes, the batch size and processing time can be constant, depending on the nature of the process, as well as on the capacity of the equipment. In some cases, the processing time can be either dependent or independent of the batch sizes. For example, the mixing time of raw materials is independent of the batch size because a quantity of them which does not use the full capacity of the equipment can be mixed. In real-life production lines, it is very common to process materials or products in a series of stages, one after another, to obtain the final products. There are numerous industries, such as pharmaceutical, textile and manufacturing that produce products using multiple stages, during which the production environment can be either similar or different, and processes such as batch or continuous production or a combination of both. Even if the production environment is continuous, a product may be produced in batches due to there being higher production capacity than demand. In these real-life situations, production disruptions are

n Corresponding author at: School of Engineering and Information Technology, The University of New South Wales, Canberra, Australia. E-mail address: [email protected] (S.K. Paul).

http://dx.doi.org/10.1016/j.cor.2014.12.003 0305-0548/& 2014 Elsevier Ltd. All rights reserved.

common events which occur for different reasons and, moreover, may take place at any time and in any stage of the system. As an organization can face a huge financial, as well as goodwill, loss due to a disruption in its system [33], it is important to develop a suitable recovery plan to minimize the effects of such a disruption. This research has been motivated by the disruption scenarios observed in a real-life pharmaceutical production line. That production line consists of three sequential processes (known as mixing, compression, and packaging) which can easily be defined as three stages of the production process. The production process starts with a discrete batch production in the mixing stage and is followed by two continuous production processes in the compression and packaging stages. The production line is sometime disrupted, mainly due to machine breakdowns that occur at any stage of the line without having any prior knowledge. Although management repairs the machines as soon as possible, it is not easy to reschedule the production line to minimize the overall loss with a minimum effect on customer goodwill. This is a common problem in many industrial units, and hence it requires a new realtime problem solving approach, such as the disruption recovery method proposed in this paper. Over the last few decades, one of the most important research topics, in operations research and computer science, has been production-inventory systems. During the early stage, researchers focused on developing inventory models under ideal conditions; for example, the development of the basic economic order quantity (EOQ) model ([12] (reprinted from 1913) and [37]) and basic

S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72

economic production quantity (EPQ) model [34] which was an extension of the EOQ model. Later, many researchers used these models in their studies; for example, Cheng [40] considered production process reliability in a single-stage imperfect production process to develop an EPQ model. Goyal [9] also applied the basic concept of the EPQ model to determine optimal lot sizes in a two-stage production system that minimized the sum of all costs. Other such extensions of EPQ models in single-stage productioninventory systems were developed by Ishii and Imori [16], Graves [11], Biskup et al. [2], Chan and Song [3], Dave [6], Chiu et al. [5], Pentico et al. [29], Paul et al. [24], Sarkar and Moon [31], and Kiesmüller et al. [17]. However, the above studies focused mainly on either single- or two-stage production systems whereas, on many production lines, products are processed in multiple stages [10]. A few researchers analysed multiple-stage production systems that included: (i) the determination of optimal production policies for single-item and multi-stage production systems [35,22], (ii) the modification of the single end-item production lot sizing model considering work-inprocess inventories [1], (iii) the determination of safety stocks in multi-stage inventory systems with normally distributed demands [15], (iv) the consideration of reworking options in a multi-stage production inventory to minimize the inventory cost of work-inprocess and finished goods [32]. Some additional studies of multistage production-inventory systems were those by Konak et al. [19], Glock and Jaber [8], and Kim and Glock [18]. Past studies of single- or multiple-stage production-inventory systems were conducted in ideal production environments but there is very little research work in the literature on managing disruptions in production-inventory systems. However, the impacts of machine breakdowns have been analysed in several studies. Some examples are: the development of a (s,S) production-inventory policy with random disruptions and exponential times between breakdowns in an unreliable bottlenecked system [23], the analysis of the impact of machine breakdown on an EPQ model for deteriorating items in a single-stage production system considering a fixed period of repair time [20], the extension of the model of Lin and Gong [20] for deteriorating items with random machine breakdowns and stochastic repair times with uniform and exponential distributions [36], the development of an EPQ model with a Poisson-distributed machine breakdown for determining an optimal production run-time [5], and the development of a robust plan for a machine breakdown and reworking failure [4]. Recently, a mathematical model for a single disruption recovery within a single-stage, single-item production system was developed by Hishamuddin et al. [13]. They proposed a heuristic for solving the disruption recovery problem that considered back orders and lost sales costs. An extension of that problem which included demand uncertainty and process reliability for a singlestage production-inventory system was investigated by Paul et al. [25]. Recently, Hishamuddin et al. [14] introduced concepts for managing a single transportation disruption in a two-echelon serial supply chain system involving both the producer and retailer. Interested readers can refer to Qi et al. [30], Xia et al. [38], Mohebbi [21], Eisenstein [7], Yang et al. [39] and Paul et al. [26] for other disruption recovery models within the context of production-inventory systems. Existing studies considered either single- or two-stage batch production-inventory systems, with disruption recovery policies for only a single disruption. In this paper, we consider a three-stage production system and deal with single as well as multiple disruptions on a real-time basis. We consider a disruption event that is not known and cannot be predicted in advance. We first develop a mathematical model for coping with a single disruption in any stage. In our experimental study, we use a uniformly random probability distribution to generate disruption parameters, such as disrupted

61

stage, and pre-disruption and disruption periods. Then, we solve the mathematical model after the occurrence of a disruption and use the estimated recovery time to develop a revised plan. We assume that the batch size in the first stage is limited by the capacity of the equipment, and that the processing time is constant and independent of the batch size but that, in the second and third stages, the processing time is proportional to the batch size. After processing in the first stage, the whole batch is transferred to the second stage for further processing and may then be split into smaller lots depending on the capacity of the transfer bucket between the production stages. In this paper, we propose a new and efficient heuristic for solving the developed mathematical model, with its results compared with the solutions obtained from a pattern search using a set of randomly generated disruption test problems. We also consider multiple disruptions, one after another in a series, that occur in any stage at any point in time and may or may not affect the plans amended after previous disruptions. If a new disruption occurs during the recovery-time window of another, a new revised plan which considers the effects of both disruptions must be derived. Accordingly, as this is a continuous process, we extend the heuristic to deal with a series of disruptions on a real-time basis by incorporating a modified version of that developed for a single disruption. This is the first quantitative model that develops a disruption recovery model for both a single and multiple disruptions, on a real-time basis, in a three-stage mixed productioninventory system. Finally, we show how the proposed methodology can be applied to real-time disruption recovery planning, with randomly generated test problems, as well as a real-world case problem from the aforementioned pharmaceutical company. The main contributions of this paper can be summarized as follows: (i) Development of a mathematical model for disruption recovery in a three-stage production-inventory system. As a disruption scenario is not known in advance and not possible to predict, the recovery plan is revised for periods after the disruption occurs on a real-time basis. (ii) Development of a new efficient heuristic for generating a revised production plan after a disruption. (iii) Extension of this heuristic to deal with multiple disruptions on a real-time basis. As any new disruptions may or may not affect the plans revised after previous ones, their scenarios may be considered dependent and independent, both of which the extended heuristic can handle. (iv) Application of the developed methodology to a real-world case problem from a pharmaceutical company. The remainder of the paper is organized as follows. The problem description and recovery strategy are presented in Section 2 and the model formulation in Section 3. The solution approaches, and experimentation and results analysis, are provided in Sections 4 and 5 respectively. A real-life case study is presented in Section 6 and, finally, conclusions are drawn and future research directions suggested in the last section.

2. Problem description and recovery strategy We consider the ideal three-stage production system, as shown in Fig. 1. In it, stage 1 processes the raw materials as a batch and the production procedures in stages 2 and 3 are continuous. As the system requires different processing techniques, we recognize it as a mixed-production environment. In Fig. 1, X 0 is the batch size in the first stage and, as it is less than or equal to the capacity of the equipment, the processing time is independent of it and, therefore,

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Batch size = X0

Quantity

Stage 1: Batch productionfixed batch size and constant processing time T0

Quantity

I1+St1

Time Stage 2: Continuous production with sub-lots (size of each sub-lot = Y0)

Usage of raw material Y0

Production

I2+St2

Time

Quantity Z0

Production

Stage 3: Continuous production with sub-lots (size of each sub-lot = Z0)

I3+St3

Time Storage

Fig. 1. Ideal three-stage production system.

fixed. After the first stage, the whole batch is transferred to the second stage for further processing into smaller lots (size Y 0 ), whereby Y 0 is limited by the capacity of the transfer bucket and is equal to X 0 =n, where n is a positive integer. Then, each batch is transferred to the third stage for further processing with a lot size of Z 0 and, finally, the finished products are transferred to storage. A disruption is any kind of interruption in a production system, for example, a machine breakdown, raw material shortage, power cut, labour strike, etc., which can happen in any stage and at any time in the process. Once the system is disrupted, it is assumed that it will be inoperable for a period of time, known as a disruption period, with the strategy taken to recover from the disruption known as a recovery strategy. In this paper, to recover from a disruption, the following two cost factors are considered. (i) Back orders: The portion of demand that cannot be fulfilled at the scheduled time but will be delivered at a later date, with a penalty, if the production system is capable. (ii) Lost sales: The portion of demand lost if customers will not wait for the required stock to be replenished following the production process not being capable of fulfilling demand. After the occurrence of a disruption in a system, its planned production quantities for all stages are revised for some periods in the future (known as recovery periods or recovery-time windows) until the system returns to its normal schedule, which is known as a recovery plan [28]. As, if a disruption occurs in any stage, it has significant impacts on other stages because they all operate in a coordinated fashion, all their production plans must be revised after the occurrence of a disruption to minimize the overall loss incurred. Although, in this paper, the recovery periods is assumed to be specified by the management of the system, it can be considered a decision variable in the model. In any industrial production environment, the system can face multiple disruptions, one after another, on a real-time basis. In this case, one disruption can occur within the recovery periods of another which is known as a dependent disruption and, as this is a complex situation, the combined effect of dependent disruptions should be considered in the development of a recovery plan. This is

achieved by re-optimizing the production schedule within the new recovery-time window under the changed production environment. The proposed heuristic (discussed earlier) for dealing with a single disruption is later extended to consider multiple disruptions on a real-time basis and is capable of handling dependent, independent and mixes of dependent and independent disruptions on a realtime basis. In this paper, we make the following assumptions: (i) The production rate in any stage is greater than the average demand rate. (ii) A single item is produced in the system. (iii) The sizes of the sub-lots in stages 2 and 3 are determined based on the capacity of the transfer bucket. (iv) The recovery-time window begins immediately after the occurrence of a disruption which can occur in any stage at any point in time. The model is developed for a single type of item and, as is common, assumes that, to fulfil demand on time, the production rate is higher than the demand rate. However, for a higher demand rate, it can easily be revised by using an option for outsourcing. In a multi-stage production environment, it is common to use a transfer bucket to transport semi-processed materials between stages. In this paper, we assume that the sub-lot sizes are determined based on the capacity of the transfer bucket to balance the production system. To make the recovery process meaningful in practice, the revised plan is generated after the disruption is experienced by the system, that is, on a real-time basis.

3. Model formulation In this section, we develop a mathematical model for a single disruption in any stage in a three-stage production system. Firstly, we derive the equations for different costs and revenue, and then formulate the disruption recovery problem as a constrained mathematical model in which the objective is to maximize total profit during the recovery periods, with the total profit function derived by

S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72

subtracting all the relevant costs from the revenue. Finally, we develop relevant capacity, delivery, demand and stage-balancing constraints to make the model realistic. The decision variables are revised production quantities in each cycle, and both the total back orders and lost sales during the recovery periods. In modelling the recovery planning problem, we consider different batch sizes for different cycles, for example, X i for cycle i in stage 1 and, similarly, Y i and Z i in stages 2 and 3 respectively. Although one lot size can be considered for all cycles in a stage, allowing different sizes makes the model more flexible (or more general) for optimization. As the model uses both of back orders and lost sales costs, its solution may suggest different quantities for different production cycles in order to maximize total profit. To establish balanced coordination among the stages, we consider that the lot sizes in stages 2 and 3 are equal. However, one can easily modify the model to have a different relationship, such as one lot size in stage 3 being equal to multiple lot sizes in stage 2 or vice versa.

3.1. Notations To formulate the mathematical model, the following notations are used. X0 T0 St1 St2 St3 A1 CP 1 CM 1 D CP 2 C2 H2 CP 3 C3 n

Y0 Z0 I1 I2 I3 Tp Td TR M Xi Yj Zj

batch size in stage 1 under ideal conditions (known) processing time of batch in first stage under ideal conditions (constant and known) set-up time in stage 1 (time per set-up) set-up time in stage 2 (time per set-up) set-up time in stage 3 (time per set-up) set-up cost in stage 1 ($ per set-up) production cost per unit in stage 1 ($ per unit) cost per unit material loss in stage 1 ($ per unit) average demand per unit time production cost per unit in stage 2 ($ per unit) capacity of machine in stage 2 (units per unit time) holding cost per unit per unit time in stage 2 ($ per unit per unit time) production cost per unit in stage 3 ($ per unit) capacity of machine in stage 3 (units per unit time) number of sub
lots in stages 2 and 3 for each full batch in stage 1 ¼ ⌈ðX 0 =capacity of transfer bucket between stages 2 and 3Þ⌉ size of each sub-lot in stage 2 under ideal conditions ¼ ðX 0 =nÞ size of each sub-lot in stage 3 under ideal conditions ¼ ðX 0 =nÞ idle time after processing batch in stage 1 ¼ ðX 0 =DÞ
T 0
St1 idle time after processing n sub-lots in stage 2 ¼ ðX 0 =DÞ
ðX 0 =C 2 Þ
St2 idle time after processing n sub-lots in stage 3 ¼ ðX 0 =DÞ
ðX 0 =C 3 Þ
St3 pre-disruption period disruption period recovery periods number of batches within recovery periods size of batch i in stage 1 after disruption; f or i ¼ 1; 2; …; M (decision variable) size of sub-lot j in stage 2 after disruption; f or j ¼ 1; 2; …; Mn (decision variable) size of sub-lot j in stage 3 after disruption; f or j ¼ 1; 2; …; Mn (decision variable)

B L m1

63

back orders cost per unit per unit time ($ per unit per unit time) lost sales cost per unit ($ per unit) mark-up of unit selling price [m1 ðCP 1 þ CP 2 þ CP 2 Þ] (must be greater than 1)

3.2. Cost and revenue calculations In this section, we derive the different cost and revenue equations for a single disruption's recovery-time window considering the set-up, holding, production back orders and lost sales costs. The total set-up cost is determined as the cost per set-up multiplied by the number of set-ups, and the production cost the per unit production cost multiplied by the quantity produced, both during the recovery periods. The average holding cost is calculated as the unit holding cost multiplied by the total inventory during the recovery periods, the back orders cost as the unit back order cost multiplied by the number of back order units and its time delay [27], and the lost sales cost as unit lost sales cost multiplied by the number of lost sales units [27]. The revenue during the recovery periods is determined as the unit selling price multiplied by the quantity produced during the recovery periods, and total profit, which is the objective function, calculated by subtracting all the costs from the revenue. 3.3. Mathematical model for disruption at first stage In some production environments, a batch of products (or semiproducts) may be unacceptable due to the effects of a disruption during processing. In others, an entire batch of products may be either unaffected or recovered by applying corrective actions. For this reason, we consider two scenarios for disruption recovery in our mathematical model: (i) no loss of materials; and (ii) 100% loss of materials. However, any further scenarios can easily be incorporated. 3.3.1. Scenario 1: no loss of materials In this scenario, as there is no loss of materials due to a disruption, pre-disruption processed materials can be used during the recovery periods. Number of batches in recovery periods, M ¼ ⌈T R =ðT 0 þ I 1 Þ⌉ Cost formulation ð1Þ

Set
up cost in first stage ¼ A1 M M

ð2Þ

Production cost in first stage ¼ CP 1 nðX 1 þ X 2 þ ⋯ þ X M Þ ¼ CP 1 n ∑ X i i¼1

Mn

Production cost in the second stage ¼ CP 2 nðY 1 þ Y 2 þ ⋯ þ Y Mn Þ ¼ CP 2 n ∑ Y j j¼1

ð3Þ Average raw material holding cost in the second stage ¼
 M H2 X1 X2 XM H2 2 n X1 þ X2 þ ⋯ þ XM n ∑ Xi ¼ 2 C2 C2 C2 2C 2 i ¼ 1

ð4Þ Mn

Production cost in the third stage ¼ CP 3 nðZ 1 þ Z 2 þ⋯ þ Z Mn Þ ¼ CP 3 n ∑ Z j j¼1

ð5Þ k

Back orders cost ¼ Bn ∑ X i ðT d þ ði
1ÞSt1
iI 1 Þ i¼1

M

Lost sales ¼ Ln MX 0
∑ X i i¼1

ð6Þ

! ð7Þ

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S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72

Revenue formulation

M

M

Revenue ¼ m1 ðCP 1 þ CP 2 þ CP 3 Þn ∑ X i

ð8Þ

i¼1

Final mathematical model The objective function, total profit¼total revenue
total costs, which is to be maximized and obtained by using Eqs. (1)–(8) and subject to constraints (9)–(17).   X i r X 0 ; 8 i to ensure delivery and transportation constraints ð9Þ   Y j rY 0 ; 8 j to ensure delivery and transportation constraints

!

Lost sales ¼ Ln MX 0
∑ X i

ð25Þ

i¼1

Revenue formulation M

Revenue ¼ m1 ðCP 1 þ CP 2 þ CP 3 Þn ∑ X i

ð26Þ

i¼1

Final mathematical model The objective function, total profit ¼total revenue
total costs, which is to be maximized and subject to constraints (27)–(35).   X i r X 0 ; 8 i to ensure delivery and transportation constraints ð27Þ

ð10Þ 

Z j r Z 0 ; 8 j to ensure delivery and transportation constraints

  Y j r Y 0 ; 8 j to ensure delivery and transportation constraints



ð28Þ

ð11Þ M   MX 0 Z ∑ X i to ensure delivery and transportation constraints

  Z j rZ 0 ; 8 j to ensure delivery and transportation constraints ð29Þ

i¼1

ð12Þ   I 1 ; I 2 ; I 3 Z0 to ensure non
negative idle time

ð13Þ

  Y ði
1Þn þ 1 þ Y ði
1Þn þ 2 þ⋯ þ Y in ¼ X i ; 8 i to balance batches and sub
lots

ð14Þ   Y j ¼ Z j ; 8 j to balance production system  T d þ ðk
1ÞSt1
kI 1 Z 0; For k ¼ 1; 2; …;

ð15Þ

  Td  to ensure non
negative delay I1

ð16Þ X i ; Y j and Z j non
negative; 8 i; j

ð17Þ

3.3.2. Scenario 2: 100% loss of materials In this scenario, as materials are completely lost due to a disruption, the pre-disruption processed materials cannot be used during the recovery periods. Therefore, we propose a recovery policy which, as the disruption period is considered a summation of the pre-disruption and actual disruption periods, it is (T d þ T p ). Cost formulation Set
up cost in first stage ¼ A1 M

ð18Þ M

Production cost in first stage ¼ CP 1 nðX 1 þ X 2 þ ⋯ þ X M Þ ¼ CP 1 n ∑ X i i¼1

ð19Þ Cost due to material loss ¼ CM 1 nX 0

ð20Þ

M   MX 0 Z ∑ X i to ensure delivery and transportation constraints i¼1

ð30Þ   I 1 ; I 2 and I 3 Z 0 to ensure non
negative idle time

ð31Þ



Y ði
1Þn þ 1 þ Y ði
1Þn þ 2 þ ⋯ þ Y in ¼ X i ; 8 i to balance batches  and sub
lots

ð32Þ

  Y j ¼ Z j ; 8 j to balance production system

ð33Þ 

T d þ T p þ ðk
1ÞSt1
kI 1 Z 0; For k ¼ 1; 2; …;   to ensure non
negative delay X i ; Y j and Z j non
negative ; 8 i; j

Td þTp I1

 ð34Þ ð35Þ

Proposition 1. The production system will be optimally recovered using only the back orders cost if (i) T d r MI 1 for no loss of materials and (ii) ðT d þT p Þ r MI 1 for 100% loss of materials. Proof. The idle time after producing a batch in stage 1 is I 1 . As there are M cycles in the recovery-time window ( T R ), the total idle time in it is MI 1 . If the disruption period is less than MI 1 , the production system is capable of managing the disruption period within T R and there will only be a delay in product delivery. Therefore, it can be said that, if T d r MI 1 for no loss of materials and ðT d þT p Þ r MI 1 for 100% loss of materials, the production process will be optimally recovered using the back orders cost.

Mn

Production costin the second stage ¼ CP 2 nðY 1 þ Y 2 þ ⋯ þ Y Mn Þ ¼ CP 2 n ∑ Y j j¼1

ð21Þ H2 Average raw material holding costin the second stage ¼ 2
 M X1 X2 XM H2 2 n X1 þ X2 þ ⋯ þ XM n ∑ Xi ¼ C2 C2 C2 2C 2 i ¼ 1

Proposition 2. Sales will be lost in the recovery policy if (i) T d 4 MI 1 for no loss of materials and (ii) ðT d þT p Þ 4 MI 1 for 100% loss of materials. Proof. This is the opposite consequence to that in Proposition 1.

ð22Þ

Proposition 3. The minimum recovery periods, without incurring lost sales in the solution, is ðT 0 þ I 1 Þn⌈T d =I 1 ⌉ for no loss of materials and ðT 0 þ I 1 Þn⌈ðT d þT p Þ=I 1 ⌉ for 100% loss of materials.

Mn

Production costin the third stage ¼ CP 3 nðZ 1 þ Z 2 þ ⋯ þ Z Mn Þ ¼ CP 3 n ∑ Z j j¼1

ð23Þ k

Back orders cost ¼ Bn ∑ X i ðT d þ T p þ ði
1ÞSt1
iI 1 Þ i¼1

ð24Þ

Proof. From the condition of the existence of only back orders for no loss of materials, T d r MI 1 ) MZ

Td I1

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As we consider M an integer, we use   T ) MZ d I1   TR T ) Z d ½inserting value of M T 0 þ I1 I1   T ) T R Z ðT 0 þ I 1 Þn d I1   Td ) T R Z ðT 0 þ I 1 Þn I1

65

4. Solution approaches

ð36Þ

 Similarly; for 100% loss of materials; T R Z ðT 0 þ I 1 Þn⌈ T d þT p =I 1 ⌉

With the help of the above propositions, a heuristic is first developed to solve the model for a single occurrence of disruption. Then, it is extended to manage multiple disruptions, as a series, on a real-time basis. A disruption scenario can be defined as the combination of a disrupted stage, and pre-disruption and disruption periods. As, in reality, these parameters follow stochastic processes, in this paper, we use uniform random variables to generate them. However, one can use different probability distributions. To judge the quality of the heuristic solutions, another standard search technique is also used to solve the model.

ð37Þ 4.1. Heuristic for single disruption 3.4. Mathematical model for disruption in the second stage The formulation of the mathematical model for a single occurrence of disruption in stage 2 is presented in Appendix A. Proposition 4. The production system will be optimally recovered using only the back orders cost if T d r MI 2 and using both the back orders and lost sales costs if T d 4 MI 2 . Proof. As there are M cycles in the recovery-time window (T R ), the total idle time in it is MI 2 . If the disruption period is less than MI 2 , the production system is capable of producing all the lost items due to the disruption within T R . So there will be a delay in product delivery, and in this case, having only back orders will be the most appropriate for generating the optimal solution. Therefore, it can be said that the production system will be optimally recovered using the back orders cost if T d r MI 2 . If the disruption period is greater than the idle time, the system is not capable of producing all the lost items during the recovery periods. In this case, the system will utilize its idle time to produce some of the lost items. So the system will experience both lost sales and back orders (delay delivery) in generating solutions if T d 4 MI 2 . Proposition 5. The minimum recovery periods, without incurring  lost sales in the solution, is ðX 0 =C 2 Þ þ I 2 n⌈T d =I 2 ⌉ Proof. From the condition of the existence of only back orders T d r MI 2 As we consider M an integer, we use   T ) MZ d I2   TR T ) Z d ½inserting value of M ðX 0 =C 2 Þ þ I 2 I2
   X0 T ) TR Z þ I 2 n d ðprovenÞ C2 I2

ð38Þ

The proposed heuristic for a single occurrence of disruption in any stage is described. Different parameters for an ideal system are input in Step 1 and disruption scenarios generated using uniformly random variables input in Step 2. If there is any disruption in stage 1, in Step 3, the model determines a recovery plan which is generated in Steps 4 and 5 after occurrences of disruptions in stages 2 and 3 respectively. In steps 4 and 5, the production system utilizes the idle time for possible recovery from a disruption. If the idle time is greater than the disruption period, then the production system is capable of producing the lost items due to the disruption, but there will be a delay in product delivery. For this reason, back orders will be the only appropriate means to generate the solution. If the idle time is less than the disruption period, then the production system is capable of producing only a part of the lost items during the recovery periods, and so the system will generate partly lost sales and partly delayed delivery (for back orders). When lost sales are appropriate for generating the solution, we determine the minimum recovery periods, without incurring lost sales, by using Propositions 5 and 7. The steps in the proposed heuristic are as follows. Step 1: Input scenarios of three stages in ideal production system. Step 2: Input disrupted stage with pre-disruption (T p ) and actual disruption (T d ) period. Step 3: For disruption in stage 1, determine disruption period. If  materials completely lost, disruption period equal to T p þ T d , otherwise, T d . 3.1: Input recovery periods (T R ). 3.2: Determine number of batches and total idle time in T R . 3.3: If total idle time Zdisruption period, obtain recovery plan using Eqs. (39)–(41). Xi ¼ X0; 8 i Y j ¼ Y 0; 8 j Zj ¼ Z0; 8 j

(39) (40) (41)

3.5. Mathematical model for disruption in the third stage The formulation of the mathematical model for a single occurrence of disruption in stage 3 is presented in Appendix B. Proposition 6. The production system will be optimally recovered using only the back orders cost if T d rMI 3 and using both the back orders and lost sales costs if T d 4 MI 3 . Proof. This is the same as that for Proposition 4. Proposition 7. The minimum recovery periods, without incurring  lost sales in the solution, is ðX 0 =C 3 Þ þ I 3 n⌈T d =I 3 ⌉ Proof. This is the same as that for Proposition 5.

Determine all costs, revenue and total profit. 3.4: If total idle time odisruption period, obtain recovery plan using Eqs. (42)–(47). X 1 ¼ 0; X i ¼ X 0 for i ¼ 2; 3; …; M Y j ¼ 0 for j ¼ 1; 2; …; n Y j ¼ Y 0 for j ¼ n þ 1; n þ 2; …:; Mn Z j ¼ 0 for j ¼ 1; 2; …; n Z j ¼ Z 0 for j ¼ n þ 1; n þ 2; …; Mn Determine all costs, revenue and total profit.

(42) (43) (44) (45) (46) (47)

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S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72

Minimum recovery periods, without including lost sales in solution, ðT 0 þ I 1 Þn⌈ðdisruption period=I 1 Þ⌉. Step 4: For disruption in stage 2 4.1: Iinput disrupted sub-lot number (l), disruption period ( T d ) and recovery periods (T R ). 4.2: Determine total idle time in T R . 4.3: If total idle time Zdisruption period, obtain recovery plan using Eqs. (48)–(50). Xi ¼ X0; 8 i Yj ¼ Y0; 8 j Zj ¼ Z0; 8 j

(48) (49) (50)

and determine all costs, revenue and total profit. 4.4: If total idle time o disruption period, Lost time (T L ) ¼ T d
total idle time, Production capacity in lost time ¼ C 2 T L and

 Number of sub-lots in lost time (m) ¼ C 2 T L =Y 0 Then, obtain recovery plan using Eqs. (51)–(58). Y j ¼ Y 0 for j ¼ 1; 2; …; 2n Y j ¼ 0 for j ¼ 2n þ 1; 2n þ 2; …; 2n þ m Y 2n þ m þ 1 ¼ Y 0
C 2 T L þ mY 0 Y j ¼ Y 0 for j ¼ 2n þ m þ2; 2n þ m þ 3; …; Mn X1 ¼ X0; X2 ¼ X0; X i ¼ Y ði
1Þn þ 1 þY ði
1Þn þ 2 þ⋯ þ Y in for i ¼ 3; 4; …; M Zj ¼ Y j; 8 j

(51) (52) (53) (54) (55) (56) (57) (58)

and determine all costs, revenue and total profit. Minimum recovery periods, without including lost sales in  solution, ðX 0 =C 2 Þ þ I 2 n⌈T d =I 2 ⌉. Step 5: For disruption in stage 3 5.1: Input disrupted sub-lot number (l), disruption period ( T d ) and recovery periods (T R ). 5.2: Determine total idle time in T R . 5.3: If total idle time Zdisruption period, obtain recovery plan from Eqs. (59) to (61). Xi ¼ X0; 8 i Yj ¼ Y0; 8 j Zj ¼ Z0; 8 j

(59) (60) (61)

and determine all costs, revenue and total profit. 5.4: if total idle time o disruption period, Lost time (T L ) ¼ T d
total idle time, Production capacity in lost time ¼ C 3 T L

 Number of sub-lots in lost time (m) ¼ C 3 T L =Z 0 Then, obtain recovery plan from Eqs. (62) to (69). Z j ¼ Z 0 for j ¼ 1; 2; …; 2n Z j ¼ 0 for j ¼ 2n þ 1; 2n þ 2; …; 2n þ m Z 2n þ m þ 1 ¼ Z 0
C 3 T L þ mZ 0 Z j ¼ Z 0 for j ¼ 2n þ m þ 2; 2n þ m þ 3; …; Mn Y j ¼ Zj; 8 j X1 ¼ X0; X2 ¼ X0; X i ¼ Y ði
1Þn þ 1 þY ði
1Þn þ 2 þ⋯ þ Y in for i ¼ 3; 4; …; M

4.2. Extended heuristic for multiple disruptions In this section, the heuristic developed for recovering from a single occurrence of disruption, is extended to manage multiple disruptions on a real-time basis. To do this, a recovery plan is obtained from the heuristic after each disruption, with the revised production scenarios saved and then used as a foundation for recovering from the next disruption. The steps in the extended heuristic for managing multiple disruptions are described below. Step A: Determine and input ideal conditions (fixed batch size, sub-lot size, machine capacity, difference cost data etc.) Step B: Input disrupted stage, disruption period and time since previous disruption Step C: Solve model with proposed heuristic developed in Section 4.1 using updated parameters, such as disruption scenario, objective function and constraints Step D: Update decision variables from Step C and record revised production plan after disruption Step E: If another disruption, go to Step B and repeat Steps B–D Step F: Stop

4.3. Alternative approach The mathematical model developed in Section 3 for a single disruption is a constrained nonlinear one which can be solved using a standard search algorithm. As there is no standard test set available for the problem considered in this research, to validate and judge the quality of the results obtained from the heuristic, we chose another approach, namely a pattern search (PS)-based technique, to solve it. Both methods were coded in MATLAB R2012a and executed on an Intel core i7 processor with 8.00 GB RAM and a 3.40 GHz CPU, with their best results from 10 independent runs compared. In the PS-based technique, the following parameters were used. Maximum number of iterations: 100*Number of decision variables Maximum function evaluation: 1,000,000 Polling order: Random X tolerance: 1e
10 Function tolerance: 1e
10 Nonlinear constraint tolerance: 1e
10 Cache tolerance: 1e
10 Search method: Latin hypercube Other parameters were set as the defaults in the optimization toolbox of MATLAB R2012a. 5. Analysis of experimentation and results

(62) (63) (64) (65) (66) (67) (68) (69)

and determine all costs, revenue and total profit. Minimum recovery periods, without including lost sales in  solution, ðX 0 =C 3 Þ þ I 3 n⌈T d =I 3 ⌉. Step 6: Record the results. Step 7: Stop.

In this section, the solutions for both the single and multiple disruption cases are analysed. To judge their quality those obtained from the proposed heuristic, we have experimented using 90 disruption test problems randomly generated using a uniform random distribution by changing the parameters for the given intervals (presented in Section 5.1) which were solved using both the proposed heuristic and PS-based approaches. As their results were consistent, we discuss only those for a few sample test problems in this section. Then, the heuristic was modified for multiple disruptions on a real-time basis, as described in Section 4.2. 5.1. Range of parameters For experimentation, we considered the following data range with a discrete uniform distribution for the disruption problem.

S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72

Disruption period in first stage: [0.0001, T 0 ] Disruption period in the second stage: [0.0001,ðX 0 =C 2 Þ] Disruption period in the third stage: [0.0001,ðX 0 =C 3 Þ] Lost sales cost: L ¼ [2, 50] Back orders cost: B ¼ [0.1, 10] Set-up cost: A1 ¼ [5, 300] Holding cost: H 2 ¼ [0.005, 2] Production cost: CP 1 ,CP 2 , CP 3 ¼ [0.5, 10]

5.2. Single disruption In this section, the solutions for a single disruption are analysed. Although we experimented on 90 random disruption test problems, for illustrative purposes, five different sample instances were used by arbitrarily changing the disruption data, with their parameters shown in Table 1. For the disruption in stage 1, the two scenarios, that is, with no loss of materials and 100% loss of materials, were considered test problems 1 and 2 respectively, with three other problems used for disruptions in stages 2 and 3, and the following data for analysing the results. Although, as production in stages 2 and 3 is continuous, we considered that no set-up was required for them, it could be considered if appropriate. D¼ 8000 units per day, X 0 ¼3000 units, T 0 ¼0.30 days, St1 ¼1 h, St2 ¼ 0, St3 ¼ 0, A1 ¼50, C 2 ¼10,000 units per day, C 3 ¼ 9000 units per day, CP 1 ¼5, CP 2 ¼3, CP 3 ¼ 1, CM 1 ¼5, H 2 ¼0.01 per unit per day, bucket capacity ¼ 1000 units, B¼ 0.5, L ¼20, m1 ¼2.5, T R ¼ 2 days The test problems shown in Table 1 were solved using the proposed heuristic and PS-based techniques, with the results presented in Table 2. Both approaches provided similar solutions to all test problems. To recover from disruptions, only the back orders cost was used for problems 1, 3 and 4, only the lost sales cost for problem 2 and both for problem 5. If any recovery plan used the lost sales cost, the minimum recovery periods without lost sales were also determined. For problems 2 and 5, the minimum recovery periods were 2.667 and 2.625 days respectively (when using only the back orders cost). If management does not specify the recovery periods, the system may use this minimum recovery periods to avoid the lost sales cost.

Table 1 Five test problems with single disruption. Disruption problem

Disrupted stage

Disruption period (days)

Pre-disruption period (days)

1

1 (no loss of materials) 1 (100% loss of materials) 2 3 3

0.15

0.00

0.23

0.035

0.20 0.10 0.28

0.060 0.085 0.008

2 3 4 5

67

To judge the quality of the solutions obtained from our proposed heuristic, 90 test problems with disruption were randomly generated using a uniform distribution. They were solved through both approaches, with the best results (out of 10 independent runs) obtained by the PS technique. Although the two approaches produced similar solution quality (with insignificant differences, as shown in Table 3), there was a significant difference in their computational times. In terms of quality, the average percentage of deviation, calculated using Eq. (70), between the results from the two approaches was only 0.00034% which could be considered negligible. Indeed, it may merely have been due to errors in rounding the values of the decision variables. Apart from its capability to produce quality solutions, the heuristic took significantly less average computational time than the PS technique (see the second column in Table 3). percentage of deviation ¼

ðtotal profit from heuristc
total profit from PSÞ  100% total profit from PS

ð70Þ 5.3. Multiple disruptions To demonstrate the usefulness of our proposed heuristic for solving different scenarios with a series of disruptions over a period of time, we used the basic data of the single disruption problem presented in Section 5.2 which could manage the first disruption. Then, if another occurred after the recovery periods of a previous one, it could be considered another single disruption that would not affect the revised plan based on the previous disruption. However, as a new disruption within the recovery periods of any previous one may affect this revised plan, its revised plan for its recovery periods must be considered a set of additional restrictions. For experimental purposes, we randomly generated a series of ten dependent disruptions, one after the other, with different conditions, as shown in Table 4. Although they could happen continuously, we present only ten as a sample representation. To maximize total profit in the system, the batch and sub-lot sizes were revised immediately after each disruption took place. The problem was then solved using the proposed heuristic for multiple disruptions, as presented in Section 4.2, with the results recorded after each disruption and shown in Table 5 for lost sales, back orders and total profit. It is observed that, in the recovery plans, there were only back orders for disruption numbers 1, 7, 8, 9 and 10, and to maximize total profit, only lost sales for disruption number 6 while both lost sales and back orders were present for disruption numbers 2, 3, 4 and 5.

Table 3 Comparison of results for 90 different disruption test problems. Approach Average computational time (s)

Average percentage of deviation (%)

Heuristic PS

0.00034

1.024 20.36

Table 2 Results obtained from heuristic and PS. Disruption problem

1 2 3 4 5

Total profit

Deviation (%)

Heuristic

PS

241,898.0 127,177.5 242,409.5 242,560.5 232,442.2

241,898.0 127,177.5 242,409.4 242,560.5 232,442.2

0.0 0.0 0.0 0.0 0.0

Computational time (s) Heuristic

PS

1.60 1.91 0.75 0.74 0.73

4.72 6.53 28.74 28.60 33.49

Recovery strategy

Minimum recovery periods without lost sales (days)

Only Only Only Only Both

– 2.667 – – 2.625

back orders lost sales back orders back orders back orders and lost sales

68

S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72

6. A real-life case study The developed model was used to solve a real-world case problem involving a pharmaceutical company's paracetamol tablet production line which had three stages, mixing, compression and packaging, and was dedicated to only one product. A flow diagram of the production line is presented in Fig. 2. In this process, the raw materials were initially blended for a fixed time and then the blended material moved to the compression stage to shape each tablet. The shaped tablets were then blistered and packed in the third stage and, finally, the finished items were transferred to storage. We collected relevant data from historical records, including a number of disruption scenarios, for all three stages in the case study. We observed that machine breakdown was a common disruption, whereby the system became inoperable for a certain period of time. We also observed that the compression and packaging stages had more machine breakdowns than the mixing stage. However, although the mixing stage suffered fewer disruptions, as the materials could be completely lost, we considered scenarios of both loss and no loss of materials for a disruption in the first stage. Moreover, it was observed that machine breakdown could occur in any stage at any time. Currently, the company uses only the lost sales cost to recover from Table 4 Disruption scenarios for series of disruptions. Disruption number

Disrupted stage

Disruption period (days)

Time since previous disruption (days)

1 2 3 4 5 6 7 8 9 10 ….

1 3 2 2 3 1 1 3 2 3 ….

0.10 0.28 0.16 0.24 0.18 0.27 0.20 0.08 0.22 0.14 ….

– 0.50 0.67 1.50 0.67 1.20 1.60 0.80 0.33 1.75 ….

The following costs of production and material loss were collected. Production cost in mixing stage ¼11.2 Taka/strip (0.14 USD) Production cost in compression stage¼7.6 Taka/strip (0.095 USD) Production cost in packaging stage¼6.5 Taka/strip (0.08125 USD) Cost per unit of material loss ¼11.2 Taka/strip (0.14 USD)

Table 5 Results obtained from heuristic for series of disruptions. Disruption number

Total lost sales

Total back orders

Total profit

1 2 3 4 5 6 7 8 9 10 ….

0 5400 5400 5400 5400 60,000 0 0 0 0 ….

212.50 1186.60 675.23 705.23 588.83 0 1420.00 1227.50 1230.00 887.50 ….

242,460 232,440 232,960 232,910 233,040 142,180 241,253 241,450 241,440 241,790 ….

Raw materials Paracetamol powder Caffeine

disruptions which means that its production capacity during these periods always leads to shortages and materials are completely lost. This disruption problem could be solved by applying our developed model and the proposed heuristic. To demonstrate this, some data was collected directly from the production line and historical records of the company while some was approximated by consulting with the plant manager. The average daily demand of 9260 strips was determined from historical records, and the capacity of the mixing cylinder was approximately 168.3 kg (equivalent to 3366 strips per batch and 10 tablets per strip) for the mixing stage. The processing time for a batch, which was independent of the batch size, was 0.3 days. The capacities of the compression and packaging stages were 10,960 strips per day and 10,230 strips per day respectively, and that of the transfer bucket used to transfer materials from the second to third stage 12,000 tables (equivalent to 1200 strips). After processing a batch in the mixing stage, it was the company's standard procedure to clean the cylinder which involved some time and cost. The time was considered the set-up time and calculated as 1.10 h following observations from a time study, and the set-up cost required for labour to prepare the raw materials and clean the cylinder was taken as 2250 Taka (28.125 USD), as the company suggested. As it was a dedicated continuous production line, the cleaning times for the compression and packaging stages could be considered negligible in comparison with the production time. Therefore, we assumed their set-up times to be 0 in this case study. The back orders and lost sales costs were also approximated as 10 Taka/strip/day (0.125 USD) and 50 Taka/strip (0.625 USD) respectively while the daily holding cost per unit for materials used in compression was approximated as 0.4 Taka (0.005 USD).

Stage 1

Stage 2

Stage 3

Mixing of raw materials

Compression

Packaging

Storage

Binders Fig. 2. Three-stage production process for pharmaceutical product.

Based on the above, seven breakdown scenarios were observed within the observation period of 59 days and relevant data collected. Of them, four were in compression, two in packaging and one in mixing, as presented in Table 6. After consulting with the plant manager, the recovery periods was set as 2 days. Our proposed heuristic was implemented to solve the disruption problem. The total lost sales, back orders and profits in the revised plan for each disruption are presented in Table 7. The benefits of implementing our proposed model are presented in Table 8 which shows that it achieved significantly better results than the company's current practice. Under ideal conditions, the company would have a profit of 253,914.41 USD within the observation period of 59 days whereas, under the disrupted environment, this would reduce to 234,138.63 USD if the company used the lost sales cost to recover from disruptions, with a lost sales cost of 10,974.87 USD. However, if our proposed model was applied, total profit would increase to 248,063.04 USD, with the back orders and lost sales costs 1129.35 and 2410.00 USD respectively. A comparison of the daily total profits obtained has been presented graphically in Appendix C. It confirms that our approach obtains better results than the company’s current practice. It is observed that the total profit reduces significantly in the case of current practice when a disruption occurs. This is because the current practice uses only lost sales cost to recover from a disruption. However, our proposed model ensures minimum

S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72

Table 6 Data for disruption scenarios. Disruption number (n)

Disrupted stage

Time since previous disruption (days)

Disruption period (h)

1 2 3 4 5 6 7

Compression Compression Packaging Compression Mixing Packaging Compression

– 10.50 1.20 21.40 7.20 15.10 1.60

6.30 4.27 6.94 5.76 4.69 3.10 5.38

Table 7 Detailed results using heuristic. Disruption number

Total lost sales (USD)

Total back orders (USD)

Total profit (USD)

1 2 3 4 5 6 7

0 0 306.26 0 2103.80 0 0

211.67 86.51 434.24 173.81 0 76.03 147.16

10,738.89 10,874.02 9989.91 10,779.41 6793.021 10,886.13 10,807.92

Table 8 Comparison of results during observation period. Proposed model

Current practice

0

1129.35

0

0

2410.00

10,974.87

observed, that for disruptions at the mixing and packaging stage, total profit decreases with lost sales cost. But the rate of decrement was much higher for disruption at the mixing stage. This is because of the presence of only lost sales in the solution for disruption at the mixing stage, and the presence of both lost sales and back order for disruption at the packaging stage. On the other hand, at the compression stage, total profit did not change with lost sales cost. This is because of the presence of only back orders in the solution. The relationship between total profit and back order cost is presented in Fig. 5. The disruption and the pre-disruption periods were taken as 2 and 0 h respectively for this analysis. Total profit decreases with back order cost for any disruption at the mixing, compression or packaging stage. This is because back order costs are always present in the solution for all cases. In the above study, our approach shows mostly better, but no worse performance, as compared to current practice. However, in some scenarios the proposed approach provides the same solution as current practice where the disruption is at the mixing stage and the disruption period is more than 3 h. This is because the proposed approach requires only lost sales cost to generate solutions, and current practice also uses lost sales in this scenario. We have proposed our approach based on a real-life problem from a pharmaceutical company. However this disruption recovery model can be applied to other similar production systems that countenance single or multiple production disruptions at any stages. We have considered fixed lot sizes at different stages throughout the planning horizon, which was based on ideal production conditions. The considered production system also has a mix of discrete and continuous batch production. However, in some production systems, the lot may be split, or may not be fixed, to meet transportation and warehouse capacities. Although this approach is applicable to three-

Total profit vs disruption period 12

248,063.04

234,138.63

reduction of the total profit for the disruption scenarios. This is because the proposed model uses both back orders and lost sales costs to recover from a disruption. The only exceptional scenario is disruption number 5 in Table 6, where the total profit both from the current practice and the proposed model is same (Fig. C1 of Appendix C). The reason for this is explained below. The variation of total profit depends on the disruption period, lost sales and back order costs. The change of total profit with duration of disruption, at all three stages, is presented in Fig. 3, which shows that total profit decreases significantly with larger disruption periods. For disruption at the mixing stage, total profit decreases suddenly at the disruption period of 3 h, because of the commencement of lost sales due to product unavailability. Before that, only back orders were used to generate the solution. For disruption at the packaging stage, total profit decreases from the disruption period of 6 h because only back orders are optimal for less than 6 h, but both back orders and lost sales are required after 6 h to generate the best solutions. The rate of decrease of total profit with disruption period is much lower for disruption at the compression stage. This is because of the presence of only back orders in the best solution when the disruption period is less than 9 h, but after that, both back order and lost sales are required to generate the solutions. The changes of total profit with lost sales cost is presented in Fig. 4. The disruption and pre-disruption periods were taken as 7 and 0 h respectively for this analysis. The remainder of the parameters have the default values of the case study. It was

Total profit (Thousand USD)

253,914.41

11 10 9 8 7 6

1

2

3

4

5

6

7

8

9

Disruption period (Hours) Disruption at compression

Disruption at mixing

10

11

12

Disruption at packaging

Fig. 3. Changes of total profit with disruption period for the case study.

Total profit vs per unit lost sales cost 12 Total profit (Thousand USD)

Total cost of back orders (USD) Total cost of lost sales (USD) Total profit (USD)

Ideal system (no disruption)

69

11 10 9 8 7 6 5 4 3 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Lost sales cost (USD/unit) Disruption at mixing

Disruption at compression

Disruption at packaging

Fig. 4. Changes of total profit with lost sales cost.

1

70

S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72 M

Total profit (Thousand USD)

Total profit vs per unit back order cost 11.2 11 10.8 10.6 10.4 10.2 10 9.8 9.6 0.1

Production cost at first stage ¼ CP 1 nðX 1 þ X 2 þ ⋯þ X M Þ ¼ CP 1 n ∑ X i i¼1

ðA3Þ Mn

Production cost at second stage ¼ CP 2 nðY 1 þ Y 2 þ ⋯ þ Y Mn Þ ¼ CP 2 n ∑ Y j j¼1

ðA4Þ 0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Average raw material holding cost at second stage ¼

Back order cost (USD/unit/year) Disruption at mixing

Disruption at compression

Disruption at packaging

(

Fig. 5. Changes of total profit with back order cost.

þ H2 n stage production systems, the concept can be used for any number of stages. The annual demand rate is assumed to be known and constant, but practically demand can fluctuate. There may also be special orders outside of the regular demand trend, and sometimes a priority should be given to produce such special orders. For all these variations, management must decide how to deal with them when the proposed approach is applied.

M H2 2 n ∑ X 2C 2 i ¼ 1 i

)

k

X p T d þ ∑ X i ðT d
ði
1ÞI 2 Þ

ðA5Þ

i¼2

Mn

Production cost at third stage ¼ CP 3 nðZ 1 þ Z 2 þ ⋯ þ Z Mn Þ ¼ CP 3 n ∑ Z j j¼1

ðA6Þ Back orders cost ¼ B ðY l þY l þ 1 þ ⋯ þ Y n ÞnT d þðY n þ 1 þY n þ 2 þ ⋯ þ Y 2n ÞnðT d
I 2 Þ þ⋯ þðY ðk
1Þn þ 1 þ Y ðk
1Þn þ 2 þ ⋯ þ Y kn Þ

nðT d
ðk
1ÞI 2 Þ

7. Conclusions The main objective of this study was to develop an appropriate recovery policy for managing disruptions in a three-stage mixedproduction environment. A mathematical model was developed, and then a new heuristic for managing both single and multiple disruptions on a real-time basis proposed. The results from the heuristic were compared with those from another search algorithm for a set of randomly generated disruption test problems. Both approaches produced very similar results, with the average percentage of deviation only 0.00034% which can be considered negligible. The proposed approach was also implemented to solve a real-world disruption problem of a pharmaceutical company. It was proven that our developed mathematical model and proposed heuristic can be easily applied to manage both single and multiple disruptions in a three-stage mixed-production system. With the help of this model, an organization could increase its profit margin and, thus, decrease its loss due to a disruption. Finally, as customer satisfaction could also be greatly increased, a company’s reputation could be enhanced. This proposed mathematical and heuristic technique offers a potentially very useful quantitative approach for helping decision-makers make prompt and accurate decisions regarding revising production plans whenever a sudden (or series of) disruption occurs in a mixed-production environment. In the future, this concept could be extended to a supply chain environment to manage disruptions in supply, production and transportation.

Mn

ðA7Þ

!

Lost sales ¼ Ln MnY 0
∑ Y j

ðA8Þ

j¼1

A.2. Revenue formulation Mn

Revenue ¼ m1 ðCP 1 þ CP 2 þ CP 3 Þn ∑ Y i

ðA9Þ

i¼1

A.3. Final mathematical model The objective function, total profit ¼total revenue
total costs, which is to be maximized and subject to constraints (A10)–(A18).   X i r X 0 ; 8 i to ensure delivery and transportation constraints ðA10Þ   Y j r Y 0 ; 8 j to ensure delivery and transportation constraints ðA11Þ   Z j rZ 0 ; 8 j to ensure delivery and transportation constraints ðA12Þ   Y j ¼ Y 0 ; j ¼ 1; 2; …; l
1 to ensure pre
disruption production constraint

ðA13Þ Appendix A. Formulation of mathematical model for disruption at second stage

  I 1 ; I 2 and I 3 Z0 to ensure non
negative idle time

A.1. Cost formulation

  Y ði
1Þn þ 1 þ Y ði
1Þn þ 2 þ ⋯ þ Y in ¼ X i ; 8 i to balance batches and sub
lots

Number of batches in recovery periods, M ¼ ⌈ðT R =ðX 0 =C 2 Þ þI 2 Þ⌉ l ¼ disrupted sub-lot number (any single number between 1 and n) If T p Z ðiY 0 =C 2 Þ, then l ¼ i þ 1 (i ¼ 0; 1; 2; …; n
1) Pre
disruption raw material level; X p ¼ X 1
T p C 2

ðA1Þ

Set
up cost at first stage ¼ A1 M

ðA2Þ

ðA14Þ

ðA15Þ   Y j ¼ Z j ; 8 j to balance production system

ðA16Þ

  T d
ðk
1ÞI 2 Z 0; For k ¼ 1; 2; …; ⌈ T d =I2 ⌉ to ensure non
negative delay

ðA17Þ X i ; Y j and Z j non
negative; 8 i; j

ðA18Þ

S.K. Paul et al. / Computers & Operations Research 57 (2015) 60–72

71

B.3. Final mathematical model

Appendix B. Formulation of mathematical model for disruption at third stage B.1. Cost formulation Number of batches in recovery periods, ⌈M ¼ ðT R =ðX 0 =C 3 Þ þI 3 Þ⌉ l ¼disrupted sub-lot number (any single number between 1 and n)

The objective function, total profit¼total revenue
total costs, which is to be maximized and subject to constraints (B9)– (B18).   X i r X 0 ; 8 i to ensure delivery and transportation constraints ðB9Þ

ðB1Þ

Set
up cost at first stage ¼ A1 M

  Y j r Y 0 ; 8 j to ensure delivery and transportation constraints

M

Production cost at first stage ¼ CP 1 nðX 1 þ X 2 þ ⋯ þ X M Þ ¼ CP 1 n ∑ X i

ðB10Þ

i¼1

ðB2Þ Mn

  Z j r Z 0 ; 8 j to ensure delivery and transportation constraints ðB11Þ

Production cost at second stage ¼ CP 2 nðY 1 þ Y 2 þ ⋯ þ Y Mn Þ ¼ CP 2 n ∑ Y j j¼1

ðB3Þ Average raw material holding cost at second stage ¼

  Z j ¼ Z 0 ; j ¼ 1; 2; …; l
1 to ensure pre
disruption production constraint

ðB12Þ

M H2 2 n ∑ Xi 2C 2 i ¼ 1

ðB4Þ

  I 1 ; I 2 and I 3 Z 0 to ensure non
negative idle time

  Z ði
1Þn þ 1 þ Z ði
1Þn þ 2 þ ⋯ þ Z in ¼ X i ; 8 i to balance batches and sub
lots

Mn

Production cost at third stage ¼ CP 3 nðZ 1 þ Z 2 þ ⋯ þ Z Mn Þ ¼ CP 3 n ∑ Z j

ðB14Þ

j¼1

ðB5Þ  Back orders cost ¼ B Z l þZ l þ 1 þ ⋯ þ Z n nT d þ ðZ n þ 1 þ Z n þ 2 þ ⋯ þ Z 2n ÞnðT d
I 3 Þ þ ⋯ þ ðZ ðk
1Þn þ 1 þZ ðk
1Þn þ 2 þ ⋯ þ Z kn Þ

nðT d
ðk
1ÞI 3 Þ Mn

  Y j ¼ Z j ; 8 j to balance production system

ðB15Þ

  T d
ðk
1ÞI 3 Z 0 for k ¼ 1; 2; …; ⌈T d =I 3 ⌉ to ensure non
negative delay

ðB17Þ

ðB6Þ

!

X i ; Y j and Z j non
negative; 8 i; j

Lost sales ¼ Ln MnZ 0
∑ Z j

ðB18Þ

ðB7Þ

j¼1

B.2. Revenue formulation

Appendix C Mn

Revenue ¼ m1 ðCP 1 þ CP 2 þ CP 3 Þn ∑ Z i

ðB8Þ

i¼1

Graphical presentation of comparison of results from different approaches (see Fig. C1).

Total profit for differnt approaches 5000 4500 4000 3500

Total Profit

ðB13Þ

3000 2500 2000 1500 1000 500 0 0

5

10

15

20

25

30

35

40

45

50

55

Days Disruption at the day number: 1, 10, 11, 33, 40, 55, 57 Ideal system (no disruption)

Current practice

Our proposed approach

Fig. C1. Comparison of daily total profits obtained from different approaches during observation period.

60

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