Joint ordering and reuse policy for reusable items inventory management

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Model 5G

pp. 1–10 (col. fig: NIL)

Sustainable Production and Consumption xx (xxxx) xxx–xxx

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Research article

Joint ordering and reuse policy for reusable items inventory management Hadi Mokhtari Department of Industrial Engineering, Faculty of Engineering, University of Kashan, Kashan, Iran

article

info

Article history: Received 4 February 2018 Received in revised form 10 July 2018 Accepted 11 July 2018 Available online xxxx Keywords: Inventory systems Reusable items Recovery process Economic order quantity Holding cost

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a b s t r a c t This paper proposes two optimal models for determining order quantity for reusable items. In most of traditional manufacturing models, it is assumed that the items can be utilized just one times, and then should be disposed. However, in today competitive markets, the limited availability of natural resources, environmental consciousness and government regulations have motivated many companies to start recovery and reusing activities. In these cases, utilizing recovery processes can reduce the need for new materials and leads to considerable cost savings. Some examples are the systems for machine tools, medical and surgical instruments, copiers, automobile parts, and office furniture. In our models, after that a new item is purchased and used, it is checked to evaluate whether the useful life amortizes. If yes, the item is disposed, and otherwise, it will be recovered and reused. The reused item is again recovered for reusing, if its useful life is not still amortized. This process is repeated till end of useful life. The aim is to determine the economic order quantity for new items (ordering policy) as well as batch quantity for recovery and reusing process (reuse policy), so as to minimize the total cost of inventory system. By utilizing different holding costs for usable and used items, two inventory model are developed. Numerical examples are presented and discussed, and then, an analysis of sensitivity is carried out to assess the sensitivity of results to parameters. © 2018 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1. Introduction Inventory control in manufacturing can be defined as coordination and supervision of the supply, storage, and distribution of materials, goods, parts and products to maintain quantities sufficient for current needs. The goal of inventory control is to maximize profits with minimum inventory costs, without decreasing customer satisfaction level. The appropriate design and planning of inventory control systems have great role and impacts on the performance of industrial systems. The economic order quantity (EOQ) model presented by Harris (1913) was the first attempt to inventory control area. The basic EOQ model is used to determine the optimal order quantity, with the aim of minimizing overall costs, including holding and ordering costs, such that demand is satisfied. It has been widely used to solve inventory planning problems under various assumptions. The EOQ model has attracted numerous researchers due to its well-established, simple and easy-to-modify nature. The number of works on EOQ model has increased rapidly over the last years. After the Harris’ work, several researchers used new assumptions for EOQ attempting to reflect real problem conditions. One of the E-mail address: [email protected].

classic EOQ assumption is that replenishments happens instantaneously at the moment. In industrial world indeed, however, due to limited production capacity of the production plants, the replenishments are done gradually. To consider this fact, economic production quantity (EPQ) was established as one of the initial extensions of EOQ. In basic EPQ the assumption is that a buyer and a producer are involved in the problem and the order is placed from buyer to the producer with a finite rate. Traditional EOQ model can be applied to most products. However, because of their inability to incorporate specific features of some product categories, particular models have been proposed for some specific types of products. For example, there is a research stream on EOQ model for deteriorating products such as food, vegetables, milk (Skouri and Papachristos, 2003; Sarkar et al., 2013; Ting, 2015; Palanivel and Uthayakumar, 2017). While, in traditional models, it is implicitly assumed that inventory items can be stored for an infinite amount of time, for deteriorating items, this assumption needs to be relaxed, which is why separate models have been proposed. In literature many authors deal with the problem of deterioration using a constant deterioration rate. Deterioration rate can also be variable along the time according to some function. Imperfect products, like electronic products, are another type of product that has received increasing attention in the last decade

https://doi.org/10.1016/j.spc.2018.07.002 2352-5509/© 2018 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

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Fig. 1. The inventory level for usable items.

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(Chang, 2004; Maddah and Jaber, 2008; Sharifi et al., 2015; Rezaei, 2016; Taleizadeh et al., 2016). For imperfect items, the implicit assumption is that not all the received (produced) items are of perfect quality. Moreover, the assumption of constant demand rate, rarely met in real life situations. Some examples of this specific category of inventories are items with price-dependent demand (Goh and Sharafali, 2002; Valliathal and Uthayakumar, 2011; Goyal and Chauhan, 2016; Melis Teksan and Geunes, 2016), stock-dependent demand (Chang and Dye, 2000; Sarkar, 2012; Glock et al., 2015; Mishra et al., 2017), time-dependent demand (Bose et al., 1995; Roy et al., 2013; Tripathi and Pandey, 2013; Manna and Chaudhuri, 2014; Aarya et al., 2017)), etc. In addition, in literature two type of quantity discounts are treated, in order to describe the common industrial policies: all-units quantity discounts and incremental quantity discounts (Schniederjans and Cao, 2000; Mendoza and Ventura, 2008; Bera and Maiti, 2012; Taleizadeh et al., 2015; Alfares and Ghaithan, 2016). In the former case the supplier provides a discount for all the items sold to the customer, if the quantity purchased exceeds predetermined quantities. In the latter case the supplier provides a discount only for the items that exceeds the predetermined level, and he sell the others at the usual price. In addition, the basic assumption that shortages are not permitted is too restrictive in real industrial situations. As a major extension of traditional EOQ models, the possibility of shortages has become a common assumption for most researchers (Chung and Tsai, 2001; Valliathal and Uthayakumar, 2011; Zhou et al., 2014; Radha and Praveen Prakash, 2016; Vandana and Sharma, 2016). For models with allowed shortages, complete backordering or complete loss sale are two extreme cases. Afterwards, many researchers have turned their attention to models that allow partial backordering (mixed backorder and lost sale). Some examples are Sicilia et al. (2009), San-José et al. (2015), Taleizadeh and Pentico (2013), Mokhtari and Rezvan (2017) and Khalilpourazari and Pasandideh (2017). In basic EPQ, on the other hand, it is assumed that the items can be utilized just one times, and then should be disposed. However, in today competitive markets, the limited availability of natural resources, environmental consciousness and government regulations have motivated many companies to start recovery and reusing activities. In these cases, natural resources can be saved and companies are contributing to sustainable development. In addition, utilizing recovery processes can reduce the need for new materials and leads to considerable cost savings. As a result, increasing attention has been paid to the recovery and reusing of used items. An important new trend in supply chain management is repair, remanufacturing and recycling of products collected from the end user after they have reached the end of their useful life. The companies produce new items, recover used items, and reuse them for the same purpose. Some examples are the inventory control systems for machine tools, medical and surgical instruments, copiers, automobile parts, and office furniture. There are

rare inventory control models in literature addressing such issue in inventory planning. We can, however, think of items that can be recovered and reused and design an inventory model for such items. The rest of the paper is organized as follows. In next section, the inventory system for reusable items proposed in current research will be described and defined in detail. In Section 3, formulation and solving approaches are discussed. Then, Section 4 presents and analyzes a numerical example to illustrate the problem. In addition an analysis of sensitivity is carried out in Section 5. Finally, Section 6 concludes the paper. 2. Inventory system description At beginning of inventory cycle, an order of new items are received where there is a deterministic and constant demand rate D. The new items, received recently, are consumed gradually, and the used items are stored simultaneously at special conditions until their quantity reach to a predefined number q. At this moment, the recovery process is implemented for the current batch of used items. The recovery brings an item up to an ‘‘as-new’’ quality. Then, all of recovered items return to the main process for reuse. It is assumed that the recovery (and reuse) process can be repeated for a pre-known number of times, i.e., m times, for each item continuously. When the reused items reach to the end of their useful life, they are disposed, and the another batch of new items enter to process. The cycle of using, recovering, reusing and disposing are repeated continuously. Fig. 1 depicts a graphical scheme of the inventory level for usable items (both new and recovered items). As can be seen, there three types of inventory cycle in this system. The smallest cycle tq is associated with the single recovering and reusing process of a batch item. The next cycle TR is related to the time interval in which a batch of items q are recovered and reused repeatedly till reach to the end of useful life. Since an item can be recovered and reused m times during its useful life, we can conclude TR = (m + 1)tq . In addition, the biggest cycle TC indicates the time interval in which all quantity of new items received via a single order Q are used and then reused completely. By assuming Q = pq, it can be concluded that TC = pTR , and therefore we reach TC = p(m + 1)tq . Since the used items can be recovered several times, they are stored at special conditions after using till recovering process is implemented. Fig. 2 depicts the inventory level for recoverable items. In addition, Fig. 3 shows total inventory level in the system including both usable and recoverable items. Indeed, the inventory level in Fig. 3 is the sum of inventory levels in Figs. 1 and 2. During a time interval tq , a batch of usable items are gradually used and its level is continuously reduced with demand rate D (Fig. 1), and at the same time, the level of used (but recoverable) items are increased with the same rate D (Fig. 2). That means the total number of usable and recoverable items are constant during interval tq . This observation, however, is

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Fig. 2. The inventory level for recoverable items.

Fig. 3. The inventory level for both usable and recoverable items.

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not true in the last interval tq of cycle TR , because this is the last number of times that an item can be reused, and then the used items should be deposed from system immediately. Hence, at this special interval, the inventory level of recoverable items is zeros (Fig. 2), and the inventory level of both usable and recoverable items is decreasing (Fig. 3). Fig. 4 shows the whole process for items in such system. After a new item is purchased and used, it will be checked to evaluate whether the useful life amortizes. If yes, the item is disposed, and otherwise, it will be recovered and reused. The reused item is again recovered for reusing, if its useful life is not still amortized. This process is repeated till end of useful life. The aim is to determine the order quantity for new items Q (ordering policy) as well as batch quantity for recovery and reusing process q (reuse policy), so as to minimize the total cost of inventory system including purchasing cost of new items, ordering cost of new items, holding cost of both new and used items, setup cost of recovery process, and operational cost of recovery process.

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3. Formulation and solving

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Fig. 4. The process of the inventory system. 20 21 22 23

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In this section, we suggest two inventory models, in which the holding cost of usable and recoverable items are assumed to be (i) equal (h), and (ii) different (hu , hr ). For convenience, the notations are summarized below. D m C Au Ar hu hr r TC TR tq Q q p TC

The total annual demand The number of times that an item can be recovered and reused The purchase cost per unit The constant ordering cost The constant recovery setup cost The holding cost for one unit of usable items per unit time The holding cost for one unit of used items per unit time The recovery cost per unit The time interval for using all items in an order Q completely The inventory cycle for single recovery and reuse of batch q The inventory cycle for recovery and reuse of batch q till reach to the end of useful life The order quantity for new items The batch quantity for recovery The ratio of order quantity for new items Q to batch quantity for recovery q (Q = pq) The total cost of system

Moreover, some important phrases are defined below. Reusable item: An item that can be used several times after recovery process. Usable item: An item that is now ready for use (a new item or an used but recovered item). Recoverable item: An item that is used recently, and can be still recovered for reuse. As mentioned before, the aim is to determine the order quantity Q as well as recovery and reusing batch quantity q. The objective is to minimize the total cost including purchasing cost of new items, ordering cost of new items, holding cost of both new and used items, setup cost of recovery process, and operational cost of recovery process. In the sequel, we formulate these costs, separately. At first step, we derive the inventory model for equal holding costs, named as Model I hereafter. Since an item can be used (m+1) times, including 1 time when it is new and m times when it is recovered and reused, the purchasing

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cost of new items (PC) can be formulated as follows. PC = C

D

29 30 31 32 33 34 35 36

37

38 39

40

41

42

(1)

(m + 1)

where D/(m + 1) indicates total number of items should be purchased to satisfy the annual demand D. Furthermore, the ordering cost of new items (OC), is expressed as follows. OC = Au

(m + 1)Q

D

(3)

(m + 1)pq

SC = Ar p

(

Q

SC = Ar

D q

) (5)

m+1

We purchase D/(m + 1) new items at one year, each of which is recovered m times. So, total number of times that recovery process is implemented is obtained m × D/(m + 1) per year. Therefore, the operational cost of recovery process (RC), as a function of number of times that recovery process is implemented, is computed as follows. RC = rD

m

) (6)

m+1

where r denotes the recovery cost per item unit. To calculate total holding cost (HC), we should obtain the average inventory level I. To this end, we need to calculate the inventory level held during time horizon. It can be attained by calculating the area under inventory curve. Since the holding cost is equal for usable and recoverable items, the inventory level at Fig. 3 is used for this purpose. Before doing so, the three inventory cycles, tq , TR and TC are determined as: tq =

pq2

2D

[(m + 1) (p + 1) − 1]

2D

(11) 43

(12)

Therefore, the average inventory can be obtained by I = I /TC as follows. q I = [(p + 1) − 1/(m + 1)] (13) 2

HC = h

q 2

[(p + 1) − 1/(m + 1)]

q

( ) D D m + Au + Ar (m + 1) (m + 1)pq q m+1 ( ) m q + rD + h [(p + 1) − 1/(m + 1)] m+1 2

(14)

TC (Q ) = CD + Au

D Q

Q

+h , 2

(16)

which is total cost of basic EOQ. To find the optimal design of inventory system, we should set the partial derivation of total cost TC (p, q) with respect to p and q to zero and solve the resulting system of equations, as follows:

∂ TC (p, q) D q = −Au +h =0 2 ∂p (m + 1)p q 2 ( ) D D m ∂ TC (p, q) = −Au − A r 2 ∂p (m + 1)pq2 q m+1 +

h 2

[(p + 1) − 1/(m + 1)] = 0

I = TR [Q + (Q − q) + (Q − 2q) + · · · qtq + Q − (p − 1) q] − p 2

(10)

51 52 53

54

55

By solving this system, the closed form of optimal solution are achieved as:

√

1

∗

p =

(m + 1)

√ q =

(

2Ar D

Au Ar

) (19)

(20)

Q∗ =

Hence, the area under inventory curve at interval TC can be calculated as:

50

(18)

and the interval TC is attained as: (9)

49

(17)

and, therefore, the optimal order quantity for new items Q ∗ = p∗ q∗ is obtained as:

D

47

(15)

Since an item can be used (m + 1) times, the interval TR can be achieved as: q TR = (m + 1) tq = (m + 1) (8) D

TC = pTR = (m + 1)

46

48

The above inventory model is a generalization of basic EOQ model. In basic EOQ, there is no reuse option, and hence, we have m = 0, p = 1 and Q = q. Therefore, the total cost TC (p, q) is simplified as:

∗

pq

45

D

TC (p, q) = C

(7)

D

44

Considering all costs in the problem, the total cost is given as follows.

(4)

m+1

m

q2

which can be simplified as:

)

m

By substituting Q = pq, the SC is rewritten as:

(

[Q + (Q − q) + (Q − 2q) + · · ·

and total holding cost (HC) is expressed as:

As we know, the number of times that each item is recovered at each interval TR is m. Moreover, by assuming Q = pq, we can conclude the number of recovery process implementation at each interval TC is pm. Hence, by knowing total number of times that new items should be ordered is D/((m + 1)Q ), the total number of recovery process implementation is obtained pm × D/((m + 1)Q ) = p(D/Q )(m/(m+1)) per year, and the setup cost for recovery process (SC) is calculated as presented below. D

q D

+ Q − (p − 1) q] − p

(2)

where D/((m + 1)Q ) represents total number of times new items should be ordered. By assuming Q = pq, it is rewritten as: OC = Au

I = (m + 1)

I=

D

( 28

or equivalently:

h

√

2Au D (m + 1)h

(21)

As it is clear, the above solution is reduced to that of EOQ model, when m = 0. Here, we are going to extend the above model to the case with different holding costs for usable and recoverable items, named as Model II hereafter. For this purpose, we should define new total holding cost (HC) which consists of total holding cost for usable items (HCu ) and total holding cost for recoverable items (HCr ). To

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Fig. 5. Schematic of optimal design of inventory system (Model I).

this end, the average inventory level for two types of items are calculated, from Figs. 1 and 2, as follows. Iu = Ir =

√

pq

(22)

(2

m+1

)

m

q

(23)

2

Therefore, the holding costs are given as: HCu = hu

pq

(2

HCr = hr 1

2

(24)

m+1

)

m

q

(25)

2

total holding cost (HC) is calculated as: HC = HCu + HCr = hu

pq 2

( + hr

m+1 m

)

q 2

(26)

and, finally, the total cost of inventory for Model II is summarized as:

( ) D D m + Au + Ar (m + 1) (m + 1)pq q m+1 ( ( ) ) m m q pq + rD + hr + hu m+1 m+1 2 2

TC (p, q) = C

3

D

(27)

It is clear that this is a generalization of Model I and basic EOQ. To find the optimal solution, the partial derivation of total cost TC (p, q) should be set to zero and the system of equations should be solved, as follows: D q ∂ TC (p, q) = −Au + hu = 0 ∂p (m + 1)p2 q 2 ( ) ∂ TC (p, q) D D m = −Au − Ar 2 ∂p (m + 1)pq2 q m+1 ( ) hr m p + + hu = 0 2 m+1 2

(28)

(29)

The closed form of optimal solution are derived by solving above system as follows:

√ p∗ = q∗ =

1

(m + 1)

√

and, therefore, the optimal order quantity for new items Q ∗ = p∗ q∗ is obtained as:

2Ar D hr

(

hr hu

)(

Au Ar

) (30)

(31)

Q∗ =

2Au D

(32)

(m + 1)hu

The above solution is reduced to that of Model I, when hr = hu = h, and that of EOQ model, when m = 0. 4. Numerical examples

p = 4, q = 547.72, Q = 2190.89, TC p , q ∗

∗

5

6

7 8

9

In order to evaluate the proposed inventory system, we design two numerical examples, in this section, by setting values for the parameters. The characteristics of the first example are as follows. The annual demand rate is predicted D = 12 000 per year, each item can be recovered three times m = 3, the purchase cost is C = 50$ per item unit, the operational cost of recovery process is r = 20$ per item unit, the ordering cost for new items is Au = 1600$ per order, the setup cost for recovery process is Ar = 100$ per order, the holding cost (Model I) is predicted as h = 12$ per item unit per year, and the unit holding cost for usable and recoverable items are set to hu = 2$ and hr = 8$ per item unit per year, respectively. Using Eqs. (19)–(21), the optimal solutions are obtained for Model I, as follows. p∗ = 2, q∗ = 447.21, Q ∗ = 894.42, TC (p∗ , q∗ ) = 344758.04 which means that the economic order quantity for new items is Q ∗ = 894.42 (ordering policy), each of which is used for the first time till the quantity of used items reach to q∗ = 447.21 (reuse policy). At this moment the recovery process is carried out and the recovered items returned to the system for reuse. This process is repeated m = 3 till end of useful life. Then second order of new items Q ∗ = 894.42 is received, and the same process is continued. Moreover, the optimal cycle times are calculated as tq∗ = 0.037, TR∗ = 0.149 and TC∗ = 0.298. The optimal total cost is calculated 344758.04$. Fig. 5 shows the inventory level for optimal solution achieved. Furthermore, using Eqs. (30)–(32), the optimal solutions are achieved for Model II, as follows. ∗

4

(

∗

∗

)

= 337668.11

which means that the economic order quantity for new items is Q ∗ = 2190.89 (ordering policy) and the batch quantity for recovery process is q∗ = 547.72 (reuse policy). The optimal cycle times are calculated as tq∗ = 0.046, TR∗ = 0.182 and TC∗ = 0.730. Moreover, the optimal total cost is calculated 337668.11$. Figs. 6

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Fig. 6. Schematic of optimal design of usable inventory system (Model II).

Fig. 7. Schematic of optimal design of recoverable inventory system (Model II).

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and 7 depict the inventory level of usable and recoverable items for optimal solution of Model II, respectively. Now we consider second example to illustrate another aspect of our inventory model. For this purpose, we use the annual demand rate D = 22 000 per year where each item can be recovered three times m = 2, the purchase cost C = 85$ per item unit, the operational cost of recovery process r = 10$ per item unit, the ordering cost for new items Au = 4500$ per order, the setup cost for recovery process Ar = 150$ per order, the holding cost (Model I) h = 18$ per item unit per year, and the unit holding cost for usable and recoverable items as hu = 3.5$ and hr = 28$ per item unit per year, respectively. By using Eqs. (19)–(21) and (30)–(32), the solutions are obtained for Models I and II, as p = 3.162, q = 605.53, Q = 1914.85 and p = 8.944, q = 485.50, Q = 4342.48, respectively. However the parameter p should be, in our models, an integer with value greater than 1. To this end, the largest integer number less than p and the smallest integer number greater than p are selected, and the values of total cost function TC (p, q) are calculated correspondingly. Due to the convexity of TC (p, q) with respect to p, the minimum value of TC (p, q) will be the optimal solution. By doing so, the optimal solutions are calculated for Models I and II, as follows.

24

p∗ = 3, q∗ = 638.28, Q ∗ = 1914.85, TC p∗ , q∗ = 811743.82$

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

(

Table 1 The total cost function for different values of p. Values of Parameter p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a

The optimal solution for Model I.

b

The optimal solution for Model II.

Total Cost Function Model I

Model II

817105.41 812509.76 811743.82a 811935.30 812509.76 813275.70 814151.06 815094.81 816084.15 817105.41 818149.88 819211.75 820287.02 821372.80 822467.01 823568.05 824674.72 825786.09 826901.40 828020.08

826235.13 806476.84 800228.49 797357.63 795837.76 794993.39 794535.01 794317.89 794261.60b 794317.89 794456.06 794655.64 794902.46 795186.39 795500.01 795837.76 796195.38 796569.55 796957.67 797357.63

)

5. Sensitivity analysis

32

25 26

27 28 29 30 31

p∗ = 9, q∗ = 482.49, Q ∗ = 4342.48, TC p∗ , q∗ = 794261.60$

(

)

As further analysis, Table 1 shows the values of total cost function with respect to different values of parameter p, for both models. Moreover, Fig. 8 depicts the optimal value of p for both models. It also reveals the convexity of total cost function with respect to p.

In real world, the changes in parameters of the model is inevitable and the parameters may fluctuate. As an example, the annual demand is a function of market condition and may fluctuate as market condition changes. Moreover, changes in the parameters may have a significant impact on the values of decision variables and the objective function. Sensitivity analysis is a systematic approach to assess the effect of changes in the parameters of model

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Fig. 8. The total cost function for different values of p. Table 2 The change in p∗ , q∗ , Q ∗ and TC due to change in D. Changes in D (%)

−50 −40 −30 −20 −10 0 +10 +20 +30 +40 +50

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

p∗

q∗

Q∗

TC

Model I

Model II

Model I

Model II

Model I

Model II

Model I

Model II

3 3 3 3 3 3 3 3 3 3 3

9 9 9 9 9 9 9 9 9 9 9

451.34 494.41 534.03 570.90 605.53 638.28 669.44 699.21 727.76 755.23 781.74

341.18 373.74 403.69 431.56 457.74 482.50 506.05 528.55 550.13 570.90 590.94

1354.01 1483.24 1602.08 1712.70 1816.59 1914.85 2008.32 2097.62 2183.27 2265.69 2345.21

3070.60 3363.67 3633.18 3884.03 4119.64 4342.48 4554.43 4756.95 4951.19 5138.09 5318.43

414517.34 494334.63 573925.39 653336.81 732601.67 811743.82 890781.29 969728.07 1048595.28 1127391.96 1206125.53

402155.54 480792.96 559298.71 637700.24 716016.58 794261.60 872445.78 950577.25 1028662.48 1106706.72 1184714.27

on optimal values of variables, and the objective function under various settings. In other words, the sensitivity analysis changes the value of input parameters and then evaluate the change in the outputs, to illustrate the sensitivity of output under various input values. In this research, we choose the cost parameters, i.e., total annual demand D, the constant ordering cost Au , the constant recovery setup cost Ar , the holding cost for usable items hu , and the holding cost for used items hr . As it is clear, the purchase cost and operational recovery cost are constant for all possible values of decision variables, and then the purchase cost per unit C and the recovery cost per unit r are discarded from analysis. Tables 2–6 present the changes in optimal value of decision variables p∗ , q∗ and Q ∗ , and in the optimal value of total cost TC by changing these five parameters. As further analysis, the sensitivity of objective function, i.e., total cost TC , is depicted by Figs. 9 and 10 for Models I and II, respectively. The sensitivity analysis shows that the optimal total cost TC increases when ordering cost for new items Au and recovery setup cost Ar increase, in both models. However, it increases as holding costs hu and hr increase in model II, and remains unchanged as holding costs hu and hr increase in Model I. In addition, Fig. 11 depicts sensitivity of total cost with respect to changes in annual demand D for both models. As the results show, the optimal TC is significantly affected by demand in both models. It increases remarkably when D increase.

6. Conclusions This research proposed two inventory models for reusable items. When an order of new items are received they are consumed gradually, and the used items are stored simultaneously at special conditions until their quantity reach to a predefined number. Then, the recovery process is implemented for the current batch of used items in such a way that an item up to an ‘‘asnew’’ quality. At this moment, all of recovered items return to the main process for reuse. The recovery (and reuse) process are repeated for a pre-known number of times for each item continuously. When the reused items reach to the end of their useful life, they are disposed, and the another batch of new items enter to process. The cycle of using, recovering, reusing and disposing are repeated continuously. Different holding costs are assumed for usable and used items and then two inventory control models are introduced. Moreover, the mathematical statements are developed to delineate the economic order quantity for new items as well as the economic batch quantity for recovery and reusing process. The objective is to minimize the total cost of inventory system including purchasing cost of new items, ordering cost of new items, holding cost of both new and used items, setup cost of recovery process, and operational cost of recovery process. Two numerical examples are presented, and an analysis of sensitivity is carried out to recognize and depict the most influence input parameters.

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Fig. 9. Sensitivity of total cost TC to cost parameters for Model I.

Fig. 10. Sensitivity of total cost TC to cost parameters for Model II.

Fig. 11. Sensitivity of total cost TC to annual demand for both models.

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Table 3 The change in p∗ , q∗ , Q ∗ and TC due to change in Au . Changes in Au (%)

−50 −40 −30 −20 −10 0 +10 +20 +30 +40 +50

p∗

q∗

Q∗

TC

Model I

Model II

Model I

Model II

Model I

Model II

Model I

Model II

2 3 3 3 3 3 3 3 4 4 4

6 7 8 8 9 9 9 10 10 11 11

677.00 494.41 534.03 570.90 485.50 638.28 669.44 699.21 545.82 566.42 586.30

511.77 480.52 454.15 485.50 457.74 482.50 506.05 475.69 495.12 467.10 483.49

1354.01 1483.24 1602.08 1712.70 4119.64 1914.85 2008.32 2097.62 2183.27 2265.69 2345.21

3070.60 3363.67 3633.18 3884.03 4119.64 4342.48 4554.43 4756.95 4951.19 5138.09 5318.43

801683.75 804114.51 806161.28 808107.52 810144.18 811743.82 813452.66 815098.78 816604.41 818064.91 819483.89

789822.42 790836.08 791799.08 792656.86 793497.20 794261.60 795011.04 795713.96 796393.65 797052.84 797677.33

Table 4 The change in p∗ , q∗ , Q ∗ and TC due to change in Ar . Changes in Ar (%)

−50 −40 −30 −20 −10 0 +10 +20 +30 +40 +50

p∗

q∗

Q∗

TC

Model I

Model II

Model I

Model II

Model I

Model II

Model I

Model II

5 4 4 4 3 3 3 3 3 3 3

13 12 11 10 9 9 9 8 8 8 7

382.97 478.71 478.71 478.71 638.28 638.28 638.28 638.28 638.28 638.28 638.28

334.04 361.87 394.77 434.25 482.50 482.50 482.50 542.81 542.81 542.81 620.35

1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85

4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48

809637.48 810097.05 810556.61 811016.18 811399.15 811743.82 812088.50 812433.17 812777.84 813122.52 813467.19

791609.41 792223.85 792784.21 793304.65 793805.64 794261.60 794717.56 795128.49 795533.79 795939.09 796308.20

Table 5 The change in p∗ , q∗ , Q ∗ and TC due to change in hu . Changes in hu (%)

−50 −40 −30 −20 −10 0 +10 +20 +30 +40 +50

p∗

q∗

Q∗

TC

Model I

Model II

Model I

Model II

Model I

Model II

Model I

Model II

3 3 3 3 3 3 3 3 3 3 3

13 12 11 10 9 9 9 8 8 8 7

638.28 638.28 638.28 638.28 638.28 638.28 638.28 638.28 638.28 638.28 638.28

472.40 467.18 471.84 485.50 508.60 482.50 460.04 495.52 476.08 458.76 506.52

1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85

6141.20 5606.12 5190.26 4855.04 4577.38 4342.48 4140.39 3964.12 3808.61 3670.07 3545.62

811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82

789813.23 790842.31 791782.57 792656.86 793491.27 794261.60 795016.41 795713.96 796393.65 797060.62 797685.39

Table 6 The change in p∗ , q∗ , Q ∗ and TC due to change in hr . Changes in hr (%)

−50 −40 −30 −20 −10 0 +10 +20 +30 +40 +50

1 2 3 4

p∗

q∗

Q∗

TC

Model I

Model II

Model I

Model II

Model I

Model II

Model I

Model II

3 3 3 3 3 3 3 3 3 3 3

6 7 8 8 9 9 9 10 10 11 11

638.28 638.28 638.28 638.28 638.28 638.28 638.28 638.28 638.28 638.28 638.28

723.75 620.35 542.81 542.81 482.50 482.50 482.50 434.25 434.25 394.77 394.77

1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85 1914.85

4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48 4342.48

811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82 811743.82

791615.91 792219.03 792798.03 793304.65 793811.27 794261.60 794711.93 795128.49 795533.79 795929.88 796298.33

We think that this study introduces an efficient application area for inventory models. An appropriate control of reusable items is very important area in practice, which can minimizes the inventory costs, greatly. A possible research extension for future would be to

develop this model for different inventory features like the occurrence of shortage or discount. Moreover, it would be interesting to assume that the recovered items enter to system with a finite rate instead arriving the whole batch instantly. In addition, it can

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