Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 816876, 10 pages
http://dx.doi.org/10.1155/2015/816876

Research Article

Deteriorating Inventory Model for Chilled Food

Ming-Feng Yang and Wei-Chung Tseng

Department of Transportation Science, National Taiwan Ocean University, Keelung City 202, Taiwan

Received 17 September 2014; Revised 12 March 2015; Accepted 15 March 2015

Academic Editor: Antonino Laudani

Copyright © 2015 Ming-Feng Yang and Wei-Chung Tseng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

With many aspects that affect inventory policy, product perishability is a critical aspect of inventory policy. Most goods will deteriorate during storage and their original value will decline or be lost. Therefore, deterioration should be taken into account in inventory practice. Chilled food products are very common consumer goods that are, in fact, perishable. If the chilled food quality declines over time customers are less likely to buy it. The value the chilled food retains is, however, closely dependent on its quality. From the vendor’s point of view, quantifying quality and remaining value should be a critical business issue. In consequence, we combined the traditional deterioration model and quality prediction model to develop a new deteriorating inventory model for chilled food products. This new model quantifies food quality and remaining value. The model we propose uses real deterioration rate data, and we regard deterioration rate as temperature-dependent. We provide a numerical example to illustrate the solution. Our model demonstrates that high storage temperatures reduce profits and force shorter order cycles.

1. Introduction

We know that inventory policy may affect supply chain performance from Beheshti’s [1] research. If inventory policy is not appropriate, it will burden enterprises with high costs. Hence, many enterprises would like to find out the most suitable inventory policy. Inventory policy researchers have included numerous different conditions into their models. These different conditions frequently include stochastic demand/lead time, lead time reduction, product quality, deterioration, and the learning curve (reviewed in [2, 3]). The aim of including such conditions is to make models more realistic and complete. In general, most physical goods will deteriorate during storage and customers are less likely to buy deteriorated goods such as food, drugs, and chemicals. The value of goods will decline or even be lost. Vendors therefore have to reduce the price of deteriorated goods in order to sell them. The economic loss this represents necessitates, we believe, the more detailed consideration of deterioration. Glock et al. [3] also indicated that deterioration may affect the productivity of an inventory system. Chilled food has become common in real life these years [4]. Herbon et al. [5] indicated that the value of chilled food has much to do with its quality. From the vendor’s point of view, quantifying quality and remaining value should be a critical business issue. We therefore desired to develop a model of deteriorating inventory for chilled food. We also reviewed the literature in order to identify currently neglected research factors.

2. Literature Review

2.1. Early Deterioration Inventory Model

Deterioration is defined as decay, damage, spoilage, or perishability and its effect cannot be disregarded in inventory models [6]. Goyal and Giri [7] differentiated between two types of deterioration, perishability, and decay, where perishability refers to products with a fixed shelf-life and decay to products with unlimited shelf-life. Now we would like to review some early researches that have considered deterioration in their inventory model. Ghare and Schrader [8] were the first to consider deterioration in their research. They indicated that inventories are depleted not only by demand but also by decay. They proposed an exponential decay model to illustrate how deterioration works in a normal inventory model. Many scholars continued using the Ghare and Schrader’s exponential decay model to establish further deteriorating inventory models. Shah and Jaiswal [9] developed a periodic review inventory model with deteriorating items. They considered both constant and variable deterioration rates. Sachan [10] established a deteriorating inventory model with time-dependent demand but he only considered a constant deterioration. Kim and Hwang [11] assumed deterioration rate to be time-proportional. They also showed that there was a substantial impact of deterioration on the optimal procurement policy. Datta and Pal [12] discussed a finite-horizon model and special sales of deteriorating items, while allowing shortage. The items began to deteriorate at a constant rate. Pal et al. [13] proposed a deterministic inventory model that assumed that demand rate was stock-dependent. Benkherouf [14] presented an optimal procedure to find out the solutions of an inventory model with deteriorating goods and demand rates that decreased over a known horizon. Benkherouf [15] extended his research in 1995; he assumed demand rates increased over time in the same model. Wee [16] noticed that there is less attention in the literature to price-dependent demand and varying deterioration rate. Therefore, he studied and established a model that included these factors and provided an algorithm for obtaining the optimal profit.

In the above-mentioned studies, deterioration rate is regarded as a constant, varying, or time-proportional value and the demand rate as stock-dependent, time-dependent, or price-dependent. With this type of deterioration inventory model becoming increasingly mature, researchers began to elaborate new deterioration inventory models.

2.2. Lately Deterioration Inventory Model

Mukhopadhyay et al. [6] established such a deteriorating model in which deterioration rate is taken to be time-proportional and a power law form of the price-dependence of demand is considered. Hou [17] studied optimal production run length for deteriorating products by using Markov chains. Mandal et al. [18] constrained deteriorating items; they assumed the storage space was limited and demand rate for items was finite. Adel and Balkhi [19] studied the effect of learning and forgetting on optimal lot size based on a deteriorating inventory model with time-varying demand. Feng et al. [20] built a deteriorating model with price-dependent demand. They took two commodities as examples to find out the joint optimal pricing strategies in their model. Sung and Gong [21] indicated that deteriorating items caused defects during the production process. These defective items have to be reworked immediately and so incur a cost. The model determines the final expected total cost due to the reworking. Barketau et al. [22] proposed a similar model to Sung and Gong’s [21], but they did not assume immediate reworking but allowed the defective items to continue deteriorating during the time they wait for reworking. The authors’ objective was then to determine the optimal batch size for minimizing the total cost. Nakhai and Jafari [23] emphasized the perishability of food, chemicals, and medicines made them particularly sensitive in deteriorating inventory models. They chose medicines for their research object and obtained the optimum cost of the drug supply chain.

Chen et al. [24] pointed out in-transit deterioration was seldom considered. They therefore proposed a deteriorating model under permissible delay in payments and used different deterioration rates during in-transit and on-hand periods. Molana et al. [25] also assumed products deteriorate during transportation and storage. Giri and Maiti [26] studied a supply chain model with deteriorating products. With the in-control state and out-of-control state during the production run, they considered that a coordinated policy is more beneficial than policies obtained separately from the buyer’s and the vendor’s perspectives. Sarkar [27] developed a deteriorating model under permissible delay in payments and assumed the demand rate and deterioration rate were time-varying. Panda et al. [28] thought that perishable products were an extremely important part of inventory management. They investigated the effect of price discounts and proved that dynamic pricing was better than static pricing. Kim et al. [29] studied a two-stage supply chain where returnable transport items (RTIs) were used to ship finished products from the supplier to the buyer. They indicated that the supply chain can influence both the risk of RTI stockouts at the supplier and the deterioration rate by changing the value of the return lot size of RTIs.

We discovered that price-dependent demand was a frequent basic assumption. Along with a basic assumption of demand rate or deterioration rate, researchers have developed many new deterioration models in order to make them complete. Some authors discussed storage limits and some reworking and defective items. Other authors have discussed the perishable goods of specific industries or pricing policy. We also compared the relevant works to our work in Table 1.

Table 1: Overview of deteriorating model.

Authors
Deterioration rate
Demand rate

Kim and Hwang [11]
Time-dependent
Constant

Wee [16]
Constant
Price-dependent

Mukhopadhyay et al. [6]
Time-dependent
Price-dependent

Hou [17]
Time-dependent
Price-dependent

Feng et al.[20]
Time-dependent
Price-dependent

Nakhai and Jafari [23]
Time-dependent
Constant

Sarkar [27]
Time-dependent
Time-dependent

Giri and Maiti [26]
In-control and out-of-control
Price-dependent

Iao and Hsiao [30]
Temperature-dependent
Constant

Qin et al. (2014) [31]
Temperature-dependent
Price-dependent

*Our work
Temperature-dependent
Price-dependent

2.3. Perishable Products

Inspired by Nakhai and Jafari [23] we noticed that specific perishable products such as food, drugs, or chemicals are more sensitive in deteriorating model than other goods. Perishable products are those that worsen in quality over time and loose value. Common perishable goods include foods, medicines, plants, and agricultural products. Dairy products, fashion products, blood, fruits, and vegetables are examples of time and temperature-sensitive perishable products that can rot or spoil easily. In general, perishable goods decay rapidly if not refrigerated, or if some other preservation technique is not employed.

Among these perishable products, food or food products are very common consumer goods in real life. In recent years, more and more families have replaced home meals with chilled food because chilled food is more convenient than home meals. Chilled food also saves a lot of preparation and cooking time [4]. Chilled food has been incorporated in many deteriorating inventory models [32–36]. Chilled food is therefore a very appropriate perishable product for inclusion in deteriorating inventory model.

Herbon et al. [5] used time-temperature indicators (TTI) for keeping track of the quality of perishable products. They claimed that the price of perishable products and their remaining value has much to do with their quality and this caused the vendor to pay attention to maintaining quality. It is therefore of interest to be able to quantify the quality of food and its remaining value. We also compared the relevant works to our work in Table 2.

Table 2: Overview of perishable goods works.

Authors
Claims

Bogataj et al. [37]
Appropriate control over the transportation time and storage temperature can keep the product on the required level of quality and quantity at the final delivery.

Osvald & Stirn [38]
They developed a vehicle routing algorithm for distributing perishable goods; to obtain minimum distribution cost, their algorithm planned a schedule to reduce distribution time.

Chen et al. [39]
They established a nonlinear mathematical model to consider production scheduling and vehicle routing for perishable goods; their model could determine the time to start producing and vehicle routes simultaneously.

Nakhai and Jafari [23]
Perishable goods were quite sensitive in time and temperature.

Xu & Wang [40]
Suitable facility and precise temperature controlling are two major factors to keep perishable goods.

Aiello et al. [41]
They implied that supply chain organization and operative characteristics have a significant influence on perishable goods, ensuring that the suitable temperature conditions in warehouse were important.

*Our work
We should focus on time and temperature if we consider perishable goods in our works. We therefore built an inventory model quantifying chilled food quality and remaining value.

2.4. Predictive Quality Model

For food quality, observing the growth of microorganisms is a highly reliable approach to defining food quality and, further, determining food safety. If microorganisms abundance exceeds the standard value, the food can be defined as inedible [42–45].

To observe and predict the growth of microorganisms, we learned some predictive quality models in predictive microbiology such as modified Gompertz, Baranyi, Rosso, and Gompertz models [46]. In 1993, Buchanan indicated that function graph of the Gompertz model (see Appendix A) closely fitted the growth curve of microorganisms. Researchers therefore prefer the Gompertz model to other predictive quality models. Afterwards Gibson et al. [47] advanced the Gompertz model to develop a predictive model of microorganisms. Many researchers continued using their model to predict the microorganism counts in their researches. Linton et al. [48] established the nonlinear survival curve for Listeria monocytogenes by extending the Gompertz model. Huang [49] described the growth of Clostridium perfringens on cooked beef by using the Gompertz model. Chowdhury et al. [50] predicted the growth of Pediococcus acidilactici by using logistic model and Gompertz model and also compared the predictive ability of these two models. They regarded the Gompertz model as appropriate model for predicting food quality. We also compared the relevant works to our work in Table 3.

Table 3: Overview of predictive quality model.

Authors
Predictive quality model

Gibson et al. [47]
Gompertz model

Buchanan et al. [51]
Gompertz model

Linton et al. [48]
Gompertz model

Huang [49]
Gompertz model

Chowdhury et al. [50]
Logistic model and Gompertz model

*Our work
Gompertz model

Qin et al. [31] formulated a model of the pricing and lot-sizing problem for food products where the quality and physical quantity deteriorated simultaneously. They indicated that temperature has the most profound effect on the deterioration rate. Thus, deterioration rate should vary with temperature when considering food products as the research object.

Based on the discussion above, we developed a deteriorating model for chilled food including the Gompertz model. This predictive model describes the growth rate of microorganisms over time. We also regard deterioration rate as temperature-dependent. We take pork sandwich, a kind of chilled food, as research object in our proposed model. Our new model treats pork sandwich deterioration and uses real deterioration rate data to illustrate the solutions [30].

3. Assumptions and Notations

The following assumptions and notations are used to formulate the problem and model.

3.1. Notations

  is selling price per unit item, a decision variable.  is the duration of each cycle, a decision variable.  is the price-dependent demand rate, equal to .  is purchase cost per unit item.  is ordering cost per cycle.  is inventory holding cost per unit per unit time.  is deterioration rate.  is initial bacterial count of food.  is growth bacterial count of food.  is end bacterial count of food at time , [30].  is the remaining ratio of value after deterioration per unit item.  is time at which maximum growth rate occurs [30].  is storage temperature of product.  is inventory level at time , .  is the loss in stock due to deterioration in time .  is the total cost per cycle.  is the total cost per unit time.  is the total profit per unit time.

3.2. Assumptions

(1)A single product and a single vendor are assumed.(2)We take the pork sandwiches as products in this research.(3)Pseudomonas spp. is the observed microorganisms on pork sandwich [30].(4)The deterioration rate of pork sandwiches is different with temperature , [30].(5)The time at maximum growth rate of pork sandwiches is different with temperature , [30].(6)Shortages are not allowed.(7)Lead time is assumed to be 0.(8)The time horizon is infinite.(9)The price-dependent demand rate is equal to , where is scale parameter and is the shape parameter of the demand curve, , .(10)Once the Pseudomonas spp. exceed the 7 logCFU/g, most meat products will be inedible [30, 42, 52–57]; we can set to be equal to ; hence the value of will decrease if increases.

4. Modeling and Solution

According to research by Huang and Wang [58] and Mukhopadhyay et al. [6], the inventory level can be described by the differential function:

with the boundary condition

And the solution of (1) is

is the inventory level at any time , represents the initial inventory level, and represents the end inventory level. Now letting be the loss in stock due to deterioration in time , with , the loss in stock at the end can be expressed as (see Appendix B)

means the difference between the inventory levels with and without deterioration at the end and the quantity ordered in each cycle obtained in the following function:

We may know the equals from (3) and (5), after determining the inventory level as discussed above, and letting be the total cost per cycle, we have

And the total cost per unit time is

As far as selling the product is concerned, we applied the predictive quality model [30, 47] to determine the remaining value of pork sandwiches after deterioration:

And the remaining value of every pork sandwich after deterioration will be as follows:

Now let be the profit per unit time and obtain the profit function as follows:

Replacing with , we have

To maximize the profit function , there must be a concave function for and , as already proved in Appendix C. Now take the first partial derivation of with respect to . We have

Let (12) , and it can be solved to obtain the optimal price by the following function:

Algorithm 1.  
Step 1. Set to be 0.5, 1, 2, 3, 4, 5, 6, and 7, and set to be 0°C to solve (13).
Step 2. Determine value by different under = 0°C.
Step 3. Substitute determined value into (11) to obtain corresponding .
Step 4. We can determine optimal value of under = 0°C.
Step 5. Reset to be 0°C, and repeat Step 1 to Step 4 to find out the optimal value of under = 7°C.
Step 6. Follow Step 1 to Step 5, to determine the optimal value under different = 0°C, 7°C, 16°C, and 25°C.

5. Numerical Example

To illustrate the preceding model, we present an example in this section and consider the following data (provided by a top 3 enterprise that produce chilled food in Taiwan).

Relevant Parameters. Consider

  = 160000000;  = 3.21;  = $40 per unit item;  = $250 per order cycle;  = $1.5 per unit per unit time;  = 3.13 logCFU/g;  = 6.87 logCFU/g.

First, we solved (13) by different temperature , and the solution results are in Table 4. With the results in Table 4, we have observed that increased in and ; in other words, higher storage temperature and/or longer order cycle will cause higher pricing.

Table 4: Optimal value of under different .

t
K

0.5
1
2
3
4
5
6
7

0°C
106.87
108.67
112.33
116.06
119.86
123.75
127.71
131.75

7°C
107.57
110.05
115.16
120.48
126.02
131.81
137.85
144.18

16°C
112.72
118.07
130.17
144.61
162.07
183.52
210.21
243.95

25°C
135.21
154.95
213.65
319.81
540.36
1151.92
6229.64
31980.1

Second, we substituted these values into (11) to obtain optimal value of total profit under different temperature (Figure 1). The solution results are shown in Table 5 and Figure 2.

Table 5: Optimal value of total profit under different .

0.5
1
2
3
4
5
6
7

0°C
0.03
96.33
0°C
404.56
621.78
685.32*
670.557
639.51
604.25
568.57
533.89

7°C
0.054
39.6
7°C
391.33
597.39
641.23*
609.815
564.7
517.52
471.75
428.513

16°C
0.115
12.63
16°C
278.83
447.17*
425.11
340.392
254.75
179.92
118.92
71.8915

25°C
0.245
4.03
25°C
−51.16
61.41*
3.49
−42.47
−53.66
−49.09
−41.66
−35.75

 *Optimal solution of in each .

Figure 1: Optimal value of total profit under different .

Figure 2: Optimal value of total profit under different .

Summarizing the results in Tables 4 and 5, as storage temperature = 0°C, the optimal value of , , and was 112.33, 2, and 685.32; as storage temperature = 7°C, the optimal value of , , and was 115.16, 2, and 641.23; as storage temperature = 16°C, the optimal value of , , and was 130.17, 1, and 425.11; as storage temperature = 25°C, the optimal value of , , and was 154.95, 1, and 61.41. We summarized the above results into a concise table as follows.

As shown in Table 6, best profit occurred at = 0°C and value of and was 112.33 and 2; worst profit occurred at = 25°C and value of and was 154.95 and 1. We may conclude that higher temperatures lead to less total profit. If buyers are unable to maintain low storage temperatures, they must shorten the order cycle if they are to obtain optimal total profit.

Table 6: Optimal value of , , and under different .

°C
°C
°C
°C

*
112.33
115.16
130.17
154.95

*
2
2
1
1

*
685.32
641.23
425.11
61.41

 *Optimal value.

6. Conclusions

In this research, we discussed the issue of deteriorating inventory model. After reviewing Glock’s [2] research, we discovered that deterioration is always popular in inventory issues. Product perishability is a critical aspect of inventory policy, most goods will deteriorate during the storage period, and the value of goods will decline or even be lost. So the deterioration should be taken into account in inventory practices.

We noticed that food products are those whose quality worsens the most over time. Tracking the quality of perishable products such as food products, chemicals, or medicines is an essential approach to observe their remaining value [5]. It is interesting to know how to quantize quality and remaining value; hence we would like to apply predictive quality model to determine quality and remaining value.

In the proposed model, we took pork sandwiches as our research object. First, we used the Gompertz model to determine the quality of pork sandwiches after deterioration has started. The Gompertz model closely fits the growth curve of microorganisms on food products and has been widely used in relevant researches. We also set value to represent remaining value of pork sandwiches. Second, we combined traditional deteriorating inventory model with Gompertz model to develop a new model in Section 4. Finally, we illustrated an example to operate our proposed model with different storage temperatures for products.

Our proposed model was able to quantify quality and remaining value of pork sandwiches. It could help vendors to assess the whole system profit under different storage temperature. We discovered that higher storage temperatures lead to less profit and shorten the order cycle. These are main contributions of this paper.

Finally, it is our hope that this work will encourage future works in this area and related area. And we will improve our further research in more real-world complexities. In real life, storage temperature will change depending on the environment, refrigeration technology, refrigeration device, and similar factors. In other words, temperature fluctuation during storage is common. We may apply fuzzy theory to simulate temperature fluctuation.

Appendices

A. Gompertz Function

See Figure 3.

Figure 3: Gompertz function example.

B. Inventory Level with and without Decay

The inventory level at any time () is

And the initial inventory level is

Let be the inventory level at any time () with no decay; we have

The loss in stock at the end is equal to ; then

According to (B.2) and (B.4), we have

C. Hessian Matrix

We need a concavity condition of the profit function to confirm the existence of a unique point of maximum for ; we apply Hessian matrix as follows:

Now the Hessian matrix of will be

And the function will be concave if

We need to prove and (C.5) will be , if are arbitrary positive value and must be positive value. We therefore have and :

Using (C.4)–(C.6) and (C.5)-(C.6), we can obtain

Hence must be ; with and , there exists a unique point of maximum for .

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgment

The authors thank the Ministry of Science and Technology for funding this research (Project’s serial no. 103-2410-H-019-006-).

References

H. M. Beheshti, “A decision support system for improving performance of inventory management in a supply chain network,” International Journal of Productivity and Performance Management, vol. 59, no. 5, pp. 452–467, 2010.

C. H. Glock, “The joint economic lot size problem: a review,” International Journal of Production Economics, vol. 135, no. 2, pp. 671–686, 2012.

C. H. Glock, E. H. Grosse, and J. M. Ries, “The lot sizing problem: a tertiary study,” International Journal of Production Economics, vol. 155, no. 1–3, pp. 39–51, 2014.

T. J. Fang, Q. K. Wei, C. W. Liao, M. J. Hung, and T. H. Wang, “Microbiological quality of 18°C ready-to-eat food products sold in Taiwan,” International Journal of Food Microbiology, vol. 80, no. 3, pp. 241–250, 2003.

A. Herbon, E. Levner, and T. C. E. Cheng, “Perishable inventory management with dynamic pricing using time-temperature indicators linked to automatic detecting devices,” International Journal of Production Economics, vol. 147, pp. 605–613, 2014.

S. Mukhopadhyay, R. N. Mukherjee, and K. S. Chaudhuri, “Joint pricing and ordering policy for a deteriorating inventory,” Computers and Industrial Engineering, vol. 47, no. 4, pp. 339–349, 2004.

S. K. Goyal and B. C. Giri, “Recent trends in modeling of deteriorating inventory,” European Journal of Operational Research, vol. 134, no. 1, pp. 1–16, 2001.

P. M. Ghare and G. P. Schrader, “A model for an exponentially decaying inventory,” Journal of Industrial Engineering, vol. 14, pp. 238–243, 1963.

Y. K. Shah and M. C. Jaiswal, “Periodic review inventory model for items that deteriorate continuously in time,” International Journal of Production Research, vol. 15, no. 2, pp. 179–190, 1977.

R. S. Sachan, “On (t, s//i) policy inventory model for deteriorating items with time proportional demand,” Journal of the Operational Research Society, vol. 35, no. 11, pp. 1013–1019, 1984.

K. M. Kim and H. Hwang, “Simultaneous decision model for production systems with deteriorating raw materials,” International Journal of Systems Science, vol. 16, no. 7, pp. 909–915, 1985.

T. K. Datta and A. K. Pal, “Finite-horizon model for inventory returns and special sales of deteriorating items with shortages,” Asia-Pacific Journal of Operational Research, vol. 8, no. 2, pp. 179–188, 1991.

S. Pal, A. Goswami, and K. S. Chaudhuri, “A deterministic inventory model for deteriorating items with stock-dependent demand rate,” International Journal of Production Economics, vol. 32, no. 3, pp. 291–299, 1993.

L. Benkherouf, “On an inventory model with deteriorating items and decreasing time-varying demand and shortages,” European Journal of Operational Research, vol. 86, no. 2, pp. 293–299, 1995.

L. Benkherouf and Z. T. Balkhi, “On an inventory model for deteriorating items and time-varying demand,” Mathematical Methods of Operations Research, vol. 45, no. 2, pp. 221–233, 1997.

H. M. Wee, “A replenishment policy for items with a price-dependent demand and a varying rate of deterioration,” Production Planning and Control, vol. 8, no. 5, pp. 494–499, 1997.

K.-L. Hou, “Optimal production run length for deteriorating production system with a two-state continuous-time Markovian processes under allowable shortages,” Journal of the Operational Research Society, vol. 56, no. 3, pp. 346–350, 2005.

N. K. Mandal, T. K. Roy, and M. Maiti, “Inventory model of deteriorated items with a constraint: a geometric programming approach,” European Journal of Operational Research, vol. 173, no. 1, pp. 199–210, 2006.

A. A. Adel and Z. T. Balkhi, “The effects of learning and forgetting on the optimal production lot size for deteriorating items with time varying demand and deterioration rates,” International Journal of Production Economics, vol. 107, no. 1, pp. 125–138, 2007.

Y. Feng, X. Q. Cai, and F. S. Tu, “A two-commodity deteriorating inventory model with price-dependent demand,” in Proceedings of the IEEE International Conference on Industrial Engineering and Engineering Management (IEEM '07), pp. 1589–1593, December 2007.

J.-C. Sung and D.-C. Gong, “An integrated production-inventory model for deteriorating production with defective item rework and joint material replenishment policy,” Journal of the Chinese Institute of Industrial Engineers, vol. 25, no. 4, pp. 326–336, 2008.

M. S. Barketau, T. C. E. Cheng, and M. Y. Kovalyov, “Batch scheduling of deteriorating reworkables,” European Journal of Operational Research, vol. 189, no. 3, pp. 1317–1326, 2008.

I. Nakhai and S. Jafari, “New inventory model for perishable valuable materials such as special diseases' drugs,” in Proceedings of the 5th IEEE International Conference on Management of Innovation and Technology (ICMIT '10), pp. 600–605, June 2010.

F. Chen, T. Jia, and Y. Zheng, “Ordering policy for deteriorating items with in-transit deterioration and permissible delay in payments,” in Education and Management: International Symposium, ISAEBD 2011, Dalian, China, August 6-7, 2011, Proceedings, Part III, vol. 210 of Communications in Computer and Information Science, pp. 302–309, Springer, Berlin, Germany, 2011.

S. M. H. Molana, H. Davoudpour, and S. Minner, “An (r, nQ) inventory model for packaged deteriorating products with compound Poisson demand,” Journal of the Operational Research Society, vol. 63, no. 11, pp. 1499–1507, 2012.

B. C. Giri and T. Maiti, “Supply chain model for a deteriorating product with time-varying demand and production rate,” Journal of the Operational Research Society, vol. 63, no. 5, pp. 665–673, 2012.

B. Sarkar, “An EOQ model with delay in payments and time varying deterioration rate,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 367–377, 2012.

S. Panda, S. Saha, and M. Basu, “Optimal pricing and lot-sizing for perishable inventory with price and time dependent ramp-type demand,” International Journal of Systems Science, vol. 44, no. 1, pp. 127–138, 2013.

T. Kim, C. H. Glock, and Y. Kwon, “A closed-loop supply chain for deteriorating products under stochastic container return times,” Omega, vol. 43, pp. 30–40, 2014.

L. C. Iao and H. I. Hsiao, Application of Continuous Temperature Information on Shelf-Life Prediction and Inventory Control: A Case of 18°C Ready-to-Eat Food, National Taiwan Ocean University Research System, 2013.

Y. Qin, J. Wang, and C. Wei, “Joint pricing and inventory control for fresh produce and foods with quality and physical quantity deteriorating simultaneously,” International Journal of Production Economics, vol. 152, pp. 42–48, 2014.

M. Zhu, M. Du, J. Cordray, and D. U. Ahn, “Control of Listeria monocytogenes contamination in ready-to-eat meat products,” Comprehensive Reviews in Food Science and Food Safety, vol. 4, no. 2, pp. 34–42, 2005.

M. de Giusti, C. Aurigemma, L. Marinelli et al., “The evaluation of the microbial safety of fresh ready-to-eat vegetables produced by different technologies in Italy,” Journal of Applied Microbiology, vol. 109, no. 3, pp. 996–1006, 2010.

D. H. Jang and K. T. Lee, “Quality changes of ready-to-eat ginseng chicken porridge during storage at 25°C,” Meat Science, vol. 92, no. 4, pp. 469–473, 2012.

H. Cui, C. Xue, Y. Xue, W. Su, Z. Li, and H. Cong, “Development of shelf-stable, ready-to-eat (RTE) shrimps (Litopenaeus vannamei) using water activity lowering agent by response surface methodology,” Journal of Food Science and Technology, vol. 50, no. 6, pp. 1137–1143, 2013.

Ş. Pamuk, Z. Gürler, Y. Yildirim, and N. Ertaş, “The microbiological quality of ready to eat salads sold in Afyonkarahisar, Turkey,” Kafkas Universitesi Veteriner Fakultesi Dergisi, vol. 19, no. 6, pp. 1001–1006, 2013.

M. Bogataj, L. Bogataj, and R. Vodopivec, “Stability of perishable goods in cold logistic chains,” International Journal of Production Economics, vol. 93-94, pp. 345–356, 2005.

A. Osvald and L. Z. Stirn, “A vehicle routing algorithm for the distribution of fresh vegetables and similar perishable food,” Journal of Food Engineering, vol. 85, no. 2, pp. 285–295, 2008.

H.-K. Chen, C.-F. Hsueh, and M.-S. Chang, “Production scheduling and vehicle routing with time windows for perishable food products,” Computers & Operations Research, vol. 36, no. 7, pp. 2311–2319, 2009.

Z.-F. Xu and Y. Wang, “Research on auto-monitoring system for cold storage temperature based on VB,” in Proceedings of the 4th International Conference on Intelligent Computation Technology and Automation (ICICTA '11), vol. 1, pp. 689–691, Shenzhen, China, March 2011.

G. Aiello, G. La Scalia, and R. Micale, “Simulation analysis of cold chain performance based on time-temperature data,” Production Planning and Control, vol. 23, no. 6, pp. 468–476, 2012.

S. Bruckner, A. Albrecht, B. Petersen, and J. Kreyenschmidt, “A predictive shelf life model as a tool for the improvement of quality management in pork and poultry chains,” Food Control, vol. 29, no. 2, pp. 451–460, 2013.

C. J. P. McGinn, “Evaluation of shelf life,” IFST Proceedings, vol. 15, pp. 153–161, 1982.

T. A. McMeekin and T. Ross, “Shelf life prediction: status and future possibilities,” International Journal of Food Microbiology, vol. 33, no. 1, pp. 65–83, 1996.

T. P. Labuza and B. Fu, “Growth kinetics for shelf-life prediction: theory and practice,” Journal of Industrial Microbiology, vol. 12, no. 3-5, pp. 309–323, 1993.

R. C. Whiting and R. L. Buchanan, “A classification of models in predictive microbiology,” Food Microbiology, vol. 10, no. 2, pp. 175–177, 1993.

A. M. Gibson, N. Bratchell, and T. A. Roberts, “The effect of sodium chloride and temperature on rate and extent of growth of Clostridium botulinum type A unpasteurized pork slurry,” Journal of Applied Bacteriology, vol. 62, no. 6, pp. 479–490, 1987.

R. H. Linton, W. H. Carter, M. D. Pierson, and C. R. Hackney, “Use of a modified Gompertz equation to model nonlinear survival curves for Listeria monocytogenes Scott A,” Journal of Food Protection, vol. 58, no. 9, pp. 946–954, 1995.

L. Huang, “Description of growth of Clostridium perfrigens in cooked beef with multiple linear models,” Food Microbiology, vol. 19, no. 6, pp. 577–587, 2002.

B. R. Chowdhury, R. Chakraborty, and U. R. Chaudhuri, “Validity of modified Gompertz and Logistic models in predicting cell growth of Pediococcus acidilactici H during the production of bacteriocin pediocin AcH,” Journal of Food Engineering, vol. 80, no. 4, pp. 1171–1175, 2007.

R. L. Buchanan, L. K. Bagi, R. V. Goins, and J. G. Phillips, “Response surface models for the growth kinetics of Escherichia coli O157:H7,” Food Microbiology, vol. 10, no. 4, pp. 303–315, 1993.

M. Mastromatteo, A. Danza, A. Conte, G. Muratore, and M. A. Del Nobile, “Shelf life of ready to use peeled shrimps as affected by thymol essential oil and modified atmosphere packaging,” International Journal of Food Microbiology, vol. 144, no. 2, pp. 250–256, 2010.

P. S. Taoukis, K. Koutsoumanis, and G. J. E. Nychas, “Use of time-temperature integrators and predictive modelling for shelf life control of chilled fish under dynamic storage conditions,” International Journal of Food Microbiology, vol. 53, no. 1, pp. 21–31, 1999.

L. S. Briones, J. E. Reyes, G. E. Tabilo-Munizaga, and M. O. Pérez-Won, “Microbial shelf-life extension of chilled Coho salmon (Oncorhynchus kisutch) and abalone (Haliotis rufescens) by high hydrostatic pressure treatment,” Food Control, vol. 21, no. 11, pp. 1530–1535, 2010.

H. Tyrer, P. Ainsworth, Ş. Ibanoǧlu, and H. Bozkurt, “Modelling the growth of Pseudomonas fluorescens and Candida sake in ready-to-eat meals,” Journal of Food Engineering, vol. 65, no. 1, pp. 137–143, 2004.

C. Stannard, “Development and use of microbiological criteria for foods,” Food Science and Technology Today, vol. 11, no. 3, pp. 137–176, 1997.

S. K. Sagoo, C. L. Little, G. Allen, K. Williamson, and K. A. Grant, “Microbiological safety of retail vacuum-packed and modified-atmosphere-packed cooked meats at end of shelf life,” Journal of Food Protection, vol. 70, no. 4, pp. 943–951, 2007.

Y. Huang and M. Wang, “Deterministic lot-size inventory models with shortages and deterioration,” International Journal of Advancements in Computing Technology, vol. 5, no. 4, pp. 930–937, 2013.