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 temperaturedependent. We provide a numerical example to illustrate the solution. Our model demonstrates that high storage temperatures reduce profits and force shorter order cycles.
We know that inventory policy may affect supply chain performance from Beheshti’s [
Deterioration is defined as decay, damage, spoilage, or perishability and its effect cannot be disregarded in inventory models [
In the abovementioned studies, deterioration rate is regarded as a constant, varying, or timeproportional value and the demand rate as stockdependent, timedependent, or pricedependent. With this type of deterioration inventory model becoming increasingly mature, researchers began to elaborate new deterioration inventory models.
Mukhopadhyay et al. [
Chen et al. [
We discovered that pricedependent 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
Overview of deteriorating model.
Authors  Deterioration rate  Demand rate 

Kim and Hwang [ 
Timedependent  Constant 
Wee [ 
Constant  Pricedependent 
Mukhopadhyay et al. [ 
Timedependent  Pricedependent 
Hou [ 
Timedependent  Pricedependent 
Feng et al.[ 
Timedependent  Pricedependent 
Nakhai and Jafari [ 
Timedependent  Constant 
Sarkar [ 
Timedependent  Timedependent 
Giri and Maiti [ 
Incontrol and outofcontrol  Pricedependent 
Iao and Hsiao [ 
Temperaturedependent  Constant 
Qin et al. (2014) [ 
Temperaturedependent  Pricedependent 
^{*}Our work  Temperaturedependent  Pricedependent 
Inspired by Nakhai and Jafari [
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 [
Herbon et al. [
Overview of perishable goods works.
Authors  Claims 

Bogataj et al. [ 
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 [ 
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. [ 
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 [ 
Perishable goods were quite sensitive in time and temperature. 
Xu & Wang [ 
Suitable facility and precise temperature controlling are two major factors to keep perishable goods. 
Aiello et al. [ 
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. 
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 [
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 [
Overview of predictive quality model.
Authors  Predictive quality model 

Gibson et al. [ 
Gompertz model 
Buchanan et al. [ 
Gompertz model 
Linton et al. [ 
Gompertz model 
Huang [ 
Gompertz model 
Chowdhury et al. [ 
Logistic model and Gompertz model 
^{*}Our work  Gompertz model 
Qin et al. [
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 temperaturedependent. 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 [
The following assumptions and notations are used to formulate the problem and model.
A single product and a single vendor are assumed.
We take the pork sandwiches as products in this research.
The deterioration rate of pork sandwiches
The time at maximum growth rate of pork sandwiches
Shortages are not allowed.
Lead time is assumed to be 0.
The time horizon is infinite.
The pricedependent demand rate is equal to
Once the
According to research by Huang and Wang [
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).
Optimal value of




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
Optimal value of total profit




 

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
Optimal value of total profit
Optimal value of total profit
Summarizing the results in Tables
As shown in Table
Optimal value of



 

^{*}

112.33  115.16  130.17  154.95 
^{*}

2  2  1  1 
^{*}

685.32  641.23  425.11  61.41 
^{*}Optimal value.
In this research, we discussed the issue of deteriorating inventory model. After reviewing Glock’s [
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 [
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
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 realworld 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.
See Figure
Gompertz function example.
The inventory level at any time
We need a concavity condition of the profit function
The authors declare that they have no conflict of interests.
The authors thank the Ministry of Science and Technology for funding this research (Project’s serial no. 1032410H019006).