Weibull Deterioration Rate on Integrated Cold Chain Inventory Performance

. Different from the previous articles on multi-echelon deteriorating inventory, this paper explains the economic implications of Weibull parameters for cold chain inventory rather than restricting to mathematical sense. A case study shows that DC warehousing activities and total throughput of the scale factor has a significant effect on the total profit compared with the storage technology and warehousing environment of shape factor. The improvements of DC warehousing standard and total throughput and preservation technology and warehousing environment contribute to the reduction of some DC operation costs and the increase of DC processing and customizing profit.


Introduction
On the deteriorating rate of cold chain product, most of the current works thought of as a constant rate or instantaneous deterioration ( (Huang et al., 2018). Thus, some cold chain products, such as, food, fruits, and vegetables are not decayed instantaneously. Weibull non-instantaneous deterioration function is used to depict the deteriorating characteristics of cold chain products (Yang 2012; Dye 2013; Sanni, Chukwu 2016).
In the comparison of the performance of three-echelon cold chain system with two echelon system, threeparameter The distribution center as a third-party logistics that efficiently links the logistics of producers and retailers and consolidates all parties into a system, can not only enhance the core competencies of DC warehousing, storage and distribution, but also improve the system profitability as a result of perfectly applying with customers ' requirements for small quantity with large variety and joint operation for all customers, with its value added services of product processing and customizing, picking and assorting.
Motivated by the above background, following questions would be proposed: How do the DC deterioration parameters affect the integrated cold chain inventory performance? And how do the DC deterioration parameters affect the DC cost structure and its gross profit.
The rest of this paper organized as follows. Section 2 presents the problem description. Section 3 provides numerical analysis. Finally, it summarizes the findings and concludes the potential impacts of this work in section.

Problem description
In this paper, three-echelon integrated cold chain inventory system composed of three manufacturers (expressed as), one DC and two retailers (expressed as), with non-instantaneous Weibull deterioration rate items in a finite horizon is investigated and proposed to obtain the optimal order quantity with an objective of maximizing the system profit.
Different from two parameter Weibull distribution, three parameter Weibull function can more comprehensively fit the deterioration rate of different conditions. The density function and cumulative distribution function of three parameter Weibull distribution for cold chain products can be written as ,respectively, and the noninstantaneous Weibull deterioration rate is , which is the deterioration trend of the cold chain product at time t . Of which,  is the scale parameter ( 0   ),  is the shape parameter ) and  is the position parameter ( 0   ) respectively. The curve shape is shown in Fig. 1  Through the deterioration rate function of   While the position factor  determines the starting position of product deterioration. If 0   , it means that deterioration occurs immediately once the cold chain products enters the distribution center (DC). If 0   , it means that there is still a period of freshness-keeping period after the goods enter the DC, that is, there is a noninstantaneous deterioration. Similarly, if 0   , it means that the product has occurred deterioration for a period of time before its entering the DC. Due to the different combinations of the three parameters in Weibull distribution, the deterioration rate presents different function curves. Compared with fixed or linear deterioration rate, Weibull function can better describe the deterioration characteristics of different cold chain products. In this paper, the parameters discussed are as Especially, we will analyse the effect of dk  and dk  on DC cost and system profit.

Case study and analysis
This paper is based on the study of three-echelon cold chain inventory system (Wang et al., 2017) to examine the effect of Weibull distribution factors of DC on system profit, so, inventory model formulation and numerical calculation are ignored. Through investigating a local cold chain system, the regarding parameters and system profit are listed in Table 1  In a cold chain, as is defined, the effect of DC warehousing activities and its total throughput on deterioration is called a scale factor of Weibull distribution and DC storage technology and warehousing environment on the effect of deterioration is called a shape factor of Weibull distribution. Here, the key parameters of the DC deterioration rate dk  and dk  are chosen for sensitivity analysis.
The optimal profits are obtained in Table 3 and Table  4 by varying the parameters' values of dk  and dk  from -40% to 40%, respectively.  Through Table 3 and Table 4, it can be found that the total profit goes inversely with the change of DC deterioration rate. The increase of the deterioration rate caused by the warehousing activities and total throughput may be from frequent and improper handling and large throughput. The increase of the storage technology and warehousing environment factor may be from an improper or outdated DC preservation technology. All of which will lead to loss and damage of the deteriorating items, in turn, a decrease of total system profit. On the contrary, the improvements of DC warehousing standard and total throughput, and its preservation technology increase the system profit and certainly help realize the industry structure optimization.
By comparing the effects of dk  and dk  on the total profit, it can be found that DC warehousing activities and total throughput factor has a higher impact on the total profit than that of storage technology and warehousing environment factor. Thus, DC warehousing standard and process as well as the design of total throughput should be made and executed to guarantee the quality of the perishable products as a result that losses and damages are reduced simultaneously. The effects of dk  and dk  on DC cost and its gross profit are illustrated in Fig.2-5.     Fig.2 and Fig.4 demonstrate that the decrease of dk  and dk  results in the decrease of the total deteriorated cost, the total variable holding cost, the decrease of total replenishing quantity and the increase of DC picking and assorting cost. It is worth noting that the increase of DC picking and assorting cost may be from the careful handling of the perishable products to assure the decrease of dk  and may be from keeping an orderly and clean environment to assure the decrease of dk  . Fig.3 and Fig.5 show that DC gets more gross profit from the product processing and customizing service for the increase of available products owing to the decrease of dk  and dk  .
It is worth noting that, when dk  is increased by 40%, DC total profit is the highest. The possible reason is that DC does not invest advanced preservation technology in the warehousing system, the cost could be reduced, in turn its profit is increased in a short term. But from the long run, distribution center should invest in preservation technology to devote to the improvements of the warehousing condition and the realization of industry structure optimization. Thus, from the long run, both the improvements of DC warehousing standard and total throughput, and innovations in its preservation technology and environment could also highlight DC performance of the value-added services.

Conclusion
This paper proposes an integrated three-echelon cold chain inventory model with three-parameter Weibull deteriorating items. The major contributions are: (1)the economic implications of Weibull deterioration parameters are described to better display the effects of warehousing activities and total throughput, preservation technology on the deterioration rate and on the total system profit. That is, with the decrease of DC deterioration rate, the system profit is increased, of which,  dk has a greater effect than that of dk  . (2) The improvements of DC warehousing standard and total throughput and preservation technology and warehousing environment contribute to the reduction of some DC operation costs and the increase of DC processing and customizing profit.
These conclusions present good theoretical and managerial insights that DC warehousing activities should be carefully operated without rough handling, and the total throughput should be adaptable rather than overcapacity, DC processing and customizing service for all kinds of products and all logistics services including both the traditional service and value added services for high value products should be strengthened, meanwhile cost savings must be achieved from picking/assorting service to devote to higher profits for both total system and all members. More attention should also be paid to preservative technology innovation in DC and DC processing and customizing cost control.
This study could be extended to inventory-joint distribution policy and coordination inventory policy.