Improvement for Amelioration Inventory Model with Weibull Distribution

Most inventory models dealt with deteriorated items. On the contrary, just a few papers considered inventory systems under amelioration environment. We study an amelioration inventory model with Weibull distribution. However, there are some questionable results in the amelioration paper. We will first point out those questionable results in the previous paper that did not derive the optimal solution and then provide some improvements.We will provide a rigorous analytical work for different cases dependent on the size of the shape parameter. We present a detailed numerical example for different ranges of the sharp parameter to illustrate that our solution method attains the optimal solution. We developed a new amelioration model and then provided a detailed analyzed procedure to find the optimal solution. Our findings will help researchers develop their new inventory models.


Introduction
Several published theses have addressed the importance of the deterioration phenomenon in field applications; as a result, many deterioration models have been consequently developed.Yet we have not observed much appreciation of ameliorating consideration.Due to lack of considering the influence of demand, the ameliorating items assuming duration for the amount of inventory will gradually increase; meanwhile, in the traditional inventory model dealing with deteriorating items the amount of inventory will gradually decrease.Amid the published literatures, scholars and researchers do not pay much attention to the ameliorating problems and items.To cope with this deficiency, lately a few studies are concerned with the problems of amelioration, because they do exist in the real world such as the farming, fishery, and poultry industries.The fast growing animals like ducks, pigs, and broilers in poultry farms, highbred fishes in ponds, and the cultivation of vegetables and fruits in farms are typical field applications.This is quite different from the deteriorating items and deserves a comprehensive study.
Weibull distribution has been applied to the problems of product life cycle in recent years, because it can describe the different life spans effectively by utilizing the changes of parameters.Many scholars have applied the inventory model that follows Weibull distribution.Jalan et al. [1], Chakrabarty et al. [2], and Wu et al. [3] considered the deteriorating inventory model.Chang and Dye [4], Wu [5], and Giri et al. [6] extended the inventory model with deteriorating items in several directions.Hwang [7,8] and Mondal et al. [9] developed the inventory models with ameliorating items under various situations.Since World War II, inventory systems with the deterioration satisfying Weibull distribution have been successfully modeled in field operations; therefore, many theses are published to address the importance.Each of these extended models is applicable to certain cases in real operating applications.Some are more generally suitable than others, but most of them carry the weakness of being too complicated to control, both algebraically and numerically.Consequently, some modelers must trade off precision with respect to tractability.Hwang [7] is one of first authors to consider the remodeling from deterioration to amelioration.However, his published literatures did not present the ultimately optimal solution.Instead, he used a graphical procedure of Gupta [10] to estimate the optimal cycle time, which could not derive to a convincingly optimal solution.We hereby generate a complete analytical framework to develop the optimal solution, in which both precision and tractability will be secure.
For the recent development of amelioration inventory models, we provide a list to compare 12 papers in Table 1.

Notation and Assumption
This note uses the same notation and expressions as Hwang [7] together with two auxiliary functions and several new expressions to clearly indicate our results, except that in Hwang [7], he assumed that   >   .However, the purchase cost of baby fry or chicken or vegetable seeds is usually less than that of feed and fertilizer, implying that   <   .For this note, we will discuss inventory model for items of ameliorating items with Weibull distribution, under the restriction   <   .For a detailed graphical expression of Weibull distribution, please refer to Wikipedia.

Review of Previous Results
Hwang [7] first found the initial inventory level,  0 , and then the inventory level, (), as follows: We cite Figure 1 from Hwang [7] to illustrate the inventory level, ().
He expressed the objection function, TC(), as Hwang used Taylor's series expansion for the exponential function to derive He plugged (4) into (3) and then computed TC()/ to imply From ( 5), he used a graphical procedure of Gupta [10] as follows: He claimed that (6) can be solved by a computer search.

Discussion of Hwang's Results
We first review his results.From the differential equation for the inventory level, we know that where the ending inventory level is zero as () = 0; then From the inventory level in (8), we obtain the initial inventory level as Comparing ( 2) and ( 8), we have a simple expression.For the sake of completeness, we point out that (5) should be revised as Moreover, (6) should be modified as Hwang [7] did not explain why (5) has solutions or the solution is unique.He just sketched a graph to show that there is a unique solution for (5) without further explanation.We will derive a series of lemmas and theorems to find out whether (5) has a unique solution.

Our Improvement
We plug ( 9) into (3) to take the first derivative; then it yields that with an auxiliary function, say (), where Studying the properties of TC()/ is equivalent to examining the properties of ().We obtain that with another auxiliary function, say (), where According to the expression in (15), we divide the inventory model into the following three cases: (a) 0 <  < 1, (b)  = 1, and (c)  > 1.
To help researchers absorb our findings, we sketch four different cases from Theorem 4 in Figure 2. Now, we begin to discuss the second case with  = 1.From ( 15), we know that To simplify the expression, we assume that We know that () > 0 for  <  5 and () < 0 for  5 < .
To help researchers absorb our findings, we sketch four different cases from Theorem 5 in Figure 3.
To help researchers absorb our findings, we sketch four different cases from Theorem 6 in Figure 4.
Theorem 7.For our amelioration model with   >   , we obtain that first solve () = 0 to yield the unique solution  # .Checking the value of ( # ), we divide the solution procedure into two different cases.

Mathematical Problems in Engineering
TC(T 7) TC(T 6) TC(T 8 ) Figure 3: For  = 1, four cases are illustrated.The sign of ( 5 ) is first considered for branching, and the sign of (∞) is then considered for further branching.The value of TC( * ) is lastly considered to result in cases (b2_1) and (b2_2).

Numerical Examples
To be comparable with the results in Hwang [7], we consider the same numerical example as Example 1 in it with the following data:  = 1000,  ℎ = 400,   = 300000,   = 10000, and  = 0.05, except that we change   from 4000 to 40000 such that we have   >   and  from 0.05 into three different cases: in Example 1,  = 0.05, in Example 2,  = 1, and in Example 3,  = 2 such that it provides three examples for each case, respectively.
Under our classification to different cases, we derive the optimal cycle time and the optimal total cost that will help the researcher and practitioners to obtain the best decision making.
For practitioners easily applying our results, we provide a detailed algorithm for the optimal solution and the optimal value.
In the following, we provided a comparison between two restrictions: (a) In Hwang [7] and Chou et al. [13] with   >   and (b) in our manuscript with   >   .For eel, until now, no one can produce baby eel from eggs of a pregnant female eel.Hence, the baby eel must be caught from the wild and then sold by tail, for example, $2/per tail, and then the purchasing cost (per kilogram)   will be very large.The selling price for an adult eel, for example, is $10/per tail such that (in kilogram) is valid for eel that is consistent with Hwang [7] and Chou et al. [13].
On the other hand, from Wikipedia, there is a picture that shows "Expressing eggs from a female rainbow trout" indicating that rainbow trout can be reproduced from eggs such that the purchasing cost for rainbow trout will be low and then our assumption   >   can be applied for this kind of fish farming.
Hence, amelioration inventory models can be used to measure (a) weight gain or (b) pregnant mother fish delivering new born baby fish.

Direction for the Future Research
There are two amelioration inventory models in Hwang [7].For the first model, the demand rate is greater than the amelioration rate.For the second model, the amelioration rate is greater than the demand rate.Under this restriction, Hwang [7] developed his "partially selling quantity model" which can be treated as sustainable management.Hence, we can predict that researchers may consider the exact objective function and then try to derive the optimal solution that will be an interesting research topic in the future.

Conclusion
With careful classification based on theoretical constructions, this paper developed an analytical framework for Weibull ameliorating in inventory models and provided the optimal solution of cycle time and the minimum value of average total cost.Numerical examples were also provided to illustrate our findings.

Figure 2 :
Figure 2: For 0 <  < 1, four cases are illustrated.The sign of ( 1 ) is first considered for branching, and the sign of (∞) is then considered for further branching.The value of TC( * ) is lastly considered to result in cases (a2_1) and (a2_2).

Figure 4 :
Figure 4: For  > 1, four cases are illustrated.The sign of ( $ ) is first considered for branching, and the sign of (∞) is then considered for further branching.The value of TC( * ) is lastly considered to result in cases (c2_1) and (c2_2).

Table 1 :
Pertinent literature on amelioration inventory models with various setting.