# Improvement for Amelioration Inventory Model with Weibull Distribution

Authors: Han-Wen Tuan; Shu-Cheng Lin; Peterson Julian
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

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.


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## Body

## 1. 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 Table1.Table 1
Pertinent literature on amelioration inventory models with various setting.
Authors
Inventory model
Objective function
Solution
Type
Distribution
Restriction
Hwang [7]
EOQ
Approximated
Graphical
Amelioration
Weibull
C
p
>
C
a
Mondal et al. [9]
EOQ
Exact
Exact
Amelioration/deterioration
Weibull
Demand depends on price
Moon et al. [11]
EOQ
Exact
Exact
Amelioration/deterioration
Constant
Time-value of money
Law and Wee [12]
EPQ
Approximated
Approximated
Amelioration/deterioration
Weibull
Chou et al. [13]
EOQ
Exact
Exact
Amelioration
Weibull
C
p
>
C
a
Ji [14]
EOQ
Exact
Numerical
Amelioration/deterioration
Constant
Time-value of money
Valliathal and Uthayakumar [15]
EPQ
Exact
Numerical
Amelioration/deterioration
Linear
Time-value of money
Chen [16]
EOQ
Exact
Exact
Amelioration/deterioration
Constant
De Zoysa and Rupasinghe [17]
EOQ
Exact
Numerical
Amelioration
Linear
Linear demand
Nodoust et al. [18]
EPQ
Exact
Approximated
Amelioration/deterioration
Constant
Linear demand
Our manuscript
EOQ
Exact
Exact
Amelioration
Weibull
C
a
>
C
p


## 2. 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 Cp>Ca. However, the purchase cost of baby fry or chicken or vegetable seeds is usually less than that of feed and fertilizer, implying that Cp<Ca. For this note, we will discuss inventory model for items of ameliorating items with Weibull distribution, under the restriction Cp<Ca. For a detailed graphical expression of Weibull distribution, please refer to Wikipedia.R: the constant demand rate (kg/day)t: the parameter to denote the time of amelioration (day)Co: the ordering cost (₩)Ca: the amelioration cost (₩/kg)Cp: the purchase cost (₩/kg), with Cp<CaCh: the holding cost (₩/kg/day)At: the deterioration rate with At=αβtβ-1 (1/day), where α is the shape parameter and β is the scale parameterIt: the inventory level (kg)T: the cycle time (day)TCT: the average total cost per unit time that consists of ordering cost, purchasing cost, holding cost, and the ameliorating cost (₩/day)fT: an auxiliary function with dTCT/dT=R/T2fTgT: another auxiliary function with d/dTfT=αβTβe-αTβgTT#: the unique positive solution for d/dTgT=0 with T#=21-β/αβ1/βT∗: the optimal solution (day)


## 3. Review of Previous Results

Hwang [7] first found the initial inventory level, I0, and then the inventory level, It, as follows:(1)I0=R∫0Te-αtβdt,(2)It=eαtβ-R∫0te-αxβdx+I0.We cite Figure 1 from Hwang [7] to illustrate the inventory level, It.Figure 1
Inventory cycle of EOQ model [reproduction from Hwang [7]].He expressed the objection function,TCT, as(3)TCT=I0Cp-CaT+Ch2+RCa+CoT.Hwang used Taylor’s series expansion for the exponential function to derive (4)I0=R∑k=0∞-αkTkβ+1k!kβ+1.He plugged (4) into (3) and then computed dTC(T)/dT to imply(5)dTCTdT=Cp-CaR∑k=0∞-αkkβk!kβ+1Tkβ+1+Ch2R∑k=0∞-αkkβ+1k!Tkβ-CoT2.From (5), he used a graphical procedure of Gupta [10] as follows:(6)T=RCo∑k=0∞-kkTkβ-3k!Cp-CakβTkβ+1+Ch2.He claimed that (6) can be solved by a computer search.


## 4. Discussion of Hwang’s Results

We first review his results. From the differential equation for the inventory level, we know that(7)ddtIt=αβtβ-1It-R,where the ending inventory level is zero as IT=0; then(8)It=eαtβR∫tTe-αxβdx.From the inventory level in (8), we obtain the initial inventory level as(9)I0=I0=R∫0Te-αxβdx.Comparing (2) and (8), we have a simple expression. For the sake of completeness, we point out that (5) should be revised as(10)dTCTdT=Cp-CaR∑k=0∞-αkkβk!kβ+1Tkβ-1+Ch2R∑k=0∞-αkk!Tkβ-CoT2.Moreover, (6) should be modified as(11)T=RCo∑k=0∞-αkTkβ+3k!Cp-Cakβkβ+1T+Ch2.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.


## 5. Our Improvement

We plug (9) into (3) to take the first derivative; then it yields that(12)dTCTdT=RT2fTwith an auxiliary function, say fT, where(13)fT=Ca-Cp∫0Te-αxβdx-Te-αTβ+Ch2T2e-αTβ-CoR.Studying the properties of dTCT/dT is equivalent to examining the properties of fT. We obtain that(14)ddTfT=αβTβe-αTβgTwith another auxiliary function, say gT, where(15)gT=ChαβT1-β-Ch2T+Ca-Cp.According to the expression in (15), we divide the inventory model into the following three cases: (a) 0<β<1, (b) β=1, and (c) β>1.For the first case with0<β<1, motivated by (15), it follows that (16)ddTgT=Ch1-βαβ1Tβ-αβ21-β.To simplify the expression, we assume the unique positive root of (16) as T#, which means(17)T#=21-βαβ1/β.From (16), we know that d/dTgT>0 for 0<T<T# and d/dTgT<0 for T>T#. Hence, gT increases for 0<T<T# and gT decreases for T>T#.Fromg0=Ca-Cp>0, gT increases to gT# then gT decreases to g∞=-∞, such that there is a unique point, say T1, with gT1=0 as (18)ChαβT11-β+Ca-Cp=Ch2T1.It yields that gT>0 for 0<T<T1 and gT<0 for T1<T. From (13), we obtain that f0=-C0/R<0. By (14), it implies that fT increases for 0<T<T1 and fT decreases for T1<T.Using (18), we obtain that(19)fT1=Ca-Cp∫0T1e-αxβdx+ChαβT12-βe-αT1β-CoR.Depending on the value of fT1 and f∞, we will divide case (a) into three cases: (a1) fT1≤0, (a2) fT1>0 and f∞<0, and (a3) fT1>0 and f∞≥0.Lemma 1.
For case (a1),fT1≤0 and then the optimal solution T∗=∞.
FromfT≤0 for T>0, by (12), dTCT/dT≤0 for T>0 so the minimum value will occur when T→∞.Next, we consider case (a2). SincefT1>0, then fT increases from f0=-C0/R<0 to fT1>0 for 0<T<T1 and fT decreases from fT1>0 to f∞<0. Hence, there are two points, say T2 and T3, where 0<T2<T1<T3 with fT2=0 and fT3=0. It yields that fT<0, for 0<T<T2, fT>0 for T2<T<T3, and fT<0, for T3<T. Consequently, by (12), TCT decreases for 0<T<T2, then TCT increases for T2<T<T3, and then TCT decreases for T3<T. Hence, we derive that T=0 and T=T3 are local maximum points. On the other hand, T=T2 and T=∞ are local minimum points. We summarize our finding in the next lemma.Lemma 2.
For case (a2), the minimum value ofTCT is min⁡TCT2,TC∞.Third, we consider case (a3). FromfT1>0 and f∞≥0, we know that there exists a unique point, say T4 such that fT4=0 satisfying 0<T4<T1. It follows that fT<0, for 0<T<T4 and fT>0 for T4<T. By the same argument, TCT decreases for 0<T<T4 and then TCT increases for T4<T such that T4 is the unique local minimum point.Lemma 3.
For case (a3), the minimum value ofTCT is TCT4.We combine all previous results in the next main theorem.Theorem 4.
Under the condition0<β<1, for the minimum point, T∗, and minimum value, TCT∗, there are three different cases:(1)
For case (a1), withfT1≤0, then the optimal solution T∗=∞.(2)
For case (a2), withfT1>0 and f∞<0, the minimum value of TCT is min⁡TCT2,TC∞.(3)
For case (a3), withfT1>0 and f∞≥0, the minimum value of TCT is TCT4.To help researchers absorb our findings, we sketch four different cases from Theorem4 in Figure 2.Figure 2
For0<β<1, four cases are illustrated. The sign of f(T1) is first considered for branching, and the sign of f(∞) is then considered for further branching. The value of TCT∗ is lastly considered to result in cases (a2_1) and (a2_2).Now, we begin to discuss the second case withβ=1. From (15), we know that (20)gT=Chα+Ca-Cp-Ch2T.To simplify the expression, we assume that(21)T5=2α+2Ca-CpCh.We know that gT>0 for T<T5 and gT<0 for T5<T. It implies that fT increases for T<T5 and fT decreases for T5<T.Depending on the value offT5 and f∞, we will divide case (b) into three cases: (b1) fT5≤0, (b2) fT5>0 and f∞<0, and (b3) fT5>0 and f∞≥0.By the same argument for the case of0<β<1 to replace T1 by T5, we may derive the similar theorem as follows, where T6 and T7 are the roots for fT=0 under case (b2) where 0<T6<T5<T7 and T8 is the root for fT=0 under case (b3).Theorem 5.
Under the conditionβ=1, for the minimum point, T∗, and minimum value, TCT∗, there are three different cases:(1)
For case (b1) withfT5≤0, then the optimal solution T∗=∞.(2)
For case (b2), withfT5>0 and f∞<0, the minimum value of TCT is min⁡TCT6,TC∞.(3)
For case (b3), withfT5>0 and f∞≥0, the minimum value of TCT is TCT8.To help researchers absorb our findings, we sketch four different cases from Theorem5 in Figure 3.Figure 3
Forβ=1, four cases are illustrated. The sign of fT5 is first considered for branching, and the sign of f(∞) is then considered for further branching. The value of TCT∗ is lastly considered to result in cases (b2_1) and (b2_2).At last, we begin to discuss the third case withβ>1. From (15), we know that (22)ddTgT=-Chβ-1αβ1Tβ+αβ2β-1<0,so gT is a decreasing function from limT→0+⁡gT=∞ to limT→∞⁡gT=-∞; then there is a unique point, say T$, with gT$=0 and gT>0 for 0<T<T$ and gT<0 for T>T$. It implies that fT increases for 0<T<T$ and fT decreases for T>T$.Depending on the value offT$ and f∞, we will divide case (c) into three cases: (c1) fT$≤0, (c2) fT$>0 and f∞<0, and (c3) fT$>0 and f∞≥0.By the same argument for the case of0<β<1 to replace T1 by T$, we may derive similar theorem as follows, where T9 and T10 are the roots for fT=0 under case (c2) where 0<T9<T$<T10 and T11 is the root for fT=0 under case (c3).Theorem 6.
Under the conditionβ>1, for the minimum point, T∗, and minimum value, TCT∗, there are three different cases:(1)
For case (c1) withfT$≤0, then the optimal solution T∗=∞.(2)
For case (c2), withfT$>0 and f∞<0, the minimum value of TCT is min⁡TCT9,TC∞.(3)
For case (c3), withfT$>0 and f∞≥0, the minimum value of TCT is TCT11.To help researchers absorb our findings, we sketch four different cases from Theorem6 in Figure 4.Figure 4
Forβ>1, four cases are illustrated. The sign of fT$ is first considered for branching, and the sign of f(∞) is then considered for further branching. The value of TCT∗ is lastly considered to result in cases (c2_1) and (c2_2).Motivated by Figures2–4, for 0<β<1, β=1, and β>1, respectively, we know that graphs of gT are different for three different cases. However, we can abstractly combine these three different graphs of gT to imply that there is a unique point, say T#, such that gT>0, for 0<T<T#, and gT<0, for T>T#. Consequently, we observe that cases (a1), (b1), and (c1) are of the same type. Similarly, cases (a2), (b2), and (c2) looked like each other. Moreover, cases (a3), (b3), and (c3) are similar. Hence, we claim that there is a unique point, say T#, satisfying gT#=0 and there is a unique point, say TΔ, with 0<TΔ<T# and fTΔ=0; we can merge Theorems 4–6 into a general result as follows.Theorem 7.
For our amelioration model withCa>Cp, we obtain that first solve gT=0 to yield the unique solution T#. Checking the value of fT#, we divide the solution procedure into two different cases.(a)
IffT#≤0, then the optimal solution T∗=∞.(b)
IffT#>0, then there is a unique point, say TΔ, with 0<TΔ<T# and fTΔ=0; then the minimum value is min⁡TCTΔ,TC∞.


## 6. 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: R=1000, Ch=400, Co=300000, Cp=10000, and α=0.05, except that we change Ca from 4000 to 40000 such that we have Ca>Cp and β from 0.05 into three different cases: in Example 1, β=0.05, in Example 2, β=1, and in Example 3, β=2 such that it provides three examples for each case, respectively.Example 1.
Under the conditionβ=0.05, that is suitable for case (a), we have that T1=1.153×1058, fT1>0, and f∞>0; by case (a3), the optimal solution is T4=1.0889806138 and the optimal value is TCT4=1.1883967×107.Example 2.
Under the conditionβ=1, that is suitable for case (b), we obtain that T5=190, fT5>0 and f∞>0; from case (b3), the optimal solution is T8=0.56785328 and the optimal value is TCT8=1.1062167×107.Example 3.
Under the conditionβ=2, that is suitable for case (c), we find that T$=150.133, fT$>0, and f∞>0; from case (c3), the optimal solution is T11=0.611864431 and the optimal value is TCT11=1.0798061×107.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.Step 1.
SolvegT=0 to derive the unique root, say T#.Step 2.
ComputefT#.Step 3.
IffT#≤0, then the optimal solution is T∗=∞.Step 4.
IffT#>0, then solve fT=0 to locate the unique root, say TΔ, satisfying 0<TΔ<T#; then the optimal value =min⁡TCTΔ,TC∞.In the following, we provided a comparison between two restrictions: (a) In Hwang [7] and Chou et al. [13] with Cp>Ca and (b) in our manuscript with Ca>Cp. 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) Cp will be very large. The selling price for an adult eel, for example, is $10/per tail such that Cp>Ca (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 assumptionCa>Cp 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.


## 7. 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.


## 8. 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.



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