the **demand** rates are proportional **with** **time**. In this paper, he did not suggest an optimal solution to the **model** but, rather, an approximate method by assumed equal replenishment periods. Goswami and Chauduri [1991] considered a **model** **with** linear **demand** rates. They also suggested an approximate replenishment schedule. This is concerned **with** the development of **ameliorating** **inventory** models. The **ameliorating** **inventory** is the **inventory** of foods whose utility increases over the **time** by **ameliorating** activation. This study is performed according to areas; one is an economic **order** quantity (EOQ) **model** for the **items** whose utility is **ameliorating** in accordance **with** Weibull distribution, and the other is a partial selling quantity (PSQ) **model** developed for selling the surplus **inventory** accumulated by **ameliorating** activation **with** linear **demand**. Numerical examples to show the effect of **ameliorating** **rate** **on** **inventory** polices are illustrated.

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Weiss (1980) first developed an **inventory** **model** by considering continuous review system and assumed that **demand** follows a Poisson distribution. Later, Liu and Lian (1999) generalized the main results of Weiss. According to their assumption **demand** shortage is fully backordered and they generalized the **model** to a stationary renewal process instead of a Poisson **demand**. Gurler and Ozkaya (2003) made a necessary amendment of Liu and Lian results. Later, Gurler and Ozkaya (2008) developed their own **model** by considering the life span of a batch as a random variable. Berk and Gurler (2008) developed a general approach known as (Q, r) policy which is an optimal policy for many continuous review **inventory** systems of nonperishable **items**. Tekin (2001) ameliorated the problem to some extent by making necessary revisions of the (Q, R) policy by proposing a (Q, R, T) policy. According to this policy, a refill **order** of amount Q is placed every **time** the available **inventory** level falls to r, or when T amounts of **time** have passed since the last occasion the **inventory** position reaches Q, whichever happens first. Chiu and Wang (2007) developed an EPQ **model** **with** the consideration of scrap, rework and stochastic machine breakdowns. They assumed random breakdown of machine and no resumption (NR) policy in their proposed **model**. Then total production-**inventory** cost functions were derived respectively for both EPQ models **with** breakdown and without breakdown and these cost functions were integrated and renewal reward theorem was used to cope **with** the variable cycle length. The authors concluded that the optimal runtime falls within the range of bounds and is determined by using the bisection method that is based **on** the intermediate value theorem. Chiu et al. (2011) derived a mathematical **model** for solving manufacturing runtime problem **with** the consideration of constant **demand** **rate**, constant production **rate**, random defective **rate** and stochastic machine breakdown.

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variable **rate** **dependent** **on** the duration of waiting **time** up to the arrival of next lot. In both the models, the deterioration **rate** follows a three-parameter Weibull distribution and the transportation cost is considered explicitly for replenishing the **order** quantity. The corresponding models have been formulated and solved by considering the transportation cost for replenishing the **items**. Two numerical examples have been given to illustrate the results and the significant features of the result are discussed. Finally, based **on** these examples, the effects of different parameters **on** the initial stock level, shortage level (in case of **second** **model** only), cycle length along **with** the optimal profit, sensitivity analyses have been performed considering one parameter at a **time** keeping other parameters at their original values.

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The economic **order** quantity (EOQ) **model** is widely used by practitioners as a decision making tool for the control of **inventory**. In general, the objective of **inventory** management deals **with** minimization of the **inventory** carrying cost. Therefore it is important to determine the optimal stock and optimal **time** of replenishment of **inventory** to meet the future **demand**. An **inventory** **model** **with** stock at the beginning and shortages allowed, but then partially backlogged was developed by Lin et al. [15]. Urban [23] developed an **inventory** **model** that incorporated financing agreements **with** both suppliers and customers using boundary condition. Yadav et al. [26] established an **inventory** **model** of deteriorating **items** **with** two warehouse and stock **dependent** **demand**. Wu et al. [25] applied the Newton method to locate the optimal replenishment policy for EPQ **model** **with** present value. Roy and Chaudhuri [18] established an EPLS **model** **with** a variable production **rate** and **demand** depending **on** price. Huang [11] developed an EOQ **model** to compare the interior local minimum and the boundary local minimum.

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element in a set while later is concerned **with** the degree of uncertainty by which an element belongs to a set. In an **inventory** control **model**, Petrovic and Sweeney (1994) fuzzified the **demand**, lead **time** and **inventory** level into triangular fuzzy numbers. They used the fuzzy proposition method to obtain the optimal **order** quantity. Ishii and Konno (1998) introduced fuzziness in shortage cost by an L- shape fuzzy number when **demand** is stochastic. Gen et al. (1997) expressed the input data by fuzzy numbers, where they used interval mean value concept to solve an **inventory** problem. Yao and Chiang (2003) considered an **inventory** **model** **with** total **demand** and storing cost as triangular fuzzy numbers. They performed the defuzzification by centroid and signed distance methods. Mondal and Maiti (2002) applied genetic algorithms (GAs) to solve a multi-item fuzzy EOQ **model**. Maiti and Maiti (2006) dealt **with** a fuzzy **inventory** **model** **with** two warehouses under possibility constraints. Mahapatra and Maiti (2006) formulated a multi-item, multi-objective **inventory** **model** for deteriorating **items** **with** stock- and **time**-**dependent** **demand** **rate** over a finite **time** horizon in fuzzy stochastic environment. Halim et al. (2008) developed a fuzzy **inventory** **model** for perishable **items** **with** stochastic **demand**, partial backlogging and fuzzy deterioration **rate**. The **model** is further extended to consider fuzzy partial backlogging factor. Goni and Maheswari (2010) discussed the retailer’s ordering policy under two levels of delay payments considering the **demand** and the selling price as triangular fuzzy numbers. They used graded mean integration representation method for defuzzification.

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Researchers started to develop **inventory** systems allowing **time** variability in one or more than one parameters. Dave and Patel [5] discussed an **inventory** **model** for replenishment. This was followed by another **model** by Dave [4] **with** variable instantaneous **demand**, discrete opportunities for replenishment and shortages. Bahari- Kashani [2] discussed a heuristic **model** **with** **time**-proportional **demand**. An Economic **Order** Quantity (EOQ) **model** for deteriorating **items** **with** shortages and linear tend in **demand** was studied by Goswami and Chaudhuri [8]. **On** all these **inventory** systems, the deterioration **rate** is a constant.

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The Economic **Order** Quantity (EOQ) **model** developed by Harris in 1915 was popularized by Wilson in 1934 (Hariga, 1995), while Taft introduced the Economic Production Quantity (EPQ) **model** in 1918. Ghare and Shrader (1963) was the first to extend the classical EOQ formula to include exponential decay, wherein a constant fraction of **on** hand **inventory** is assumed lost due to deterioration. Misra (1975) developed the first production lot size **model** in which both constant and variable **rate** of deterioration were considered (Raafat, 1991). In subsequent models, the deterioration rates of the **items** vary from exponential distribution to gamma, normal or Weibull distributions. According to Nahmias (1982), the exponential decay can be derived by assuming that a constant fraction of **on**-hand stock is lost (i.e. deteriorates) each period regardless of the age distribution of **inventory**. Hence, those models **with** exponential deterioration **rate** are classified as having constant deterioration **rate** while those **with** Weibull, normal or other distribution are considered as having variable **rate** of deterioration (Urban 2005). There are also different Production-**inventory** models **with** constant deterioration **rate** (Maity & Maiti, 2005; Hedjar et al., 2004; Yang & Wee, 2003; Yu, 2007; Alfares et al., 2005; Jaggi et al., 2011). A generalized **model** in which deterioration and production varies continuously **with** **time** was developed by Balkhi (2001) while Lo et al. (2007) developed an integrated production **inventory** **model** **with** varying **rate** of deterioration, imperfect production processes and inflation. Other models **with** varying deterioration **rate** include Pal et al., (2008) and Sridevi et al. (2010).

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mal replenishment **time** for a linearly increasing **demand** pattern. Barbosa and Friedman[6] and Henery[7] respec- tively extended the **demand** pattern to a power **demand** form and a log-concave function. Hariga and Goyal[8] and Teng[9] extended Donaldson’s work by considering various types of shortages. For deteriorating **items** such as medicine, volatile liquids and blood banks, Dye[10] developed the **inventory** **model** under the circumstances where shortages are allowed and backlogging **rate** linearly depends **on** the total number of customers in the waiting line during the shortage period. However, there still re- main many problems associated **with** replenishment poli- cies for retailers that should theoretically be solved to provide them **with** eﬀective indices. We focus **on** a case where special display goods[11, 12, 13] are dealt in. The special display goods are heaped up in the end displays or special areas at retail store. Retailers deal in such special display goods **with** a view to introducing and/or exposing new products or for the purpose of sales promotions in many cases. They are sold at a fast velocity when their quantity displayed is large, but are sold at a low veloc- ity when their quantity becomes small. Baker[14] and Baker and Urban[15] dealt **with** a similar problem, but they expressed the **demand** **rate** simply as a function of a polynomial form without any practical meaning. Our previous work has developed an **inventory** **model** for the special display goods **with** a seasonal **demand** **rate** over a ﬁnite **time** horizon to determine the optimal re- plenishment policy, which maximizes the retailer’s total proﬁt[16]. However, the salvage value which is the dis- posal value of unsold **inventory** at the end of the season was assumed to be equal to the purchase cost. In this study, we relax this restriction **on** the salvage value in **order** to derive a more general solution. Numerical ex- amples are presented to illustrate the theoretical under- pinnings of the proposed **model**.

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fuzziness of ordering cost and holding cost. Roy and Maiti (1997) presented a fuzzy EOQ **model** **with** **demand**-**dependent** unit cost under limited storage capacity considering different parameters as fuzzy sets **with** suitable membership function. Kao and Hsu (2002), Dutta, Chakraborty, and Roy (2005) studied single period **inventory** **model** **with** fuzzy **demand** and fuzzy random variable **demand**, respectively, and developed models for optimum **order** quantity in terms of cost. Syed and Aziz (2007) modeled **inventory** **model** without shortage under fuzzy environment. Ordering and holding costs were considered as fuzzy triangular numbers, and optimum **order** quantity was developed using signed distance method. Wang et al. (2007) developed the **model** of fuzzy economic **order** quantity without backordering. Holding cost and set-up cost were considered as fuzzy in nature and the **model** was developed for keeping the credibility of total cost in the planning period below certain budget level. Vijayan and Kumaran (2008) investigated continuous review and periodic review **inventory** models under fuzzy environment, where the membership function distribution took a trapezoidal form. Gani and Maheswari (2010) discussed the retailer’s ordering policy under two levels of delay payments considering the **demand** and the selling price as triangular fuzzy numbers. They used graded mean integration representation method for defuzzification. Singh et al. (2011) and Malik and Singh (2011) utilized soft computing techniques for modeling of **inventory** under price **dependent** **demand** and variable **demand**, respectively. In the same year, Mahata and Mahata (2011) applied fuzzy EOQ **model** to supply chains and Rong (2011) developed EOQ **model** by treating the holding cost, shortage cost and ordering cost per unit as uncertain variables. Based **on** above mentioned situations, this paper considers the retailer’s optimal policy for non- instantaneous deteriorating **items** **with** permissible delay in payments under different scenarios in fuzzy environment. The paper discusses all the possible cases which may arise and yet not considered in the previous **inventory** models under permissible delay in payments. Further, this paper also considers the price **dependent** **demand** and the possibility of higher earning interest **rate** than interest payable. The components of **demand** function are assumed as triangular fuzzy number. The arithmetic operations are defined under the function principle and for defuzzification, signed distance method is employed to evaluate the optimal cycle length T, markup **rate** and payoff **time** which maximize the total profit in all possible cases. Finally, numerical examples are presented to show the validity of the **model** followed by the sensitivity analysis. Results have shown significant effect in real life.

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In this paper, we develop a multi-period lot-sizing **model** for deteriorating **items** **with** varying **demand** patterns when the **time** horizon is unknown or unspecified. There are three main reasons for our assumptions. (i) The first reason is to present a multi-period **inventory** **model** for deteriorating **items** using a general ramp-type **demand** pattern **with** full backlogging of shortages. The general ramp-type **demand** function allows three-phase variation in **demand**, representing the growth, the steady and the decline phases of **demand** commonly experienced by many products. This will be more suitable for practical applications than single period models that assume a single replenishment to cover all phases of **demand**. (ii) The **second** reason is to make the developed **model** suitable for unknown **time** horizon by extending the Silver-Meal approach to a general ramp-type **demand** pattern. This makes the **model** to be suitable for situations, discussed earlier, when the **time** horizon is neither fixed nor infinite. (iii) Finally, the third reason is to examine various possible replenishment patterns when shortages and **demand** pattern variation occur in a multi-period **inventory** **model**. The replenishment intervals are allowed to vary from one period to another along the cycle and a replenishment policy to generate optimal replenishment schedules, **order** quantity and costs is proposed. An additional solution procedure based **on** trust region methods is also presented to complement the usual direct implementation of derivatives. This paper is organized as follows: Section 2 contains a brief literature review and the proposed **model** of this paper is presented in section 3. Solution procedure to obtain the optimal replenishment policy, numerical illustrations and conclusions are also presented in sections 4 to 6.

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In real-life situations, exact data are often inadequate for a mathematical **model**. **Inventory** is a physical stock that a business keeps **on** hand in **order** to promote the smooth and efficient running of its affairs. But in practice, the effects of deterioration, shortages, holding cost, ordering cost etc. are important for **inventory**. Various types of uncertainties are involved in any **inventory** system. Historically, probability theory has been the primary test for representing uncertainty in mathematical models. Because of this, all the uncertainty was assumed to follow the characteristics of random uncertainty. A random process was one where the outcome of any particular realization of the process is strictly a matter of chance, and prediction of a sequence of events is not possible. Fuzzy set theory is an excellent tool for modeling the

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Determining the optimal **inventory** control and selling price for deteriorating **items** is of great significance. In this paper, a joint pricing and **inventory** control **model** for deteriorating **items** **with** price- and **time**-**dependent** **demand** **rate** and **time**-**dependent** deteriorating **rate** **with** partial backlogging is considered. The objective is to determine the optimal price, the replenishment **time**, and economic **order** quantity such that the total profit per unit **time** is maximized. After modeling the problem, an algorithm is proposed to solve the resulted problem. We also prove that the problem statement is concave function and the optimal solution is indeed global.

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The concept was also used **on** organizational communication (Castor, 2017) and in interpersonal communications studies to codify and analyze interaction between individuals at the relational level (Rogers & Escudero, 2004). In political studies, (Simons, 1994) metacommunication is a rhetorical tactic used in political debates e.g. politicians striving to cope **with** provocative questions by challenging the journalist’s legitimacy for asking them. In the international relations and political science domains (Rich & Craig, 2012), metacommunication is used as a key concept in Digital Media studies. Jensen (2016: 7), for instance, emphasizes, “the importance of meta-communication for contemporary communication theory, examines mediated presence as an instance of metacommunication, and addresses the implications of digitally mediated presence for current issues of surveillance”.

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In a wide variety of practical applications, the characteristics of the plant dynamics are changing through **time** due to aging of components or the variation of the reference. When the dynamics of the system is changing at a slow pace, the least squares algorithm (**with** and without forgetting factor) is quite effective when it is used to follow this type of behavior. Nevertheless, the efficiency of the algorithm diminishes when the dynamics is changing rapidly. In the past years, systems **with** fast changing dynamics are being used more often in modern engineering applications; in particular, linear periodic **time** varying systems (LPTVS) have appeared in different areas such as Control, Power Electronics, Aeronautics, Communications and Signal Processing [1]. For example, in aeronautics, the equations that describe the flight dynamics in aircrafts have coefficients that are functions of **time**. Recently, the high speeds and accelerations that modern aircrafts can handle, have contributed to show that the parameters of the system that depend **on** speed will change in the same fashion. For instance, the dynamic of a main rotor in a helicopter, which can be modeled by a linear periodic system, is important from the control perspective due to the fact that a precise control of certain variables in the system can help to attenuate vibrations in the rotor components, that eventually may cause high levels of stress and fatigue [2]. Thus, there

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The PGIS-II factors had stronger relations **with** the presence of meaning in life than **with** the other variables. The presence of meaning in life refers to the extent to which people comprehend and see significance in their lives, as well as the degree to which they perceive themselves to have a purpose or overarching goal in life (Steger, Shin, Shim, & Fitch-Martin, 2013). Theoretically, it is a future- oriented, cognitive, and motivation-based construct. PGI, in turn, refers to active and intentional engagement in the process of self-improvement (Robitschek et al., 2012). People will use their PGI skills in life domains of greatest importance to them (Freitas, Damásio, Tobo, Kamei, & Koller, 2016). Growth and development that are personally important should lead to an increased sense of purpose in life. Although not tested in this study, it is possible that better developed PGI skills promote a greater sense of meaning in life.

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I published in 1951 a study of the geographic vari- ation of Amphisbaena fuliginosa Linnaeus, 1758, concluding that five subspecies could be recognized, based mainly **on** color pattern, but **with** substantial support from scale counts. The paper was writ- ten in the flush and heat of the Mayrian paradigm (Mayr 1942): if two taxa are similar and allopatric, they should be considered as geographical races or subspecies. I have since come to question the un- qualified application of the paradigm. The concept of subspecies involves more than likeness and al- lopatry; it entails too the presence of broad areas of morphological stability connected by relatively nar- row belts of integradation (Vanzolini and Williams 1970). Additionally, I came to worry about the sta- tistical tools: the methods then available in text- books within my reach were not ideally suited to the

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SAGE II version 6.3 to more confidently avoid the introduc- tion of systematic error. We have retained the cutoff for two reasons. Firstly, although the error function is defined as the deviation between the calculated and true transmission, in practice the “true” transmission must be approximated. Specifically, we use a “best guess” profile, constructed by smoothing the data using a boxcar average and a running median, the same algorithm that is used to generate the fi- nal profile from the transmission scatter data. It is possible that errors in the best guess transmission profile could lead to errors in the **time** **dependent** correction, and these errors are more likely where the atmosphere has a high degree of natural variability. There is also some risk of falsely “cor- recting” or attempting to correct real atmospheric variability by the method described here. It is possible that future de- velopments will allow the algorithm to distinguish between real atmospheric variability and **time** **dependent** calibration errors.

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Abstract: This study considered reorder point **on** the continuous review **inventory** **model** under controllable lead **time** **with** mixture of backorder price discounts and partial lost sales. We developed a continuous review **inventory** **model** where the lead **time**, the **order** quantity, backorder discount and safety factor were considered as the decision variables of a mixture of backorders and lost sales **inventory** **model**. The objective was to minimize the expected total annual cost **with** respect to related decision variables. The purpose **model** **with** lead **time** **demand** distribution was unknown. The author applies a minimax distribution free procedure to find the optimal solution and numerical example was included to illustrate the solution procedure of the proposed algorithms.

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