This paper is developed a three layer supply chain **production**-inventory model for supplier, producer and retailer. The model speculates **exponential** **demand** and **production** **rate** for all member of three-layer supply chain. The model deliberates the effect of commercial policies such as **exponential** **demand** **rate**; **production** **rate** is **demand** dependent, perfect order size of raw material, unit manufacturing price and idle times in assorted sectors on integrating marketing framework. The basic assumption of the model is that manufacturing items are of perfect and imperfect quality. This technique used in different types of **production** industries like electrical and electronics, paper, pharmaceutical, automobiles etc. Mathematical modeling is used to derive the **production** **rate** and order size of raw material for maximum total profit of supply chain. A numerical example including the sensitivity analysis is validated to the outcomes of the **production**- inventory model.

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In this paper, we proposed two level **production** inventory models with **exponential** **demand** and time dependent deterio- ration **rate**. The objective is to determine optimal **production** policy and minimize the total inventory cost. The variation in **production** **rate** provides a way resulting consumer satisfaction and earning potential profit. The present model differs from the existing models, as two-level **production** and exponentially increasing **demand** are considered here. A numerical example is provided to demonstrate means practical usage. This model can be extended by taking more realistic assumptions such as shortage, other types of **demand** patterns etc.

developed a two warehouse inventory model for deteriorating items having linear **demand** with inflation and permissible delay in payments. Also sensitivity with respect to the parameters have been carried out.[12]Sharmila and Uthayakumar examined the partial trade credit financing in a supply chain by EOQ-based model for decomposing items together with shortages. [5]Lakshmidevi discussed a continuous **production** inventory model for time dependent deteriorating items with shortages in which three different rates of **production** and quadratic **demand** **rate**. [3]Jhuma and Samanta assumed that the **demand** and **production** rates are constant and the distribution of the time to deterioration of an item follows the **exponential** distribution. Also results are illustrated by numerical examples.[4]Krishna and Bani developed a mathematical model of an inventory system in which **demand** depending upon stock level and time with various degrees , gave more flexibility of the **demand** pattern and more general to the study dine so far with the condition to minimize the total average cost of the system.

Joglekar (2003) used a linear **demand** function with price sensitiveness and allowed retailers to use a continuous increasing price strategy in an inventory cycle. He derived the retailer’s optimal profit by ignoring all the inventory costs. His findings are not restricted to growing market only, which is neither for stable market nor for a declining market. By dividing the **demand** **rate** into multiple segments, Shukla and Khedlekar (2010) introduced three-component **demand** **rate** for the newly launched deteriorating item. Qi, Bard and Yu (2004) analyzed the supply chain-coordination with **demand** disruption in a deterministic scenario. Expenditure sources like ordering cost, safety features, lead time and numbers of lots are the integral parts of decision making. An integrated inventory model focusing on these issues and aspects has been discussed by Lo (2007).

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inventory model for deteriorating item. Sharma et al. [8] solved a deterministic **production** inventory model for deteriorating products with exponentially declining **demand** and shortages. Mishra and Shah [9] presented inventory management of time dependent deteriorating items with salvage value. Baten and Kamil [10] studied the inventory management systems for hazardous items of two-parameter **exponential** distribution with constant **production** and **demand** rates. Li and Mawhinney [11] presented a review on deteriorating inventory study. Baten and Kamil [12] studied optimal fuzzy control with application to discounted cost **production** inventory planning problem. Mishra and Singh [13] gave computational approach to an inventory model with ramp-type **demand**. Sharma and Muhammad [14] developed an EOQ Model for Hazardous Items of Two- Parameter **Exponential** Distribution with Uniform **Rate** of **Demand** & Finite **Rate** of Replenishment. Sharma and Muhammad

JUNE-JULY, 2015, VOL. 2/10 www.srjis.com Page 2415 Khanra,S. Ghosh, S.K. and Chaudhuri, K.S. (2011). An EOQ model for a deteriorating item with time- dependent quadratic **demand** under permissible delay in payment. Applied Mathematics and Computation, 218, pp.1-9.

lowed. Abad [20] discussed regarding optimal pricing and lot sizing under conditions of perish ability and par- tial backordering. Sing and Pattanak [21]-[23] developed the model for deterioration and time dependent qua- dratic **demand** under permissible delay in payment, whereas we have used the **demand** of linear trend but ignor- ing the payment aspect. Amutha and Chandrasekaran [24] formulated the inventory model with deterioration items, quadratic **demand** and time dependent holding cost, but in our proposed model, we have emphasized on the **production** **rate**, linear type of **demand** and constant holding cost. Ouyang and Cheng [25] explained the in- ventory model for deteriorating items with **exponential** declining **demand** and partial backlogging. Dave and Pa- tel [26] introduced an inventory model for deteriorating Items with time proportional **demand**, but we have con- sidered the **demand** which is level dependent and a type of linear trend. Teng et al. [27] developed the model with deteriorating items and shortages assuming that the **demand** function was positive and fluctuating with re- spect to time, but in the proposed model, the **demand** was considered as a linear function and **production** starts with a buffer stock as a reserve. The previous model established various types of inventory models considering several parameters which did not consider the **production** **rate**, linear **demand** along with the buffer stock. Here comes the necessity to build the proposed model.

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deteriorating items where time to deterioration has **Exponential** distribution and with time-dependent quadratic **demand**. In this model, shortages are not allowed and holding cost is time-dependent. Also, Gothi and Kirtan Parmar (2015) have extended above deterministic inventory model by taking two parameter Weibull distributions to represent the distribution of time to deterioration and shortages are allowed and partially backlogged. Kirtan Parmar and Gothi (2015) developed an economic **production** model for deteriorating items using three parameter Weibull distributions with constant **production** **rate** and time varying holding cost. The consideration of PT is important due to rapid social changes, and the fact that PT can reduce the deterioration **rate** significantly. By the efforts of investing in preservation technology, we can reduce the deterioration **rate**. So in this paper, we made the model of Mishra and Singh (2011) more realistic by considering the fact that use of preservation technology can reduce the deterioration **rate** significantly, which help the retailers to reduce their economic losses. We have analyzed an inventory system for deteriorating items under quadratic **demand** using preservation technology and time dependent IHC. The assumptions and notations of the model are introduced in the next section. The mathematical model and Analysis is derived and numerical illustration is presented. The article ends with some concluding remarks and scope of a future research.

DOI: 10.4236/ajor.2018.81001 2 American Journal of Operations Research Consequently, the **production** and inventory problem of deteriorating items has been extensively studied by researchers. Some of the researchers include Ghare and Schrader [2] who are the first researchers to derive an economic or- der quantity model by assuming **exponential** decay for the item. Later, Covert and Philip [3] extended Ghare and Schrader’s [2] model by considering the de- terioration **rate** to be a two-parameter weibull distribution. Later, Shah and Jaiswal [4] presented an order-level inventory model for deteriorating items with a constant **rate** of deterioration. Aggarwal [5] corrected the analysis in Shah and Jaiswal’s model [4]. Dave and Patel [6] considered an inventory model for dete- riorating items with time-proportional **demand** when shortages were not allowed. Authors such as Hollier and Mark [7], Hariga and Benkherouf [8], Wee [9] [10] all developed their models by considering **demand** to be an **exponential** **demand**.

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Jamal et al. (2004) developed a manufacturing system with rework process consisting two cases of rework process to minimize the total **production** cost. At the first instant, they executed reworking in the same cycle. At the second instant, they executed reworking after N cycles. Chiu et al. (2007) considered a manufacturing system with rework, including optimal lot sizing decision, random scrap **rate** and service level constraint. They derived that the expected total cost of a manufacturing system is less if backlogging is allowed or equal if backlogging is not allowed. Hafshejani et al. (2012) presented a multi-product economic **production** quantity model with imperfect quality items and reworking, it is a nonlinear programming problem solved by genetic algorithm with limited warehouse space.

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Abstract—Product reliability and uniformity play significant roles in manufacture. This article described an integrated preventive maintenance inventory model with restoration activities. When defective parts are produced, perfect repair, inspection, and rework are conducted after the **production** run period. Two types of preventive maintenance (PM) are performed after the “in-control” **production** run period. Additionally, we considered how number of shipments from producer to purchaser affected the whole inventory cost. The minimal total cost could be determined by an optimal inventory cycle. Finally, a numerical example and sensitivity analyses were illustrated in the end of this paper.

In the literature of inventory after the development of classical economic order quantity (EOQ) model researchers extensively studied several aspects of inventory modeling by assuming constant **demand** **rate**. But in a real market **demand** of a product is always dynamic state due to the variability of time, price or even of the instantaneous level of inventory displayed in retail shop. This impressed researchers and marketing practitioners to think about the variability of **demand** **rate**. The ramp type **demand** is very commonly seen in real life situations when some fresh come to the market. In case of ramp type **demand** **rate**, the **demand** increases linearly at the beginning and then the market grows into a stable stage such that the **demand** becomes a constant until the end of the inventory cycle.

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In a practical way, simulations considered consist of a shock related to the reduction of the customs duties, 25% or 100%, on all the imported goods. In order to neutralize the effect of the degradation of the public saving consecutive to the losses of customs receipts on the labor market, in particular female, we supposed that the shortfall due to the dismantling of the tariff barriers is compensated by taking away corresponding to the title of the indirect taxation, primarily the value- added tax. An alternative consist to be supposed, following Dessus and Suwa- Eisenmann (1998), that these budgetary losses are counterbalanced by an increase in the tax household income. Nevertheless, in the case of Morocco, the income taxes seem capped; in any case, it is what arises from the observations of the last mission of the technical assistance of the IMF. For these reasons, the public saving is maintained constant due to the levies on the receipts of the indirect taxation, in particular the VAT When the customs duties it lower by 25%, the sectors where the female employment **demand** increases more, consecutively with liberalization, are those of agribusiness industries, the textile and leather. Like we have announced, it concern there the principal exporting sectors of the Moroccan economy. Such an assertion is perfectly in agreement with the theory of Hecksher, Ohlin and Samuelson, since the fall of the **production** costs allows in fine to increase exports significantly and to thus limit the losses pulled by the increased competition of the products imported on the domestic market.

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Inventory may be considered as an accumulation of a product that would be used to satisfy future demands for that product. An carrying cost and shortage cost. An important problem confronting a supply manager in any modern organization is the control and maintenance of inventories of deteriorating items. ys, glassware, hardware, etc. There is little requirement for considering deterioration in the determination of economic lot size. So in this paper, an inventory model is developed for technology the retailer can reduce the deterioration **rate** by which he can reduce the economic losses, improve the customer service level and increase business competitiveness. In reality, the t. Time also plays and important role in the inventory system. Recently, Mishra and Singh (2016) developed a deteriorating inventory model with partial backlogging when **demand** and deterioration **rate** is constant. Vinod kumar Mishra developed an inventory model of instantaneous deteriorating items with controllable deterioration **rate** for time dependent developed deteriorating inventory model with controllable deterioration 2015) developed EOQ model with formulated a deteriorating inventory model dependent **demand** by allowing preservation technology cost as a decision variable in conjunction with replacement by assuming that the preservation technology cost developed an order level inventory quadratic **demand** and partial backlogging. Sarala Pareek and Garima Sharma (2014) developed an inventory model with Weibull distribution deteriorating item with **exponential** declining **demand** and partial

In most of the available literature researchers developed their inventory model without considering the deterioration, while in practical situations deterioration occurs for almost every product during the storage period, for example vegetables, fruits, chemical, medicines, and electronic equipments. The earliest model of inventory, incorporating the deteriorating items is in the inventory was made by the Ghare and Schrader in 1963. They developed an economic lot size model for the decaying items with constant **demand** **rate**. For more insights concerning decaying things the survey paper of Goyal and Giri (2001) is the best source of information. For a very long period it is expected by the analysts that the retailer disburses the costs of the material to the supplier instantly. But in a real environment it is not always possible to pay the amount immediately. Usually supplier offers his client a grace period for the payment and allows him to settle the account without charging the interest. The phenomenon is known as the trade credit policy. The strategy appeals to the consumer to by the product. The earliest approach it the field of delay in payment, was made by the Goyal, (1985). Thereafter, several researchers have developed their model to extend the Goyal’s (1985) model. Another model was developed by Aggarwal, and Jaggi (1995), he has extended the model of Goyal, (1985) to find the optimal replenishment quantity for the decaying items under the effect of trade credit. The model was developed without shortages and the **demand** and deterioration **rate** were considered as constant. Furthermore, Singh and Singh (2009, 2010), Yang and Wee (2006), Benkherouf, (1995) and Singh and Kumar (2010) have also considered the permissible delay in their paper. Kumar and Singh (2012) have explored a stock model with volume adaptability under the effect of inflation. Dem, et al., (2014) researched an EPQ stock control framework with seasonal **demand** under volume adaptability. Kumar, et al., (2015) have built up an EPQ model with admissible delay in imbursement with preservation technology. Saxena, et al., (2017) contemplated a green supply chain framework with reasonable delay in payment. Yadav, et al., (2019) presented

Inventory holding refers to producing ahead of **demand** and sales realizations [1]. The total investment in inventories is enormous and accounts for nearly half of the total logistics cost [2]. In view of this high cost, the management of in- ventory offers high potential for improvement and results in a relatively rich li- terature on theoretic inventory models. In inventory planning and control, the performance measures adopted should encourage the positive aspects of holding inventory such as providing flexibility, providing resources for **production**, pro- viding responsive customer service. We observe that inventory arises in many How to cite this paper: Ophokenshi, N.P.,

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The overexpression of the RIB genes ensures that the cell is devoid of bottlenecks in the riboflavin biosynthetic pathway. This provides a cell model system to identify new mutations or target genes improving riboflavin pro- duction that otherwise would be masked by a limiting riboflavin biosynthetic pathway. The overexpression of the riboflavin pathway is also a strategy of great value for further improvement of current industrial strains because it is highly unlikely that those strains may have accumu- lated all the necessary mutations required for overexpres- sion of all six RIB genes. In this regard, we found that the underexpression of the ADE12 gene has an additive effect over the overexpression of five RIB genes, provid- ing the highest yield of riboflavin **production** obtained by metabolic engineering in A. gossypii so far. According to our results, metabolic engineering of the IMP hub and improvement of the riboflavin biosynthetic pathway may

vanizing furnace are limited by one of two factors. The first and more easily understood is the maximum heat output of the burner system. The second, more critical, factor is that the ultimate heat **demand** may be limited by the maximum acceptable heat flux through the kettle wall. As described above, the alloying reaction between the molten zinc and the kettle wall causes the kettle wall to be eroded over time. This **rate** of wear is exponentially dependent on the heat flux through the wall [15, 16] and becomes unaccep- tably high above q q _ crit ¼ 29 kW/m 2 [17].

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This study develops an integrated **production** inventory model from the perspectives of vendor, supplier and buyer. The **demand** **rate** is time dependent for the vendor and supplier and buyer assumes the stock dependent **demand** **rate**. As per the **demand**, supplier uses two warehouses (rented and owned) for the storage of excess quantities. Shortages are allowed at the buyer’s part only and the unfulfilled **demand** is partially backlogged. The effect of imperfect **production** processes on lot sizing is also considered. This complete model is studied under the effect of inflation. The objective is to minimize the total cost for the system. A solution procedure is developed to find a near optimal solution for the model. A numerical example along with sensitivity analysis is given to illustrate the model.

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A continuous **production** inventory model for time dependent deteriorating items with shortages in which three different rates of **production** and quadratic **demand** **rate** is considered. The case of change of **production** is very useful in practical situations. By starting at a low **rate** of **production**, a large quantum stock of manufactured item, at the initial stage is avoided, leading to reduction in the holding cost. The variation in **production** **rate** provides a way resulting consumer satisfaction and earning potential profit.The total cost of the system and the optimal values for ݐ ଵ , ݐ ଶ , ݐ ସ , ܶ is derived for quadratic **demand** **rate** and time dependent deterioration **rate**.