Replenishment Cycle Length Research Articles

Research on an Optimal Maintenance and Inventory Model Based on Carbon Tax Policy

The equipment in a factory will gradually deteriorate during production, leading to the production of defective products. Without appropriate maintenance, the defect rate will increase over time. Consequently, the production cost will rise, the inventory quality will be affected, the profit will decrease, and the risk of carbon emissions will increase, leading to more customer complaints and damaging the corporate image. In addition to focusing on preventive maintenance to ensure the quality of products, companies should also take carbon emissions into consideration. Furthermore, the frequency of maintenance must be carefully considered, as both carbon emissions and maintenance costs will increase if the frequency is too high; conversely, if the maintenance frequency is too low or non-existent, the defect rate may increase cumulatively, or production may be suspended due to equipment failure. Therefore, this research explores preventive maintenance and inventory management issues within an imperfect production system and develops an extended economic production quantity model that incorporates defective products as well as taking carbon tax and preventive maintenance into consideration. The main purpose is to determine the optimal maintenance frequency, production, and replenishment cycle length, so as to maximize the total profit under the carbon tax policy. This study demonstrates a computing process with relatively impractical product data based on the actual business situation of a disposable diaper manufacturer. Furthermore, a sensitivity analysis is implemented to the model parameters in the proposed model. The managemental insights are illustrated based on the results of theoretical analysis to provide a reference to policy makers during decision making, hence, to secure the sustainability and green transitions of corporates. The results of this study not only help to reduce environmental impact but can also improve the competitiveness and sustainable development of enterprises.

Shelf-life extension under implementation costs

The strategy of shelf-life extension has the potential to yield environmental benefits and monetary savings associated with decreased waste. In particular, shelf-life extension may reduce shortages of crucial commodities, such as drugs and food, in times of crisis. We develop a demand model for a perishable product that considers not only the multiplicative effect of price and time until expiration, but also a parameter representing the degree of shelf-life extension. Since there are costs associated with extending the shelf-life of a perishable product (i.e., investing in technologies), which might be incurred by the retailer, our model, aside from deriving the optimal pricing and ordering policy, determines whether and for how long the shelf-life of a product should be extended assuming that the technology is available. Thus the three decision variables in our model are price, extension duration ratio (the extension period divided by the original shelf-life), and replenishment cycle length. We demonstrate the applicability of the proposed model and its significance. In particular, we show that increasing the shelf-life of a product has the potential to create additional demand, although the demand only increases up to a particular value of the extension ratio, after which it decreases. We argue that the application of the model to decision-making regarding shelf-life extension creates a win–win situation for the retailer and society; the retailer increases her profit as a result of shelf-life extension, while society achieves social welfare gains by consuming a product with longer expiration than would otherwise be possible.

A Study of a Two Storage Single Product Inventory System with Ramp Type Demand, N-Phase Prepayment and Purchase for Exigency

This model considers a two-warehouse inventory system of deteriorated items with ramp-type demand and a constant rate of deterioration. It is maintained a rental warehouse (RW) of infinite capacity to load the excess items of replenished goods after filling the items of finite capacity in the own warehouse (OW). Retailers are encouraged to opt for the prepayment option of paying their purchase cost in equal installments prior to the delivery of the ordered items with a considerable discount, which will ensure the purchase guarantee of their orders. The slotted backlog interval of the stock out period is handled in two different ways to retain the customers and ease their impatience. Customers in the first slot of the stock out period are satisfied by the emergency purchases from local suppliers with high purchasing costs to avoid losing customers. Customers in the next slot are satisfied immediately after the next replenishment point. Essential measures of the system are derived: optimal ordering quantities from both regular and local suppliers; replenishment cycle length; and a partitioned backlog interval. A numerical example is given along with the optimal solutions for a particular environment with sensitive analysis in order to validate the model’s efficacy.

Multi-period dynamic pricing model for deteriorating products in a supply chain with preservation technology investment and carbon emission

In this study, a multi-period dynamic pricing model in a two-echelon supply chain consisting of a retailer and a manufacturer is developed for a deteriorating product with a fixed shelf life. The demand at the retailer in each period is considered to be a function of the retail price, inventory on stock, reference price, and product freshness. Preservation technology investment (PTI) is considered to retard the rate of deterioration with carbon emission to keep the process green. Centralised and decentralised models are developed considering dynamic pricing for the retailer and static pricing for the manufacturer with a variable replenishment cycle length. Algorithms have been developed to obtain the optimal replenishment cycle length, retailer’s end of inventory level, retail price, PTI and discounted total profit for the retailer/manufacturer/supply chain. We study the generated trade-off between PTI, benefits of reduced deterioration rate and cost of carbon emission. The numerical studies suggest that supply chain profit can be greatly improved with efficient PTI and dynamic pricing policy. We also find that the initial reference price plays a key role in improving the profit.

A Profit Maximization Inventory Model: Stock-Linked Demand Considering Salvage Value with Tolerable Deferred Payments

Every business owner wishes that all sales were made on a cash basis, but in a cutthroat market, it is not always feasible. To entice buyers to purchase their goods, dealers may need to offer sales with credit terms. Unfortunately, selling with deferred payment conditions introduces a completely new facet of business management. Additionally, the salvage amount or value is significant for determining depreciation and can have an impact on the company’s overall depreciable amount used in its depreciation schedule. This study looks at an inventory model with the best possible pricing and ordering policy for retailers. Research is derived from when inventory is subjected to a constant deterioration rate and associated with appropriate salvage value. A perishable payments strategy inventory model is created, with the demand as a function of the stock level as well as selling price. Manufacturers provide to the retailer a tolerable deferred payment scheme to repay against the purchase products. That the cycle length and order size will rise under tolerable deferred period is refuted. The study’s goal is to determine the optimal replenishment cycle length and selling price to optimize retailer’s net income. With reference to cycle length and selling price, we developed an algorithm with a numerical example to optimize the net profit. The results are mathematically proven, and data is provided to validate the aforementioned model. Numerical examples are used to validate the model, and sensitivity analysis was performed. Using mathematical tools, a 3D graph will be used to demonstrate the concavity of the objective function.

An EPQ Model for Delayed Deteriorating Items with Two-Phase Production Period, Exponential Demand Rate and Linear Holding Cost.

The economic production quantity (EPQ) model for delayed deteriorating items considering two-phase production periods, exponential demand rate and linearly increasing function of time holding cost is proposed to solve a production problem similar to the one caused by the Covid-19 pandemic. Without shortages, the necessary and sufficient conditions for optimality of this model are characterized through a theorem and lemmas while a solution methodology based on differential calculus is adopted. This paper determines the best replenishment cycle length corresponding to the optimal total variable cost and production quantity of imperfect production industry. To illustrate this model, a numerical experiment is conducted. The results demonstrate that a higher carrying charge decreases the production quantity and a longer demanding period decreases the total variable cost of an industry with a distracted production period. Finally, managerial insights are discussed using sensitivity analysis and future research directions are exposed.

An EOQ model for deteriorating items analyzing retailer’s optimal strategy under trade credit and return policy with nonlinear demand and resalable returns

This paper presents an EOQ model where demand is dependent upon time and selling price. In the proposed model of inventory, the retailer allows its unsatisfied customers to return their product whereas the manufacturer offers a full trade credit policy to the retailer. To make our model realistic, we have assumed that the product returned can be resold with the same selling price. Number of returns is a function of demand. In this proposed inventory model considering deterioration, the retailer does not fully reimburse its customers for the returned product. The primary purpose of this inventory model is to determine the optimal selling price, optimal order quantity, and optimal replenishment cycle length in order to maximize the retailer’s total profit earned per unit time. A numerical example is also presented and a sensitivity analysis is carried to highlight the findings of the suggested inventory model.

PARTIAL BACKORDERING INVENTORY MODEL WITH LIMITED STORAGE CAPACITY UNDER ORDER-SIZE DEPENDENT TRADE CREDIT

This study formulates an inventory model with limited storage capacity under the condition of order-size dependent trade credit. Shortages are allowed and partially backlogged. The objective of this study is to determine the optimal replenishment cycle length, the optimal fraction of no shortage, and whether retailers should choose to rent an extra warehouse to store more items, such that retailers’ total annual profit is maximized. We prove the global optimally of objective functions and derive the closed-form optimal solution. Some numerical examples are presented to illustrate the applicability of the proposed model. Sensitivity analysis is carried out and managerial insights are obtained. We find that if retailers’ own warehouse capacity is relatively small, they always benefit from enlarging order quantity and renting an extra warehouse; meanwhile, suppliers further prolong the credit period is beneficial for both parties. On the contrary, as retailers’ own warehouse capacity increases and exceeds the optimal order quantity under that of without capacity constraints, adopting the same replenishment strategy as that without capacity constraints is profitable for retailers. Our results also reveal that other model parameters (e.g., ordering cost, inventory holding cost, shortages cost, backordering rate, etc.) have a significant impact on retailers’ optimal decisions.

Optimal lot-size decision for deteriorating items with price-sensitive demand, linearly time-dependent holding cost under all-units discount environment

Due to the competitive market dynamics and globalisation, it gradually becomes challenging for the manufacturers or suppliers to entice any potential customer and turn into actual one. In this connection, to attract the retailers’ consideration, manufacturers or suppliers offer an order size dependent discount policy where the per unit purchase cost is measured as a descending step function of the amount of order quantity. A profit maximising inventory model is constructed for a deteriorating product where demand of the product is dependent on selling price, per unit carrying cost as linearly time varying and also proportional to the per unit purchase price under all-units discount environment. After examining the optimality of the decision variables a solution algorithm is suggested to achieve the optimal values of the selling price per unit and replenishment cycle length along with the optimal order size for optimising the net profit. Two computational experiments are executed along with observing the applicability of the proposed algorithm. Finally, a sensitivity analysis is performed for the model by changing the values of all parameters and hence a conclusion is made regarding the current study.

Controlling Waste and Carbon Emission for a Sustainable Closed-Loop Supply Chain Management under a Cap-and-Trade Strategy

Considering the increasing number of end-of-life goods in the context of improving the ambience and health of a population and their destructive impacts, recycling strategies are important for industries and organizations. In this article, a closed-loop supply chain management containing a single manufacturer, a single retailer, and a third party is introduced in which the manufacturer first propagates newly finished goods and then dispatches some of the finished goods to the retailer considering a single-setup multi-delivery policy. Due to shipping, carbon emission is taken into account as well as a carbon emission trading mechanism to curb the amount of carbon emissions by the retailer. For recycling through collection, inspection, remanufacturing, and landfill, the third party collects the end-of-life goods from its customers and ships perfect products to the manufacturer after a two-stage inspection. In this model, major sources of emissions such as shipping, replenishment orders, and inventory have been taken care of. The minimizing of the total cost relating to the container capacity, shipment numbers, and replenishment cycle length is the main objective of the closed-loop supply chain management for making the system more profitable. Expository numerical explorations, analysis, and graphic representations are conferred to elucidate this model, and it is observed that this model saves some percentage of the cost compared to the existing literature.

Optimal dynamic pricing, preservation technology investment and periodic ordering policies for agricultural products

This paper focuses on the inventory management of agricultural products, a specific type of perishable items carrying the deterioration property. In practice, the deterioration rate of agricultural products is varying with time and can be slowed downviainvesting in the preservation technology. This objective of this paper is to maximize the firm’s total profit per unit time by simultaneously determining dynamic pricing, replenishment cycle length, replenishment quantity and preservation technology investment. We first derive pricing policy by solving a dynamic optimization problem and then propose a solution procedure to obtain the optimal strategies that maximize profit. Furthermore, numerical examples and sensitivity analysis are conducted to gain more managerial insights. We find that the firm should take a penetration pricing policy. In addition, if the shelf life of products is very long, the firm should not take preservation technology investment. When the unit holding cost is relatively small or the unit purchasing cost is relatively large, the firm should increase preservation technology investment.

Determining an optimal replenishment run time and lot size for an imperfect manufacturing system and rework under markdown policy

This study proposes a continuous production inventory model for deteriorating items with preservation technology, rework and price-dependent demand under markdown policy. In this study, we have adopted the markdown policy to reduce inventory cost and clearance at the end of the season or to sell old or slow-moving inventory at the end of its life. Rework is another main factor to reduce production cost and environmental problem. The preservation technology is also considered during product deterioration period to control the deterioration rate of products. The shortages are not allowed. We have also considered that in one cycle, the production system can produce items in several determined replenishment cycles under markdown policy. The aim of this research is to derive the optimal production-run time, the optimal replenishment cycle length, the optimal length of time horizon and the optimal production quantity so as to minimize the total average cost. An example and sensitivity analysis is shown to illustrate the model.

Analyzing an inventory model with two-level trade credit period including the effect of customers’ credit on the demand function using q-fuzzy number

In this paper, an EOQ model for deteriorating items has been developed in infinite time horizon including two-level delay in payment in which one delay in payment (M) is offered to the retailer by the supplier and the another delay in payment (N) is offered by the retailer to all customers. Since the real business world is full of uncertainties and the supplier has to face different problems with the retailer, hence there may exist uncertainties in the credit period which is offered by the supplier to the retailer. Again, uncertainties may be linear or non-linear type. Till now, there is no standard fuzzy number by which linearity and non-linearity can be explored simultaneously. In this respect, a new type of fuzzy number known as q-fuzzy number has been introduced to consider linearity and non-linearity together and this is the novelty of the paper. On the other hand, the retailer intends to offer a credit period to all customers to give rise the demand of the items. So, here demand function depends on credit period and duration of offering the credit period. The aim of the retailer is that how much credit period be benefited to get maximum profit. Therefore the purpose of this model is to determine the optimal credit length for the customers and optimal replenishment cycle length. Also, the model has been discussed considering the situation when the retailer offers no credit to the customers. Then some theoretical results and an algorithm for defuzzification have been developed. Finally, some numerical examples have been carried out to interpret the model and a sensitivity analysis of the optimal solution has been provided with respect to some parameters.

Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items

In this paper, we consider a problem of the dynamic pricing and the periodic ordering for deteriorating items with a stochastic inventory level depending on the stock-dependent demand and the selling price. In the model to be established, both shortages and remains are allowed, and the backlogging rate is variable and dependent on the waiting time for the next replenishment. Combined with the dynamic pricing, a stochastic dynamic optimization model, which maximizes the total profit, is developed. Based on the dynamic programming principle, the stochastic control model is transformed into a Hamilton–Jacobi–Bellman (HJB) equation. We show that there exists an optimal replenishment cycle length with an optimal inventory level at start of cycle. Moreover, the optimal pricing strategies are given. In addition, the finite difference scheme with the semi-smooth Newton method is proposed to solve the HJB equation numerically, and some numerical simulations are presented to illustrate the theoretical results. The sensitivity analysis of the optimal solution with respect to the major parameters is carried out and some managerial insights are proposed.

Integrated inventory problem under trade credit in fuzzy random environment

This paper is concerned with an integrated inventory problem under trade credit where both the demand rate and deteriorating rate are assumed to be uncertain and characterized as fuzzy random variables with known distributions. The objective of this paper is to determine the optimal inventory policy by optimizing simultaneously the replenishment cycle length and trade credit period. At first, three decision criteria are given: (1) expected value criterion, (2) chance-constrained criterion and (3) chance maximization criterion. Then, after building the fuzzy random models based on the above decision criterion, a hybrid intelligent algorithm by integrating fuzzy random simulation and genetic algorithm is employed to deal with these models. At the end, three numerical examples are given to illustrate the benefits of the models and show the effectiveness of the algorithms.

An economic lot sizing production model for deteriorating items under two level trade credit

This paper develops an economic lot-sizing production model for deteriorating items with time dependent quadratic demand and delay in payments under two levels of trade credit policy. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides customers a permissible delay period N. Cases where M < N and are investigated. In addition, the demand rate is assumed to be time and deterioration rate dependent quadratic demand. This paper deals with the problem of determining the optimal replenishment cycle length, optimal total cost and economic order quantity for an integrated inventory system under both two-level trade credit and time dependent quadratic demand rate. Numerical examples are presented to illustrate the developed model.

A two warehouse deterministic inventory model for deteriorating items with a linear trend in time dependent demand over finite time horizon by Elitist Real-Coded Genetic Algorithm

This paper deals with a deterministic inventory model developed for deteriorating items having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linear time dependent demand (increasing) over a fixed finite time horizon. The model is formulated with infinite replenishment and the successive replenishment cycle lengths are in arithmetic progression. Partially backlogged shortages are allowed. The stocks of rented warehouse (RW) are transported to the owned warehouse (OW) in continuous release pattern. For this purpose, the model is formulated as a constrained non-linear mixed integer programming problem. For solving the problem, an advanced genetic algorithm (GA) has been developed. This advanced GA is based on ranking selection, elitism, whole arithmetic crossover and non-uniform mutation dependent on the age of the population. Our objective is to determine the optimal replenishment number, lot-size of two-warehouses (OW and RW) by maximizing the profit function. The model is illustrated with four numerical examples and sensitivity analyses of the optimal solution are performed with respect to different parameters.

Two-Level Credit Financing for Noninstantaneous Deterioration Items in a Supply Chain with Downstream Credit-Linked Demand

Trade credit financing is a useful tool in business today, which can be characterized as the agreement between supply chain members such as permissible delay in payments. In this study, we assume that the items have the property of noninstantaneous deterioration and the demand is a function of downstream credit. Then, an EOQ model for noninstantaneous deterioration is built based on the two-level financing policy. The purpose of this paper is to maximize the total average profit by determine the optimal downstream credit period, the optimal replenishment cycle length, and the optimal ordering quantity per cycle. Useful theorems are proposed to characterize the method of obtaining the optimal solutions. Based on the theorems, an algorithm is designed, and numerical tests and sensitive analysis are provided. Lastly, according to the sensitive analysis, managerial insights are proposed.

An optimal replenishment policy for EOQ models with time-varying demand and shortages

While considering time dependent demand rate, most of the researchers developed their inventory models considering time dependent linear demand rate. Some models have been developed also by considering time dependent quadratic demand because by considering quadratic demand increasing, decreasing or constant all types of situations can be explained and hence it is more realistic than that of linear demand rate. But in the present paper, we have developed an EOQ model to describe an optimal replenishment policy with shortages and general time-varying demand instead of time dependent linear or time dependent quadratic demand. The successive replenishment cycle lengths are assumed to be reduced in arithmetic progression. The model is extended to the no shortage case also. The model is solved analytically to obtain the optimal solution of the problem. It is then illustrated with the help of numerical examples. The sensitivity analysis of the optimal solution towards changes in the values of different system parameters is also studied.

Optimal pricing and lot-sizing for perishable inventory with price and time dependent ramp-type demand

Product perishability is an important aspect of inventory control. To minimise the effect of deterioration, retailers in supermarkets, departmental store managers, etc. always want higher inventory depletion rate. In this article, we propose a dynamic pre- and post-deterioration cumulative discount policy to enhance inventory depletion rate resulting low volume of deterioration cost, holding cost and hence higher profit. It is assumed that demand is a price and time dependent ramp-type function and the product starts to deteriorate after certain amount of time. Unlike the conventional inventory models with pricing strategies, which are restricted to a fixed number of price changes and to a fixed cycle length, we allow the number of price changes before as well as after the start of deterioration and the replenishment cycle length to be the decision variables. Before start of deterioration, discounts on unit selling price are provided cumulatively in successive pricing cycles. After the start of deterioration, discounts on reduced unit selling price are also provided in a cumulative way. A mathematical model is developed and the existence of the optimal solution is verified. A numerical example is presented, which indicates that under the cumulative effect of price discounting, dynamic pricing policy outperforms static pricing strategy. Sensitivity analysis of the model is carried out.