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A production inventory in bioethanol production consists of three basic segments; raw material ordering, fermentation plant and finished product inventory. A sustainable production process focuses on the optimal utilization of raw materials in order to reduce the negative impact on the environment. In most of the production inventory model, the rate of production is variable which changes with time at a constant amount. In this present paper, an effort has been made to develop a sustainable economic production quantity (EPQ) model with a variable production rate. Although the production rate is dependent on the amount of raw material used for the fermentation process and the rate of fermentation. Further, the production rate is a dependent variable of the maximum input of raw material in the production unit. Model is studied under four different environments; without green investment, with green investment, with variable labour cost and with variable labour cost together with overtime cost. Now to solve the formulated model, weighted particle swarm optimization (Weighted PSO) and constriction factor particle swarm optimization (Constriction PSO) are applied and the optimality is established through statistical analysis and Taguchi L9 design together with the graphical representations of convergence characteristics. Model is validated through numerical examples along with the sensitivity analysis of parameters and managerial implications are provided for effective decision making.

In this work, a new model is developed to determine optimum sustainable economic order quantity (EOQ) and economic production quantity (EPQ) values for multiple products of a supply chain with two echelons, consist of a retailer and a supplier, considering product return policy. Based on a literature review on previous related works in sustainable inventory area, 17 different sustainability cost functions, in three types of economic, environmental and social, are defined to form the profit functions of the retailer and the supplier as the main members of the chain. We formulate the profit function of each members of the chain using direct accounting approach as a careful method, considering sustainability costs and return policy to produce and supply multiple products. Two different scenarios for product return are studied considering return amount as a variable or a given parameter. After a numerical example according to an Iranian supply chain that supplies various petrochemical products, a sensitivity analysis related to three parameters of the problem that imposed return functions of products has been conducted. The results show that the total profit amount of each members of the sustainable supply chain is sensitive on the return parameters with different rates.

Inventory control is one of the most widely recognized issues in the reality. This investigation manages the utilization of fractional derivatives and integration on an inventory control problem. The memory of a dynamical model is a highly concerned issue which is commonly neglected by the models described in terms of integer-order differential equation. The memory capturing the power of fractional derivative (in Caputo’s sense) is utilized here to describe an economic production quantity model with deterioration when the demand depends on price and stock and production is stock dependent. Also, this study covers the integer-order model with the same assumptions as a memoryless model and a particular case of the fractional model. Due to the complex nature of the model, numerical optimization with the help of a modified artificial bee colony algorithm is done instead of the analytical approach of optimization. Finally, we have performed a sensitivity analysis in order to make a fruitful conclusion.

Inventory control is one of the most critical issues in corporate management. Many mathematical models have been developed to optimize control strategies for the companies’ inventory. Economic production quantity (EPQ) is one of the classic models for inventory control, which is widely used. To deal with the uncertainty in the real world, we need to develop new and useful models for modeling systems of inventory management. In such cases, fuzzy models play a unique role in the field of inventory management. The main contribution of this study is to apply some well-known metaheuristic to solve an extended EPQ model based on fuzzy numbers considering multiple deliveries. This study aims to develop an EPQ model by considering demand as triangular fuzzy numbers and multiple deliveries (delivering in multiple packages) and by considering limitations in warehouse space as well as the total number of orders. Given these conditions, EPQ costs are calculated, and new modeling is presented. The obtained fuzzy model has been simplified by using the $$ \alpha{\text{-cut}} $$ and changing the variables, and finally, the most well-known metaheuristic algorithms, GA, PSO, GWO, and ICA, are applied in different problem sizes, and obtained results are analyzed in terms of minimizing cost function and CPU time. The result of this paper shows that GWO has superior performance in terms of various parameters.

The lot-sizing problem with a defective processing system is one of the most widely known inventory control models that can be regarded as the generalized form of the economic production quantity (EPQ). This paper proposes a model for lot sizing problem within a defective processing system. The study assumes that the processed items are from different categories: high-quality, low-quality, reworked, and rejected items. In addition, to identify items associated with each category, an online full screening inspection of the items would be implemented. We model the system considering immediate rework after regular processing cycle where reworked items are sold at the same price as high-quality items. The rejected items are also scrapped at the end of the production cycle. Motivation of this paper is proposing a model for the EPQ problem with faulty products, scrap, and rework. Moreover, to integrate decisions for increasing the system’s profitability, the model aims to provide a practical and general framework to integrate pricing decisions, optimal processing order quantities, and optimum level of shortage with respect to maximizing system’s profits. Then, a numerical example is presented to illustrate performance of the proposed model and solution algorithm. Finally, we perform a sensitivity analysis to evaluate the influence of the model parameters on optimal decisions.

In this paper, an EPQ model for delayed deteriorating items is presented, where the demand before deterioration sets in is assumed to be time dependent quadratic demand and the holding (carrying) cost is assumed to be linearly dependent on time. Three stages are considered as follows: (1) production build up period, (2) period before deterioration starts and, (3) period after deterioration sets in. There is no demand during production build up period and the demand before deterioration begins is assumed to be quadratic time dependent while that after deterioration sets in is assumed to be constant. It is also assumed that shortages are not allowed. The purpose of this paper is to investigate the optimal set of production rates that minimizes the total inventory cost per unit time, the best cycle length and the economic production quantity. A numerical example is given to illustrate the applicability of the model and sensitivity analysis carried out on the example to see the effect of changes on some system parameters.

In the traditional economic order quantity/economic production quantity model, most of the items considered are of perfect type. But this situation rarely takes place in practice. Thus, in this paper, an economic order quantity model with imperfect-quality items is developed. $$100\%$$ screening process is performed, and the items of imperfect quality are sold as a single batch. A proportionate rate of discount for the items of imperfect quality has also been studied. Moreover, a case study has been incorporated to comprehend the model. To nullify the issues of non-random uncertainties of demand rate in business scenario, cloudy fuzzy method has been utilized here. Numerical study reveals that cloud model along with its new defuzzification methods can give maximum profit of the model all the time instead of deterministic ones. Finally, sensitivity analysis and graphical illustrations are made to justify the novelty of the model.

In the present paper we have considered a production supply chain and developed two models for production supply cycle. These models are (i) The optimum total cost model and (ii) The maximum inventory level model. We have assumed constant demand rate for deteriorating inventory and considered three parameter Weibull distributions for deterioration rate with finite time horizon. We have also considered three different level of production in one cycle with shortages allowed. In real world of production and manufacturing market, EPQ models are playing role of an effective tool. The objective of our work is to find optimal production quantity and minimize total cost per cycle. Our work will be useful for the organizations which are going to launch new product in the market or newly established production organizations where at starting organizations are not aware about the production size and changing their production level according to the market demand. These models will help to avoid surplus stock or shortages in business organizations and reduce holding cost; deterioration cost etc. The objective of this paper is to find optimal total cost and optimal production quantity for one cycle. At last numerical example and sensitivity analysis is given to illustrate the result. We have used Mathematica 0.9 to solve equations, calculation and plotting graph.

In production-inventory control, economic order quantity (EOQ) and economic production quantity (EPQ) models are used to determine the optimal order quantities for purchasing and manufacturing. Most EOQ and EPQ models are constructed assuming constant costs. Recently, however, EOQ/EPQ models assuming varying costs (i.e., holding, ordering, and purchasing costs) have been receiving considerable attention. The objective of this paper is to review and classify EOQ and EPQ inventory models formulated under the assumption of variable holding costs. The relevant papers are reviewed and classified into three main types: time-dependent holding cost, stock-dependent holding cost, and multiple dependence or other holding cost variability. Additional classification is proposed for the reviewed models according to their objectives, solution methods, and applications. The paper identifies research trends and includes several suggestions for future research directions.

In this research, an imperfect manufacturing inventory model for deteriorating items has been developed wherein a manufacturer produces items in m determined imperfect replenishment cycles. In each replenishment cycle, production facility can produce items in four different imperfect replenishment cycles and one rework-run, (m : 4 − 1) policy. Demand rate is considered as inflation rate and selling price dependent. Many researchers focused on economic production quantity (EPQ) model under the three assumptions that the items produced in replenishment cycles are all perfect, all the deteriorated items can be detached from the stock, and all the items are produced in a single replenishment cycle. In this work, these three assumptions are relaxed to create positive impact on corporate image and reduce the holding cost of the inventory model. Shortages are allowed and fully backordered. The aim of the research is to determine the optimal replenishment-run time, the economic replenishment lot size, and the optimal backordered quantity so as to minimize the total cost of the inventory model. Finally, a numerical example along with graphical representation and sensitivity analysis is discussed to validate the results of the model.

This paper develops a multiproduct economic production quantity inventory model for a vendor–buyer system in which several products are manufactured on a single machine. The vendor delivers the products to customer in small batches. The number of orders must be a discrete value. Moreover, benefitting from a just-in-time policy, the buyer decides the size of the delivered batches. Due to the fact that several products are manufactured on one machine, this makes that the production capacity be considered as a constraint. The aim of this study is to determine the optimal cycle length and the number of delivered batches for each product so that the total inventory cost is minimized. The problem under study is modeled as a mixed integer nonlinear programing problem considering maximum number of orders, capacity and budget constraints. Three different methods are developed and employed to solve this problem: an exact method, a heuristic algorithm and a hybrid genetic algorithm. Based on the results, the three algorithms have near efficiency with different running times. The results shows that the heuristic algorithm obtains a good solution in a short time and the hybrid genetic algorithm finds solutions with higher quality in an acceptable time. Finally, a sensitivity analysis is done to evaluate the effect of changes in the parameters of problem.

This paper develops an economic production quantity model for a multi-product multi-objective inventory control problem with fuzzy-stochastic demand and backorders. In this model, the annual demand is represented by trapezoidal fuzzy random numbers. The centroid defuzzification and the expected value methods are applied to defuzzify and make decisions in a random environment. In the case where the warehouse space is limited, the Lagrangian relaxation procedure is first employed to determine the optimal order and the maximum backorder quantities of the products such that the total inventory cost is minimized. The optimal solution obtained by the proposed approach is compared with that obtained by the traditional deterministic method. Moreover, a sensitivity analysis presents the rationality of the solution. Then, the model is extended to a multi-objective integer programming problem in which the optimal numbers of redundant production machines are determined to maximize the production system reliability. In the second proposed model, several constraints are considered to fit real-world situations. As the second model is developed for an NP-hard problem and hence cannot be solved using exact methods in a reasonable computational time, a multi-objective evolutionary algorithm called non-dominated sorting genetic algorithm-II (NSGA-II) is employed to provide Pareto front solutions. Due to non-availability of benchmark in the literature, another multi-objective evolutionary algorithm called non-dominated ranking genetic algorithm is implemented as well to validate the obtained results and evaluate the performance of NSGA-II. In addition, the Taguchi method is used to calibrate the parameters of both algorithms for better performance.

In this paper, a multi-product single-machine economic production quantity model with preventive maintenance, scraped and rework is studied. Shortages are permitted and a fraction of them is backlogged. Capacity and service level are limitations of the production system. It is assumed that preventive maintenance can be performed when the inventory level is positive or negative. Indeed two different scenarios are modeled and according to the comparisons between their costs, a new scenario according to the best time of preventive maintenance is investigated and modeled. The aim of this research is to determine the best time for preventive maintenance, production and back-ordered quantities of each item and common cycle length, such that the expected total cost is minimized. The objective function of the final proposed model is proved to be convex and closed-form optimal solutions are derived. Two numerical examples based on a real application of the proposed model in a turning manufactory with only one computer numerical control machine applied to lathe metal plates to different sizes are used to illustrate the applicability of extended model and proposed solution method.

This article comprises a bi-objective economic production quantity (EPQ) inventory problem for two book producer groups. In a competitive world, there are several companies which are usually producing similar items and getting hurry to launch their products first in the market. Under these circumstances two different cases may occur (1) early demand-early profit and (2) showing low marked unit selling price of each item and creating motivation by offering some discounts, as a result total demand will be increased and get benefit from that. Thus, in the inventory process, the interacting demand functions for two different producers are exclusively depending upon marked unit selling price and the duration of the production period respectively. We have developed two profit maximizing objective functions under several considerations. Then the model has been solved with the help of Goal Attainment approach as well as fuzzy goal attainment approach via some solution algorithms. Sensitivity analysis and graphical representations are given to illustrate the model also. Finally the decision is made on the basis of some numerical illustrations incorporating decision maker’s intelligence.

Most research studies on the economic production quantity (EPQ) model considered that produced items are of perfect quality. On the other hand, real production systems have some product defects. Considering the imperfect items makes the inventory model more complex, and more difficult to solve analytically rather than it is time consuming. Therefore, an efficient approach like D-optimal response surface methodology (RSM) is required since heterogeneous combination of data can be modeled to generate response surfaces and obtain optimum decision parameters values. This paper solves the EPQ model with sales return, rework, shortage and scrap by RSM optimization technique in order to optimize the long run average cost function. ANOVA analysis of data obtained from the total cost RSM quadratic model has shown that the Model is significant according to F, “Prob > F” and p-values.

The constant demand rate is the most common assumption of the basic economic production quantity model, which is not very frequent in practice. In real world situations, demand usually varies with time. With regard to the widespread necessity of power demand pattern, demand is supposed to follow a power law. Another unrealistic assumption is perfect quality of all items. This paper presents a production system with defective items to determine the optimal replenishment quantity, cycle length and backordered size with a power demand rate dependent production rate. We assume that a manufacturer may be faced with three different cases regarding to the date that defective items are drawn from inventory. The set-up, backordering, inspection, and production costs, as well as holding cost of both perfect and imperfect items are accounted in the inventory system. An algorithm is offered to optimize total inventory cost and then numerical analyses are presented to demonstrate the applicability of the proposed models. Finally, some sensitivity analyses and managerial insights are provided.

An economic production quantity (EPQ) inventory model for a multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint is proposed. This research work revisits this EPQ inventory model and identifies some technical inconsistencies in it. Additionally, some observations and corrections to the shortcomings are provided as well as the solution procedure is improved.

In this paper, an energy-aware Economic Production Quantity (EPQ) model is presented to determine optimum production run length and batch size with respect to variable energy cost. Here, variable unit production cost includes energy consumption charge which is a function of production time and time-of-use, and alternates between two prices during peak and off-peak hours. This paper addresses the above integration in order to minimize the overall cost of the system. In the first phase of this study, a new scenario-based framework is proposed to find the optimal value of production time. In the second phase, a general mixed integer nonlinear programming (MINLP) model is developed for the given framework. The energy cost defined by the framework and mathematical model depends on the number of peak periods during the production period and is calculated using floor functions. The MINLP is solved numerically and analytically, and a closed form solution is obtained for the production run length. The model is analyzed for different scenarios and the results are discussed.

The classic economic production quantity (EPQ) inventory model assumes all items produced are of perfect quality. However, in most real-life manufacturing settings, generation of defective items is unavoidable. Reworking of imperfect items reduces the overall costs. The finished products can only be delivered to customers if the whole lot is quality assured at the end of rework. In addition, in many circumstances the delivery of the products is not continuous. This paper studies pricing and production decisions in multi-product single machine manufacturing system considering discrete delivery as well as rework. We solved this problem in two situations: first, when the condition is satisfied and second, when it is not satisfied. The obtained closed-form solution is optimal when the only constraint of the problem is satisfied; otherwise, the Lagrangian relaxation method is used to determine T* and Q*. At the end, numerical example is solved to illustrate the validity of the proposed solution procedure.

One of the economic production quantity problems that have been of interest to researchers is the production with reworking of the imperfect items including waste most disposal form and vending the units. The available models in the literature assumed that the decay rate of the items is satisfied from three different points of view: (i) minimum demands of the customer’s requirement, (ii) demands to be enhanced for lower selling price and (iii) demands of the customers who are motivated by the advertisement. The model is developed over a finite random planning horizon, which is assumed to follow the exponential distribution with known parameters. The model has been illustrated with a numerical example, whose parametric inputs are estimated from market survey. Here the model is optimized by using a population varying genetic algorithm.

Mahata (Expert Syst Appl 39(3):3537–3550, 2012) developed an economic production quantity (EPQ) inventory model for exponentially deteriorating items under permissible delay in payments considering that both demand and production are constant and known. This paper, applying well-known approximation mathematical expressions, derives closed-form formulas for the time at which the production ends, the cycle length and the total cost of inventory system. Moreover, this work presents a comparison of the solutions to the numerical examples by approximation closed-form formulas and Mahata (2012)’s method. The approximated method works properly because the percent of penalty is negligible less than 0.09%.

This study presents an uncertain mathematical model for maximizing profit of the defective goods supply chain during uncertain situations by using selection of appropriate suppliers, just-in-time (JIT) logistic philosophy and minimizing total costs including costs of production, shipping, holding, defective goods, scrap goods, shortage in retailers. The selections of suppliers are based on three criteria, namely, quality, JIT delivery, and level of responsibility, which have important roles in the production of perfect goods by manufacturers. In each criterion, two kinds of indicators are considered by manufacturers. The first indicator is the importance of the weighted factor of each criterion for each manufacturer, and the second is weighted factor of each supplier with respect to each criterion. The proposed mathematical model with uncertain parameters and the probability of occurring in various scenarios is investigated. The model is studied in ten scenarios and the average amount is calculated. The mentioned mathematical model is solved by the averages of the parameters using CPLEX.12 solver and Expert Choice software. The findings of the model are maximum profit, amounts of economic production quantity, defective goods, scrap goods, and amounts of products that should be exchanged among the nodes of the supply chain. To achieve maximum benefit, the model can select the appropriate suppliers. The results obtained demonstrate the validity and efficiency of the proposed uncertain mathematical model.

In this paper, an economic production quantity model in a three levels supply chain including multiple non-competing suppliers, single manufacturer and multiple non-competing retailers for multiple products with rework process under integrated and non-integrated structures is developed. In this chain, suppliers ship raw material to the manufacturer and the manufacturer combines the fixed percentage of different kinds of raw material to produce the finished products and then delivers them to the retailers. The Stackelberg game is established among the members of the chain. Optimizing the total profit of the chain under both structures applying the optimal pricing, inventory and production policies is the purpose of the research. Eventually, a numerical example is presented for comparing the total profit of the integrated and non-integrated chains of proposed model and some sensitivity analyses are performed to study the effects of parameter changes on the decision variables and the chain profit.

In this paper, the classical economic production quantity (EPQ) model is developed for non-instantaneous deteriorating items by considering a relationship between the holding cost and the ordering cycle length. Two models are developed. First, the proposed model is considered when backorders are not permitted and this condition is waived for the second case. The cost functions associated with these models are proved to be convex and an algorithm is designed to find the optimum solutions of the proposed model. Results show that the relationship between holding cost and ordering cycle length has a significant impact on the optimal lot size and total cost in the EPQ model. Numerical examples are presented to demonstrate the utility of the models.

This paper deals with an economic production quantity (EPQ) inventory model with reworkable defective items when a given multi-shipment policy is used. In this work, it is assumed that in each cycle, the rework process of all defective items starts when the regular production process finishes. After the rework process, a portion of reworked items fails. This portion becomes scrap and only the perfect finished items can be delivered to customers at the end of rework process. A profit function is derived to model the inventory problem and it is shown that the profit function is concave. Due to the complexity of the optimization problem, an algorithm is developed to determine the optimal values of manufacturing lot size and price such that the long-run average profit function is maximized. Furthermore, two special cases are identified and explained. Finally, a numerical example is given to illustrate the applicability of the proposed inventory model.

In steel industries, iron ores are divergently transformed into several hundreds of finished goods through steel making processes in accordance with the stipulations requested by customers. Similar patterns of production can also be observed in many petrochemical industries where items are produced through refining processes of crude oil. In view of that the ordering and production policies of the above industries would be dependent on the ordering quantities of raw materials and production rate of finished goods. In this point of view, this paper addresses a single stage integrated procurement production (IPP) inventory model for defective raw material and finished goods with the objective of minimizing total inventory cost by optimizing ordering quantity of raw material, production rate of finished goods, setup and ordering cost simultaneously. In this study, a manufacturer costs from the ordering policies of an EOQ (Economic Order Quantity) for raw material and production policies of an EPQ (Economic Production Quantity) for finished goods are combined together to integrate the manufacturer’s raw material procurement and its production. In addition, two computational algorithms have been developed to find the optimal solutions and a numerical example is presented to demonstrate the results of the proposed model. This paper offers an interesting scope for application in batch industries like steel, chemical, pharmaceuticals, paints manufacture, bottling plants, etc.

This article presents an economic production quantity (EPQ) model for deteriorating items involving probabilistic deterioration where the demand is dependent on sales team’s initiatives. Here, we assumed that the production rate is variable and the unit production cost is the function of the production rate. The unit selling price is calculated by a mark-up over the unit production cost. This model aids in minimizing the total inventory cost of the manufacturer by finding the optimal cycle length and the quantity. On this backdrop, a few theorems have been developed and, the necessary and sufficient conditions of the existence and uniqueness of the optimal solutions are suggested. Moreover, numerical examples and sensitivity analysis were also carried out. The results reveal that the total cost will be minimized when the deterioration follows uniform distribution and when the initiatives of the sales team equal to one.

While there has been considerable work over the years on the basic deterministic economic production quantity (EPQ) and its derivative models, there have been few extensions of these models that recognize the potential effects of machine degradation. As maintenance activities can keep machines in good operation, it should be integrated into EPQ models to meet real situations. Due to machine degradation, this paper integrates predictive maintenance into EPQ model in which autoregressive integrated moving average model is adopted to predict system's healthy indicator. Moreover, two kinds of system out-of-control states are considered in this proposed EPQ model: in State I, the system produces non-conforming items; and in State II, the system fails. Aiming at minimizing the expected average total cost and optimizing the EPQ, suitable maintenance intervals and frequency are determined prior to any predicted failure. Finally, a case study is presented and the computational results are discussed to show the efficiency of this integrated EPQ model.

This paper develops an economic production quantity (EPQ) model with price and advertisement dependent demand under the effect of inflation and time value of money. Here, the rate of replenishment is considered to be a variable and the generalized unit production cost function is formulated by incorporating the several factors like raw material, labour, replenishment rate, advertisement and other factors of the manufacturing system. The selling price of an unit is determined by a mark-up over the production cost. In most of the inventory model for perishable items, the holding cost has been considered as a constant function. But in realistic situation this cost is varying according to time. In this model, the holding cost per unit of the item per unit time is assumed to be an increasing linear function of time spent in storage. Also in this model, shortages are allowed and we consider that shortage occurs before the starting of inventory. This type of inventory is called SFI (shortage followed by inventory) policy. In the model, the customers are viewed to be impatient and a fraction of the demand is exponentially backlogged. This fraction is a function of the waiting time of the customers. This model aids in minimizing the total inventory cost by finding the optimal cycle length, optimal production and the optimal order quantity. The model is extended to the case of non-perishable product also. The optimal solution of the model is illustrated with the help of a numerical example.

Inventory Management and production planning are essential tasks for every company in the industry. Therefore, the development of a large set of Economic Order Quantity (EOQ) models is needed. In this paper, a Fuzzy Rough (Fu-Ro) multi-item Economic Production Quantity (EPQ) model is developed with constant demand, infinite production rate having flexibility and reliability consideration in production process and demand dependent unit production cost and shortages under the restrictions on storage area. Some of the inventory parameters are considered as trapezoidal fuzzy rough numbers and the model is formulated following cost minimization principle. Then by geometric programming (GP) technique we solve the problem. The model is illustrated through numerical example.

In this study, an economic production quantity (EPQ) model is generalized by considering maintenance and production programs for an imperfect process involving a deteriorating production system with increasing hazard rate. There are two types of preventive maintenance (PM), namely imperfect PM and perfect PM. The probability that perfect PM is performed depends on the number of imperfect maintenance operations performed since the last renewal cycle. Following a failure, the delayed repair performs some restorations and reduces production rate to restore the system into an operating state (in-control state), but leaves its lower production rate until perfect PM is performed. That is, the production run period not always starts in normal production rate. This study considers backorders, as well as loss of inventory due to the lower production rate. For the EPQ model, the optimum run time, which minimizes the total cost, is discussed. Various special cases are considered, including the maintenance learning effect. Finally, a numerical example is presented to illustrate the effects of PM ability, repair cost and production decreasing rate on total costs and production period.

In this paper, we study the inventory model for defective items with trapezoidal type demand rate. This paper develops a model to determine the optimal product reliability and production rate that achieves the biggest total integrated profit for an imperfect manufacturing process and necessary and sufficient conditions for optimality of the dynamic variables are given. The Euler–Lagrange method is used to obtain optimal solutions for product reliability parameter and dynamic production rate. Finally, numerical examples are presented to illustrate the proposed model. Suggestions for further research are provided.

In this paper, an Economic Production Quantity (EPQ) model is developed with flexibility and reliability consideration of production process in an imprecise and uncertain mixed environment. The model has incorporated fuzzy random demand, an imprecise production preparation time and shortage. Here, the setup cost and the reliability of the production process along with the backorder replenishment time and production run period are the decision variables. Due to fuzzy-randomness of the demand, expected average demand is a fuzzy quantity and also imprecise preparation time is represented by fuzzy number. Therefore, both are first transformed to a corresponding interval number and then using the interval arithmetic, the single objective function for expected profit over the time cycle is changed to respective multi-objective functions. Due to highly nonlinearity of the expected profit functions it is optimized using a multi-objective genetic algorithm (MOGA). The associated profit maximization problem is illustrated by numerical examples and also its sensitivity analysis is carried out.

This case study uses a surface mine to investigate multidisciplinary logistics analysis methods for improving refinery operations. Existing resource scheduling, inventory forecasting, and economic production quantity procedures have not been able to identify how to improve productivity. The objective was to locate and demonstrate proven techniques from operations research (and other related disciplines) which could be applied to solve logistics problems. Historical operations data along with a new sample (n = 140) were utilized for the analysis. Preliminary parametric tests failed, but later a multiple server queue model was developed by integrating nonparametric techniques, waiting line theory, stochastic probabilities, and break-even scenario analysis. Quantitative and qualitative data were analyzed, resulting in a solution to increase truck arrival rates by 10% which was projected to increase refinery utilization by 7–77%, thereby generating a potential productivity savings of $161,223.31 per year.

A great deal of research has been done on economic production quantity (EPQ) model, which concern deterministic or stochastic or fuzzy demand and cost situations. In this paper, the EPQ model with imprecise demand and production preparation time is considered which are characterized as independent fuzzy variables rather than fuzzy numbers as in previous studies. Based on an expected value criterion or a credibility criterion, a fuzzy expected value model (EVM) is constructed. The purpose of the fuzzy EVM is to determine the optimal policy such that the fuzzy expected value of the total cost is minimal. In order to obtain the exact expected value directly, instead of relying on simulation procedure, the results for uncertain variables with uncertainty distribution are exploited by treating them as fuzzy variables with credibility distribution. The mathematical analysis is carried out to compute exact expected values. Numerical study is also provided to demonstrate the contribution of our model.

In general, cost constant parameters of inventory or production models are not rigid in real life rather these are imprecise in nature. In this paper we have presented imprecise cost constant parameters for an economic production quantity model. The unit production cost of this economic production quantity model is demand and reliability dependent. The imprecise parameters of the production model are considered as fuzzy in nature. The fuzzy model is solved by fuzzy parametric geometric programming technique. Finally, production model is numerically illustrated using fuzzy parametric geometric programming.

This paper discusses the production inventory model over an infinite time horizon. Here we consider demand as a function of stock and time. Deterioration is a function of time and time-varying production. Our objective is to minimize the total cost which is a function of set up cost, holding cost, shortage cost, and opportunity cost due to lost sales. The traditional costs such as purchasing cost, shortage cost and opportunity cost due to lost sales are kept constant. We consider holding cost to be a non-linear function of time. Shortages are allowed and are partially backlogged. Here, time durations are the decision variables. Numerical examples are given to illustrate the model.

The economic production quantity (EPQ) model is often used in manufacturing environments to assist firms in determining the optimal production lot size that minimizes the overall production–inventory costs. While there are some unrealistic assumptions in the EPQ model that limit its real-world applications, in this research, some of these assumptions such as (1) infinite availability of warehouse space, (2) all of the produced items being perfect, and (3) the existence of one product type are relaxed. In other words, we develop a multi-product EPQ model in which there are some imperfect items of different product types being produced such that reworks are allowed and that there is a warehouse space limitation. Under these conditions, we formulate the problem as a nonlinear integer-programming model and propose a genetic algorithm to solve it. At the end, a numerical example is presented to identify the optimal values of the genetic algorithm parameters and to illustrate the applications of the proposed methodology to more realistic real-world problems.

A single item economic production quantity (EPQ) model is discussed to analyse the behaviour of the inventory level after it’s introduction to the market. It is assumed that demand is time dependent accelerated growth-effect of accelerated growth-steady type. Unlike the conventional EPQ models, which are restricted to general production cycle over the finite or infinite time horizon, we consider the production sale scenario of the very first production cycle for newly introduced perishable product. Shortage is not allowed. Set up cost of an order cycle depends on the total amount of inventory produced. The finite production rate is proportional to demand rate. Optimal production stopping time is determined to maximize total unit profit of the system. A numerical example is presented to illustrate the development of the model. Sensitivity analysis of the model is carried out.

This paper considers the economic production quantity (EPQ) problem with backorder in which the setup cost, the holding cost and the backorder cost are characterized as fuzzy variables, respectively. Following expected value criterion and chance constrained criterion, a fuzzy expected value model (EVM) and a chance constrained programming (CCP) model are constructed. Then fuzzy simulations are employed to estimate the expected value of fuzzy variable and α-level minimal average cost. In order to solve the CCP model, a particle swarm optimization (PSO) algorithm based on the fuzzy simulation is designed. Finally, the effectiveness of PSO algorithm based on the fuzzy simulation is illustrated by a numerical example.

This paper considers the materials procurement decision-making based on the economic order quantity and economic production quantity models with backlogging permitted. Make-or-buy is one of the fundamental decisions routinely made by most manufacturing firms. Cost is always considered as the biggest determinant in the decision to make or buy. Four terms composing the total inventory cost per unit time are analyzed in this study; they are annual setup cost, purchase/manufacturing cost, holding cost, and backordering cost. The impact of different values of the unit backorder cost b i , setup cost K i, and unit cost c i to the make-or-buy decision is studied specifically in our mathematical analysis. As a result, the guidelines in terms of simplified formulas for the make-or-buy decision makings are derived. Numerical examples are provided to demonstrate their practical usages.

This paper considers the economic production quantity (EPQ) model with a random defective rate and an imperfect rework process. The assumption of perfect quality production condition of the classical EPQ model is unrealistic. Due to process deterioration or other factors, production of defective items is inevitable. This study assumes that the defective rate is a random variable and all items produced are inspected. The imperfect quality items fall into two groups, the repairable and the scrap. The reworking of the repairable defective items starts when the regular production process finishes in each cycle. The rework process is also assumed to be imperfect, with a random scrap rate. Since the scrap items are produced during a production run, it follows that the production cycle length is not a constant. The renewal reward theorem is utilized to cope with the variable cycle length. A mathematical model is developed to minimize the overall costs and to derive the optimal lot size for the imperfect quality EPQ model where shortages are not permitted. A numerical example is provided to demonstrate its practical usage.

Generally, in deriving the solution of economic production quantity (EPQ) inventory model, we consider the demand rate and deterioration rate as constant quantity. But in case of real life problems, the demand rate and deterioration rate are not actually constant but slightly disturbed from their original crisp value. The motivation of this paper is to consider a more realistic EPQ inventory model with finite production rate, fuzzy demand rate and fuzzy deterioration rate. The effect of the loss in production quantity due to faulty/old machine have also been taken into consideration. The methodology to obtain the optimum value of the fuzzy total cost is derived and a numerical example is used to illustrate the computation procedure. A sensitivity analysis is also carried out to get the sensitiveness of the tolarance of different input parameters.

We propose an interactive Goal Programming (GP) for operational recovery problems that are present in diverse areas of application. After defining the problem we discuss its relevance to scenarios taken from airline scheduling, supply chain management and call centers operations. We then construct the GP procedure for operational recovery decision making and illustrate the mechanics of the proposed procedure through two examples: an abstract example based on a minimum spanning tree problem and a more practical example of a production-inventory problem.

In order to solve the economic production quantity (EPQ) inventory model, we always consider the deterioration rate as a constant quantity. But in the real situation, the deterioration rate will have little disturbances from the actual constant value. Therefore, we have developed an EPQ inventory model with fuzzy deterioration rate and finite production rate. The effect in the loss in production quantity due to faulty/aged machine have also been considered. The methodology for obtaining the optimum value of fuzzy total cost is also derived and the solution procedure is explained with one numerical example. To study the effect of changes in the system parameters on the decision variables, a sensitivity analysis is carried out by one numerical example.

In this paper, we develop an integrated model for the joint optimization of the economic production quantity, the economic design of x¯-control chart, and the optimal maintenance level. This is done for a deteriorating process where the in-control period follows a general probability distribution with increasing hazard rate. In the proposed model, Preventive Maintenance (PM) activities reduce the shift rate to the out-of-control state proportional to the PM level. Compared to the case with no PM, the extra cost of maintenance results in lower quality control cost which may lead to lower overall expected cost. These issues are illustrated using an example of a Weibull shock model with an increasing hazard rate.

This paper analyzes a problem of optimal inspection intervals for a failure-prone batch production facility. The model makes a seemingly minor modification to similar models that have appeared recently, but which results in a significant change in the form of the optimal policy. Optimal solutions are developed for both continuous and integer valued sublot sizes. The paper also examines the effects of inspection sublots in an economic lot sizing context and shows that a simple, exact modification of the Economic Production Quantity formula is often possible in the continuous relaxation of the problem.

The Economic Production Quantity (EPQ) model is often used in the manufacturing sector to assist firms in determining the optimal production lot size that minimizes overall production-inventory costs. There are some assumptions in the EPQ model that restrict this model for real-world applications. Some of these assumptions are (1) infinite space of warehouse, (2) all of the produced items are perfect, and (3) only one type of goods is produced. In this paper, we develop the EPQ model by assuming that each produced lot contains some imperfect items and scraps. In addition, we have more than one kind of products along with warehouse space limitations. Under these conditions, we formulate the problem as a non-linear programming model and propose a genetic algorithm to solve it. At the end, we present a numerical example to illustrate the applications of the proposed methodology and identify the optimal value of the parameters of the genetic algorithm.

The classical economic production quantity (EPQ) model assumes that items produced are of perfect quality and that the unit cost of production is fixed. However, in realistic situations, product quality is never perfect but is directly affected by the production processes and the quality assurance programme. In addition, the unit production cost is not fixed but increases with quality assurance expenses. We present an EPQ model with imperfect production processes and quality-dependent unit production cost. After discussion of the procedure for determining the optimal solution, a sensitivity analysis of the impacts of the cost parameters on the optimal solution is provided. Finally, the problems associated with cost estimation are addressed.