Economic Production Quantity Problem Research Articles

Derivation of the Optimal Solution for the Economic Production Quantity Model with Planned Shortages without Derivatives

In this paper, we study a reformulation of the Economic Production Quantity (EPQ) problem. We study a more general version of the problem first and derive the conditions for an optimal solution, as well as the optimal solution itself, all without using derivatives. Then, we apply the approach to the reformulated EPQ problem. This version of the EPQ problem has been tackled by a number of researchers, wherein they have derived the conditions for the optimal solution and proposed algebraic derivations. However, their derivations for the conditions, as well as the optimal solution, have been shown to be questionable. Other than being questionable, the existing approaches are so complicated that they defeat the purpose of simplifying the optimization by using a derivative-free approach. We propose a correct and more succinct, much less complicated approach to derive the conditions and the optimal solution without using derivatives.

Solution of an Economic Production Quantity model using the generalized Hukuhara derivative approach

In this study, an economic production quantity (EPQ) model with deterioration is developed where the production rate is stock dependent and the demand rate is unit selling price and stock dependent. The low unit selling price and more stocks correspond high demand but more stock corresponds to slow production because of the avoidance of unnecessary stocks. First of all, we develop the production model by solving some ordinary differential equations having deterministic profit function under some specific assumptions. Later, we develop the fuzzy model by solving the fuzzy differential equations using Generalized Hukuhara derivative. In fact, the differential equation of the model has been split into two parts namely gH(L-R) and gH(R-L) on the basis of left(L) and right(R) α- cuts of fuzzy numbers for which the problem itself is transformed into multi-objective EPQ problem. A new formula of aggregation of several objective values obtained at different aspiration levels has been discussed to defuzzify the fuzzy multi-objective problems. We solve the crisp and fuzzy models using LINGO software. Numerical and graphical illustrations confirm that the model under Generalized Hukuhara derivative of (R-L) type contributes more profit which is one of the basic novelties of the proposed approach.

An Extension to the Economic Production Quantity Problem with Deteriorating Products Considering Random Machine Breakdown and Stochastic Repair Time

The recent advances in manufacturing systems motivate several studies to focus on Economic Production Quantity (EPQ) problem. Althuogh there are several extentions to the EPQ, this paper provides a new extension by considering some of the real world parameters like: (a) shortages in the form of partial backordering, (b) inventory can deteriorate stochastically, (c) machine can break down stochastically, and (d) machine repair time may change stochastically based on the failure status of machine. As far as we know, there is no study treated all these suppositions in an EPQ framework. In addition to this development, two forms of uniformly- and exponentially-distributed repair times are formulated and necessary convexity conditions are discussed. Then, the corresponding optimality conditions are written that lead to finding the roots of two equations. Due to difficulty of achieving a closed-form solution, the solution is obtained numerically by means of Newton-Raphson method. Finally, some sensitivity analyses are provided to explain the models’ applicability. The practicality and efficiency of the proposed method in this context lends weight to development of proposed EPQ with more complex elements and its application more broadly.

Lot-sizing problem for a defective processing system with categorized items, backordering and pricing policy

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.

The Expected Value Model of the Fuzzy Economic Production Quantity Based on Credibility Measure

Based on the credibility theory, this paper presents the expected value model of the fuzzy economic production quantity problem without shortage. In the model above, assuming the production preparation cost and holding cost per unit be fuzzy variable, we have gained many results about the optimal inventory strategy to the problem. Furthermore, if the fuzzy parameters are some special fuzzy variable such as triangular, trapezoidal and normal fuzzy variable, we have investigated the optima strategy of the economic production quantity problem. The results obtained in this paper can provide theoretical foundation for designing algorithm to solve the fuzzy inventory problem.

The Bi-objective EPQ Problem: Fuzzy Goal Attainment Approach

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.

EPQ for an unreliable production system with endogenous reliability and product deterioration

AbstractIn this paper, we study an economic production quantity problem for a production line subject to random shift from the in‐control state (with high production rate) to out‐of‐control state (with low production rate). Different from previous research, we model the expected shift time as a controllable variable, based on the fact that, by investment in resources, the reliability of the production line can be improved. In the mathematical model, we consider three possible scenarios: no shift, shift without demand shortage, and shift with demand shortage, which are determined by the values of actual shift time and shifted production rate. Combining the three possible scenarios, the goal is to minimize the expected total cost per unit time by finding the optimal production time, as well as the optimal expected shift time. In addition, we extend the model to deterioration products and study the influence of product deterioration to the optimal decisions. Numerical examples are presented to illustrate the optimal solutions, followed by the sensitive analysis on important parameters. Comparing the optimal solutions under no reliability investment, reliability investment can help companies save more cost. Some other managerial insights are also proposed based on the numerical tests.

A multi-machine multi-product EPQ problem for an imperfect manufacturing system considering utilization and allocation decisions

This paper considers a multi-product problem with non-identical machines. This manufacturing system consists of various machine types with different production capacities, production costs, setup times, production rates and failure rates. One of the major issues in the planning phase of a manufacturing system is to take the best decision about which machines must be utilized to manufacture which items. As a result, the decision makers face three critical questions: what machines must be purchased, which items should be allocated to each machine, and what is the optimal cycle length. These decisions must be made to minimize system costs including utilization, setup, production, holding and scrap costs. The multi-machine multi-product economic production quantity (EPQ) problem for an imperfect manufacturing system is formulated as a mixed integer non-linear programming (MINLP), where the convexity property of multi-product single machine EPQ model is used to convert the problem into a bi-level decision-making problem. In the first level, decisions about machine utilization and items allocation are made. After, in the second level the optimal cycle length for each machine is determined. To solve the problem at hand, a hybrid genetic algorithm (HGA) is proposed integrating genetic algorithm and derivatives method. In the proposed HGA, the solutions of the first level are obtained randomly and then, for the second level, the derivatives method is applied to obtain optimal cycle length based on solutions of the first level. Finally, the results of HGA method are compared to the results of general algebraic modeling system (GAMS) and it is found that HGA method has better and more efficient results. Also, a numerical experimentation and a sensitivity analysis of the model are done.

Heuristics for the Economic Production Quantity Problem under Restrictions on Production and Maintenance Time

This paper proposes an economic production quantity problem with the maximal production run time and minimal preventive maintenance time over a finite planning horizon. The objective is to find the efficient production and maintenance policy to minimize the total cost composed of production, maintenance, shortages, and holding costs under the restriction on the production run time and the preventive maintenance time. The production and maintenance decisions include the production and maintenance frequencies and the production run and the maintenance time. The variability and the boundedness of the production run and maintenance time make the problem difficult to solve. Two heuristic algorithms are developed using different techniques based on the optimal properties of the relaxed problem. The performance comparison between the two algorithms is illustrated by numerical examples. The numerical results show that, for the most part, there exists a heuristic algorithm which is more effective than the other.

An economic production quantity problem with backorders and fuzzy cycle times

Optimization models combining Economic Production Quantity models and fuzzy set theory are important to the process industry as they are capable of modeling the numerous uncertainties inherent in t...

Optimization of a multiproduct economic production quantity problem with stochastic constraints using sequential quadratic programming

In this paper, a multiproduct single vendor–single buyer supply chain problem is investigated based on the economic production quantity model developed for the buyer to minimize the inventory cost. The model to be more applicable for real-world supply chain problems contains five stochastic constraints including backordering cost, space, ordering, procurement, and available budget. The objective is to find the optimal order quantities of the products such that the total inventory cost is minimized while the constraints are satisfied. The recently-developed sequential quadratic programming (SQP), as one of the best optimization methods available in the literature, is used to solve the problem. Twenty numerical examples in 3 scales of small, medium, and large are solved in order to demonstrate the applicability of the proposed methodology and to evaluate its optimum performance. The results show that SQP has satisfactory performance in terms of optimum solutions, number of iterations to achieve the optimum solution, infeasibility, optimality error, and complementarity. Besides, the optimum performance of the SQP method is compared with the one of another exact method called interior point using the above numerical examples under similar conditions. The comparison results are in favor of the employed SQP. At the end, a sensitivity analysis is performed on the change rate of the objective function obtained based on the change rate of the variance of the order quantity.

An economic production quantity problem with backorders and fuzzy cycle times

Optimization models combining Economic Production Quantity models and fuzzy set theory are important to the process industry as they are capable of modeling the numerous uncertainties inherent in this context. In this paper, we incorporate backorders i.e the inventory to go below zero in a fuzzy Economic Production Quantity EPQ model. The uncertainties in the backorders for different products are modeled using triangular possibility distributions. We present an example describing a typical decision making problem in the paper industry to illustrate our model.

Net present value analysis of the economic production quantity

Using Laplace transforms we extend the economic production quantity (EPQ) model by analysing cash flows from a net present value (NPV) viewpoint. We obtain an exact expression for the present value of the cash flows in the EPQ problem. From this, we are able to derive the optimal batch size. We obtain insights into the monotonicity and convexity of the present value of each of the cash flows, and show that there is a unique minimum in the present value of the sum of the cash flows in the extended EPQ model. We also obtain exact point solutions at several values in the parameter space. We compare the exact solution to a Maclaurin series expansion and show that serious errors exist with the first-order approximation when the production rate is close to the demand rate. Finally, we consider an alternative formulation of the EPQ model when the opportunity cost of the inventory investment is made explicit.

Replenishment Lot Size and Number of Shipments for EPQ Model Derived Without Derivatives

This paper investigates optimal replenishment lot size and optimal number of shipments for economic production quantity (EPQ) model with rework process using an algebraic approach. The classic EPQ model assumes a perfect quality production for all items produced and a continuous issuing policy for satisfying customer‟s demand. However, in a real life vendor-buyer integrated-production-inventory system, a multi- delivery policy is commonly used in lieu of a continuous issuing policy and generation of defective items during a production run seems to be inevitable. A recent published work examined such an identical problem using mathematical modeling and differential calculus to derive the optimal replenishment lot size and optimal number of deliveries that minimizes overall production-inventory-shipment costs. This paper proposes a straightforward algebraic approach to replace the use of calculus on the cost function for determining optimal production-shipment policies. It also presents a simpler form for calculating the long-run average cost for such an imperfect EPQ problem.

A multi-item fuzzy economic production quantity problem with a finite production rate

Abstract 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 multi-item Economic Production Quantity (EPQ) model is developed. This paper contributes to the state-of-the-art with a theoretical study of a problem, where a company has to decide the size of some production batches under uncertain cycle times. The uncertainty will be handled with triangular fuzzy numbers and an analytical solution will be found to the optimization problem.

The economic production and pricing model with lot-size-dependent production cost

In this paper, the economic production quantity problem for a single-product single-machine system is extended. It is assumed that annual demand of the product is a function of price set by manufacturer. This extension considers sales revenue, inventory and setup costs as well as a variable cost of production which is a function of the lot size. Several linear and non-linear functions of demand and variable cost are considered in this paper and a global solution methodology is presented for the models developed. Newton's method is used to find local optima and asymptotic convergence of the solution algorithm to a global optimum is proved. Numerical studies followed by a discussion provide additional insights into the problem.

Computational Optimization of Manufacturing Batch Size and Shipment for an Integrated EPQ Model with Scrap

This paper employs mathematical modeling and algebraic approach to derive the optimal manufacturing batch size and number of shipment for a vendor-buyer integrated economic production quantity (EPQ) model with scrap. Unlike the conventional method by using differential calculus to determine replenishment lot size and optimal number of shipments for such an integrated system, this paper proposes a straightforward algebraic approach to replace the use of calculus on the total cost function for solving the optimal production- shipment policies. A simpler form for computing long-run average cost for such a vendor- buyer integrated EPQ problem is also provided.

A parameter-tuned genetic algorithm for multi-product economic production quantity model with space constraint, discrete delivery orders and shortages

In this paper, a multi-product economic production quantity problem with limited warehouse-space is considered in which the orders are delivered discretely in the form of multiple pallets and the shortages are completely backlogged. We show that the model of the problem is a constrained non-linear integer program and propose a genetic algorithm to solve it. Moreover, design of experiments is employed to calibrate the parameters of the algorithm for different problem sizes. At the end, a numerical example is presented to demonstrate the application of the proposed methodology.

The impact of a free-repair warranty policy on EMQ model for imperfect production systems

In this study, the economic production quantity problem in the presence of imperfect processes for products sold under a free-repair warranty policy is considered. In the literature, that a production facility may deteriorate with time is assumed, and the time to shift from an in-control state to an out-of-control state is assumed to be exponentially distributed, i.e., the process failure rate is a constant. However, in many practical situations, the process possesses an increasing failure rate due to cumulative wear in producing items. This study is extended to consider a process subjected to random deterioration from an in-control state to an out-of-control state with a general shift distribution. A mathematical model representing the expected total cost per item is developed to determine the optimal production policy. The objective here is to obtain the optimal production run-length (lot size) so that the expected total cost per item is minimized. Different conditions for optimality, properties, and bounds on the optimal production run-length are provided. A numerical example is used to see the adequacy of using the exponential distribution when the actual distribution is Weibull with an increasing failure rate.

Note: The economic manufacturing lot-sizing problem with imperfect production processes: Bounds and optimal solutions

In this paper, we consider the economic production quantity problem in the presence of imperfect processes. In the literature, the time to shift from the in-control state to the out-of-control state is assumed to be exponentially distributed. In this study, we consider general time to shift distributions and provide distribution-based and distribution-free bounds on the optimal cost. For the exponential case, we compare the optimal solutions to approximate solutions proposed in the literature. A numerical example is used to illustrate the analysis presented and to conduct a sensitivity analysis in order to see the effect of the input parameters on the various solutions to the problem. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 423–433, 1998