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Two-echelon inventory systems are widely used to distribute products among the customers over wide areas. Many researchers have studied their operating characteristics under a variety of conditions and assumptions. In most papers attention has been paid to the minimisation of inventory costs, but the cost of transportation is an influential factor in setting the right levels of inventories among the echelons being neglected. In this paper, we consider an inventory system consisting of a producer and N-retailers with an assumed inventory policy. Determination of optimal retailers' orders and production quantities takes into account the transportation costs. We conclude that the model can effectively be used as a tool to increase pricing transparency among producer, retailer, and third party logistics service providers. It can also reflect the environmental (social) costs of transportation.

In this article, we develop two deterministic economic production quantity (EPQ) models for Weibull-distribution deteriorating items with demand rate as a ramp type function of time. It is assumed that the finite production rate is proportional to the time-dependent demand rate and the unit production cost is inversely proportional to the production rate. The EPQ model without shortages is studied first. The EPQ model with shortages is investigated next. Results are illustrated with the help of a base example. Computational results show that the model with shortages allowed is economically beneficial. Sensitivity of the solution to changes in the value of input parameters of the base example is also discussed.

This paper discusses a deterministic finite horizon economic production lot size (EPLS) model for deteriorating items under permissible delay in payments. Shortages are allowed and backlogged partially. Three models are developed over a finite planning horizon. The existence and uniqueness of the optimal solutions of the proposed models are examined. Here, production cycle time, the time at which the shortage begins, production stopping time, shortage ending time and total number of production cycles are taken as decision variables. Numerical examples are given to illustrate the model.

The objective of this paper is to provide a framework to integrate the existence of products with imperfect-quality items, inspection errors, rework and scrap items into a single Economic Production Quantity (EPQ) model. To achieve this objective, a suitable mathematical model is defined and the optimal production lot size that minimises the total cost is obtained. Sensitivity analysis is carried out for this model. The sensitivity analysis results indicate that the model is very sensitive to defective proportions and type 1 errors of inspection. Nowadays, given the progress made, inspection errors are often ignored. But the findings of this study show that the values of EPQ and total cost are very sensitive to type 1 errors of inspection. If the existence of such errors is ignored, then the obtained results will differ considerably from the optimal outcome. This will impose additional costs to the system.

This paper developed an integrated model in considering the situations of an imperfect process with imperfect maintenance and shortage, and an imperfect rework process for the determination of economic production quantity. This imperfect process has a general deterioration distribution with increasing hazard rate. Even with periodic preventive maintenance, such a production system cannot be recovered as good as new. As the process produces non-conforming items, a percentage of them are scrapped and the others are reworked. There is a percentage of the reworking items fail the repairing and become scrap items in the imperfect rework process. We use numerical analyses to simulate the effect of changes in the various parameters on the optimal solution. And look at the optimal inspection interval, inspection frequency and production quantity that maximum total profit which the advantages of rework process were explored.

The work presented in this paper is derived from an existing model for an imperfect process model (Moon et al., 2002) under constraint of setup time, and the total production time per cycle is equal to the cycle length. This paper considers the investment in setup cost reduction and quality improvement in: imperfect production process; imperfect production process with inspection and restoration. A solution algorithm using the Lagrangian approach with the objective of minimising the expected total cost is developed to simultaneously solve the cycle length, number of inspections, setup reduction and quality improvement for different products. Numerical examples are used to illustrate comparison of the performances with investment in setup cost reduction and quality improvement. [Received 11 August 2007; Revised 20 December 2007; Accepted 4 May 2008]

This paper deals with a closed-loop recycling system in which a stationary demand is satisfied by remanufactured products as well as newly manufactured products. In this system, only those used products that satisfy a minimum allowed quality level are bought back from customers for recycling. Thus the collection rate of used products is expressed as a function of the quality level and the unit bought-back price. It is assumed that the cycle consists of a single remanufacturing run and a single manufacturing run, both in Economic Production Quantity settings. Treating the quality of recycled products as a random variable, we develop a mathematical model with the objective of minimising the production and inventory costs. The decision variables are the length of the cycle, the minimum allowed quality level and the unit bought-back price of used products. To illustrate the validity of the model, sample problems are solved.

The model deals with an economic order quantity (EOQ) or economic production quantity (EPQ) model where the effect of 'advertising', 'sales price' and 'stock-display' is investigated. It is developed both for deteriorating and ameliorating items in the light of capacity constraint for storage facility and the limitation of the budget for advertising. The associated average profit function is maximised by calculus method. It is also illustrated by some numerical data for the test problem.

This paper describes an expert system that helps designers select a manufacturing process in the earliest stage of product design. First, the paper focuses on net-shape manufacturing processes and identifies the major factors that affect the selection of an appropriate process. A versatile methodology should consider all the factors simultaneously in assessing the suitability of the candidate processes. The proposed system uses the concept of design compatibility analysis to represent the suitability of candidate processes with respect to the given product specifications. The expert system uses this knowledge to eliminate incompatible candidates and rank the compatible set of processes. A prototype system called HyperQ/Process uses HyperCard and Prolog to implement the proposed methodology. HyperQ/Process also contains information related to each process.

This paper derives an Economic Order Quantity (EOQ)/Economic Production Quantity (EPQ) model with finite replenishment/production rate, taking into account the effect of the suppliers' trade credit and the retailers' promotional effort made by the advertising and the sales-team. The rate of demand in market is assumed to be a function of the selling price and the budget for the advertising and the sales-team. The objective of this paper is to determine the optimal selling price, replenishment period and the above budget so that the average net profit is maximised. The associated constrained maximisation problem is solved by Interior-Penalty-Function Method (Fiacco and Mc Cormick). The model is illustrated with a result, using numerical example and it provides some insights into the effect of the retailers' promotional effort.

In design and operation of processing systems, two categories of uncertainty play an important role: uncertainty with respect to the realization of continuous parameters (product demands, feedstock qualities, hest transfer coefficients, etc.) uncertainty with respect to discrete states (especially those related to equipment availability). This paper presents novel analytical tools to simultaneously account for both type of uncertainties in process design and evaluation.

The purpose of this paper is to examine the effect of manufacturing transfer batch sizes to financial performance, using an unbalanced production process as an exemplified situation. According to the most recent changes in the manufacturing business environment, as well as earlier research in the field (especially the Theory of Constraints (TOC) related research), manufacturing organisations should favour decisions related to lead time improvement and ignore the proposals of cost saving models (Economic Order Quantity, Economic Production Quantity). The results of this paper support this view and it seems that manufacturing organisations using transfer batching are able to serve versatile customer orders in a profitable manner, regardless of actual order size. On the contrary traditional management methods will lead to the situation, where some potential customer groups should be ignored. Despite these interesting results, this paper also proposes further avenues of research on the topic of transfer batching. Lead time improvement creates an interesting trade off situation between two different components (Lead time improvement based purely on transfer batch size and non-constraint cycle time savings).

This paper evaluates conventional lot-sizing rules in a multi-echelon coalescence MRP system. A part explosion diagram of three levels and twenty-one nodes is simulated using FORTRAN IV Level G. Nine separate lot-sizing methods were evaluated in this analysis. These methods included Lot for Lot, Economic Order Quantity, Periodic Order Quantity, Least Total Cost, Least Unit Cost, Part-Period Balancing, the Silver-Meal Algorithm, and the Wagner-Whitin Algorithm. A hybrid rule using both the Economic Order and the Economic Production Quantity rules was also evaluated. The performance of each lot sizing rule was simulated over nine different sets of market requirements patterns over a twelve month period. The types of demand variation included a constant rate, three different patterns of normally distributed demand, a random pattern, and two cyclic patterns. A hybrid pattern was used which equally weighted components of constant, random, normal, and cyclic demand. Finally, the ninth pattern consisted of actual data obtained from a job shop manufacturing facility. Within the part explosion diagram, ratios of setup cost to carrying cost, “goes into” quantities, and lead times were assigned for each node. Assigned values were selected from uniform distributions with prespecified ranges. A computer model was developed to perform the simulation. It consisted of an executive program, a routine for data generation, and separate routines to exercise each of the different lot-sizing rules. The simulations were conducted under three operational rules. The first rule allowed for the establishment of initial inventories just large enough to “cover” those gross requirements that occurred prior to the time the first order arrived. Carrying costs for this stock were included in the computation of total costs per node. The second rule provided for the delay of application of each lot sizing rule. This avoided receiving an order in a period of zero demand. The third rule addressed the computation of costs. The total cost was computed on the basis of average inventory level and the number of required setups. The analysis required the completion of 1701 separate simulation runs (9 rules X 9 demand patterns X 21 nodes). The performance of each rule was evaluated on the basis of total annual inventory cost. The Periodic Order Quantity (POQ) rule performed best in six of the nine demand patterns analyzed. In two of the remaining three cases, it ranked second on the basis of minimizing costs. The Least Unit Cost (LUC), Least Total Cost (LTC), and Pan-Period Balancing (PPB) algorithms demonstrated identical performance in four of the demand patterns analyzed. Generally, they ranked in the upper half of the rules evaluated. The Economic Order Quantity (EOQ) and the Economic Order/Production Quantity hybrid rules performed only moderately well. On the basis of cost, the consistent worst performers were the Wagner-Whitin (WW), Silver-Meal (SM), and Lot-for-Lot (LFL) rules. It was found that gross requirements tend to occur sporadically in different levels of the system. Order policies of parent nodes often cause the policies in higher level nodes to resemble the lot-for-lot order philosophy, regardless of the rule being used. Because of this phenomenon, those rules that generate fewer orders over the planning horizon for parent nodes often tend to perform better on the basis of total inventory cost.

Classical Economic Manufacturing Quantity (EMQ) models have usually ignored the possibility of process deterioration and the existence of defective items in the production lot. Moreover, the use of machine inspection for maintenance and restoration purposes has not been considered. On the other hand, past studies on maintenance models do not consider the length of a production run to be a decision variable. This paper addresses the problem of joint control of production cycles or manufacturing quantities and maintenance by inspection. A simple relationship has been developed to determine the effectiveness of maintenance by inspection. Furthermore, when maintenance by inspection is adopted, it is shown that the optimal inspection intervals are equally-spaced. The problem of simultaneous determination of EMQ and the inspection schedules is solved by using an approximation to the cost function. The resulting EMQ is found to be an adjustment to the classical EMQ. Finally, the relationships between different parameter values and the magnitudes of the cost penalty for using the classical EMQ are examined.

In this paper, we study the effects of an imperfect production process on the optimal production cycle time. The system is assumed to deteriorate during the production process and produce some proportion of defective items. The optimal production cycle is derived, and is shown to be shorter than that of the classical Economic Manufacturing Quantity model. The analysis is extended to the case where the defective rate is a function of the set-up cost, for which the set-up cost level and the production cycle time are jointly optimized. Finally, we also consider the case where the deterioration process is dynamic in its nature, i.e., the proportion of defective items is not constant. Both linear, exponential, and multi-state deteriorating processes are studied. Numerical examples are provided to illustrate the derivation of the optimal production cycle time in these situations.

An analytical approach is discussed to study the effect of varying marketing policies/parameters on the behaviour of a two-stage production/inventory system. In most manufacturing organizations, the production system cannot be considered to consist of independent sub-systems (such as inventory, marketing, finance, etc.). Realizing the practical relevance of the impact of various marketing instruments on the economic production quantity (EPQ) of a production system, a mathematical model has been developed here to estimate the values of EPQs for different marketing situations. The basic criterion considered here to optimize the batch size is the minimization of total annual variable cost. The total annual variable cost consists of cost elements such as set-up cost, and in-process and finished-product inventory costs. A numerical example is given to illustrate the application of the model developed.

Goyal [Goyal, S. K. 1976. Note on ‘Manufacturing cycle time determination for a multi-stage economic production quantity model’. Management Sci. 23 (3, November) 332–333.] suggests an extension of the EPQ model described in [Szendrovits, A. Z. 1975. Manufacturing cycle time determination for a multi-stage economic production quantity model. Management Sci. 22 (3, November) 298–308.]. His simultaneous optimization of both the lot size (Q) and the number of sub-batches (b) is valid provided that the transportation cost of sub-batches through all stages can be established and that the transportation cost function is of the form he suggests. We should, however, examine the basic nature of transportation costs in a plant and their usual allocation. In practice, any system of transportation equipment and personnel must be designed to cope flexibly with unscheduled activities. The timing of transport operations and the variability in loading the equipment have a great influence on costs. In addition, this transportation system must handle a whole spectrum of products, and it is difficult to allocate the costs of the system to particular product lots. As a consequence, transportation costs in a plant are usually regarded as sunk costs and are treated accordingly in most of the multi-stage inventory models.

The formulation of the classic economic production quantity (EPQ) model is extended to include setup cost as a function of capital expense. Additional capital will buy reduced setup cost. Thus, the objective now is to balance holding, setup, and capital expenses. This new formulation is solved under conditions where setup cost varies exponentially and linearly as a function of capital expense. Decision rules are formulated to indicate under what conditions setup cost reduction reduces total cost. For the linear function, it is shown that once the decision to reduce setup cost is justified, the optimal choice is the minimum setup cost that is technologically feasible.

This paper seeks to demonstrate that lower setup costs can benefit production systems by improving quality control. It does so by introducing a simple model that captures a significant relationship between quality and lot size: while producing a lot, the process can go "out of control" with a given probability each time it produces another item. Once out of control, the process produces defective units throughout its production of the current lot. The system incurs an extra cost for rework and related operations for each defective piece that it produces. Thus, there is an incentive to produce smaller lots, and have a smaller fraction of defective units. The paper also introduces three options for investing in quality improvements: (i) reducing the probability that the process moves out of control (which yields fewer defects, larger lot sizes, fewer setups, and larger holding costs); (ii) reducing setup costs (which yields smaller lot sizes, lower holding costs, and fewer defects); and (iii) simultaneously using the two previous options. By assuming a specific form of the investment cost function for each option, we explicitly obtain the optimal investment strategy. We also briefly discuss the sensitivity of these solutions to changes in underlying parameter values. A numerical example illustrates the results.

This paper analyzes the behaviour of a specific type of production system considering the decay of raw materials and in-process materials. The material which perishes with time will register a decrease in its stock level. The decrease in inventory level has considerable impact on inventory policy of a production system. Realising the importance of the effect of perishability of the products, an attempt has been made in this paper to investigate the effect of decaying materials on the Economic Production Quantity (EPQ) and Economic Order Quantity (EOQ). The product is manufactured in batches and the raw materials are obtained from outside suppliers. The proposed model integrates the inventory problem of raw materials and the finished product of a two stage production system manufacturing a single product. The production system considered here allows the procurement of several batches of raw materials at the same time. The basic criterion to estimate the EPQ and EOQ is the minimization of total variable cost.

The classical economic production quantity (EPQ) model assumes that items are produced by a perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be perfect without a price, its set-up cost can be reduced with investment in flexibility improvement. An EPQ model with a flexible and imperfect production process is proposed in this paper. This inventory optimization problem is then formulated as a geometric program (GP) and solved to obtain closed-form optimal solutions. After the theoretical treatment, a numerical example is provided to illustrate how the GP theories are applied to solve a given problem. Finally some aspects of post-optimality sensitivity analysis based on the GP approach are discussed.

Production lot-sizing has a special significance in supply chain taking into account the fact that majority of the lot-sizing problems are associated with NP-hard scheduling and sequencing problems. The complexity increases exponentially when multi-item capacitated dynamic lot-sizing is considered. The basic economic production quantity (EPQ) model minimizes the sum of setup and holding cost under certain favorable assumptions. However, when assumptions are removed by introducing more complex constraints, the solution procedure becomes extremely difficult to solve. As a result NP-hardness arises which necessitates the use of heuristics. The objective of this paper is to minimize the sum of setup and inventory holding costs over a time horizon subject to constraints of capacity limitations and elimination of backlogging. As reports reveal, algorithm for an optimal solution exists in case of a single item production. But for multi-item problems, no algorithm exists which can provide global optimality. This paper develops a model for multi-item problem with setup time. Based on the model a program has been executed and feasible solutions have been obtained for the bench-mark data.Keywords: Capacitated scheduling; heuristics; multi-item; production lot-sizing.DOI: 10.3329/jme.v40i2.5352Journal of Mechanical Engineering, Vol. ME 40, No. 2, December 2009 110-118

This paper formulates a two-echelon singleproducer multi-buyer supply chain model, while a single product is produced and transported to the buyers by the producer. The producer and the buyers apply vendormanaged inventory mode of operation. It is assumed that the producer applies economic production quantity policy, which implies a constant production rate at the producer. The operational parameters of each buyer are sales quantity, sales price and production rate. Channel profit of the supply chain and contract price between the producer and each buyer is determined based on the values of the operational parameters. Since the model belongs to nonlinear integer programs, we use a discrete particle swarm optimization algorithm (DPSO) to solve the addressed problem; however, the performance of the DPSO is compared utilizing two well-known heuristics, namely genetic algorithm and simulated annealing. A number of examples are provided to verify the model and assess the performance of the proposed heuristics. Experimental results indicate that DPSO outperforms the rival heuristics, with respect to some comparison metrics.

In a recent paper, Chiu et al. (2014) utilized mathematical modeling and differential calculus to determine the common production cycle time that minimizes total production, inventory, and delivery costs for a multi-item economic production quantity (EPQ) model with scrap, rework, and multi-delivery. The present study proposes an algebraic approach substituting the use of differential calculus on the system cost function for deriving the optimal common cycle time in the aforementioned multi-item EPQ model. This simplified approach may enable managing practitioners to resolve real multi-item EPQ models more effectively. All Rights Reserved © 2015 Universidad Nacional Autonoma de Mexico, Centro de Ciencias Aplicadas y Desarrollo Tecnologico. This is an open access item distributed under the Creative Commons CC License BY-NC-ND 4.0.

Recently, a cost minimization method to determine the lot size for the EOQ/EPQ models with backorders was published. This method is based on the well-known arithmetic-geometric mean inequality. Although the cost minimization method is correct and interesting, it does not focus on deriving the backorders level. This paper proposes another simple approach. The proposed method finds both the lot size and the backorders level. © 2009 Elsevier Ltd. All rights reserved.

A multi-product production system with several identical facilities is considered over a finite planning horizon. The horizon consists of H periods during each of which production and demand rates are constant, and product inventory changes during any period are assumed to occur linearly. All demands must be met without allowing backorders. The scheduling criterion is to minimize the sum of setup. production and inventory carrying costs. The problem is formulated as a pure integer programming, and single pass allocation method is given when the special case \( {\psi _{B\;}}\mathop > \limits_ = \;{\psi _{A\;}} + \;{S_A}\; + \;{S_B} \) holds in 2-product problem. The branch and bound algorithm is used to general cose. At each lower bounding stage, the primal-dual approach is applied to obtain an LP solution of candidate problem with the advantage of the special structure of coefficient matrix.

Note on “Manufacturing Cycle Time Determination for a Multi-Stage Economic Production Quantity Model” [Szendrovits, A. Z. 1975. Manufacturing cycle time determination for a multi-stage economic production quantity model. Management Sci. 22 (3, November) 298–308.].