A note on “The EPQ with partial backordering and phase-dependent backordering rate”

A note on the deterministic EPQ with partial backordering

Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost

We give an alternative proof of the optimality of the solution for the deterministic EPQ with partial backordering (EPQ-PBO) [Omega, vol. 37, no. 3, pp. 624–636, 2009]. Our proof is based on the mathematical programming theory. We also demonstrate the determination of the optimal decision policy through solving the corresponding mathematical programming problem. We indicate that the same approach can be used within other inventory models with partial backordering, and we consider additional models.

Economic production lot size model for deteriorating items with partial back-ordering

Purpose To examine the effect of order-size dependency on all-unit discount and the trade credit period at two levels, and formulate a total combined profit function for the manufacturer and the supplier. Also, to test this model by taking a numerical example and performing a sensitivity analysis to gain managerial insights and results. Design/methodology/approach This work presents an integrated inventory model with a bi-level trade credit period and an all-unit discount dependent on order size under capacity constraints. To address stockouts, this model uses a partial backorder approach. The concavity of the profit function is shown and an algorithm is developed to find the optimal decision policy for the manufacturer and supplier. Findings The supplier places more frequent orders to take benefit of trade credit whenever demand and interest earned rate increase. Joint profit generated by both the manufacturer and supplier grows, even though the manufacturer offers a price discount to the supplier. The supplier should order an amount of inventory that can be accommodated in their own warehouse instead of renting an additional warehouse. Originality/value Till now, as per our best knowledge, there is no existing literature that has simultaneously considered order size-dependent two-level trade credit, capacity constraint, all-unit discount linked to order purchase price and partial backordering. Combining these factors, this paper derives an integrated inventory model to maximize the total combined profit of the manufacturer and supplier. These unique factors set this paper apart from existing literature, providing distinct advantages.

Most of the inventory models discuss two extreme situations with respect to the demand in stock-out period. In this research, the demand rate within stock-out period is assumed as a diminishing function of backorders. Shortage of inventory will lead to the opportunity cost of lost sales. This research considers backorders and lost sales to formulate an inventory model with partial backorders. The objective of the inventory model is to determine the time scales of inventory depletion and stock-out periods in order to minimise total relevant cost. An efficient algorithm is developed to find the optimal solution. Finally, a sensitivity analysis is performed to study the effects of the model parameters on the optimal solution. This research apparently makes an improvement compared to the previous works, and further fits the requirements of practical environment.

THE PURPOSE of the dissertation is to integrate financial intermediation into a theoretical model of real economic output and growth. The methodology involves construction of a model in which capital accumulation is based upon an interest-elastic demand for the capital stock. This model is shown to be capable of adjusting to an equilibrium capital intensity and saving share. To this growth model is added a financial intermediary sector to form a two-sector model of an economy. The production sector uses real inputs of capital and labor to produce a homogeneous good which is then allocated to consumption and real capital accumulation. The financial intermediary sector uses real inputs of capital and labor to produce financial intermediary services. Financial intermediary output is described by a production function for the financial intermediary in which real capital and labor inputs are related to the units of real capital placed by the financial intermediary with production sector firms. That is, for each unit of real capital used to produce the homogeneous good, the financial intermediary expends real resources of capital and labor to gather capital from surplus households in order to place this capital with deficit production units. It is assumed that at the firm level the financial production functions exhibit the usual decreasing average product after some point, which leads to increasing average unit costs. At the aggregate sector level the financial production function exhibits constant returns to scale and constant unit costs, as expansion is carried out by the introduction of new and similarly efficient financial intermediaries. The financial intermediary thus employs real capital and labor in profit-maximizing quantities and pays these factors the competitive factor returns. The intermediary pays deposit rates to attract deposit capital which it then lends to production sector units and earns a loan rate equal to the marginal product of capital. The financial intermediary earns unit revenues equal to the difference between the loan rate and deposit rate. This quantity, called the unit cost of finance, is a per period rate per unit of capital. Total revenues of the intermediary are then composed of these per unit charges. When competition prevails in the financial intermediary market for deposit capital the intermediary is forced to pay a high enough deposit rate so that the intermediary merely covers its real factor input costs. The intermediary thus is capable of adjustments in deposit rates and factor inputs; and in a competitive equilibrium situation it produces financial services at minimum average costs. The deposit rate in a competitive equilibrium is then equal to the production sector marginal product of capital (loan rate) less the minimum average unit cost of financial services. The solution for the variables of the system are determined by a simultaneous solution of the sectoral production functions, the capital demand function (saving function), and capital and labor allocation equations (perfect substitutability). The model is solved for the sectoral capital intensities, the distribution of labor and capital between sectors, the real rates in the system (deposit rate and marginal product of capital), as well as aggregate and per capita income, investment and consumption.

This research presents an inventory system with partial backordering and imperfect process with a stochastic numbers of products that are defective per order. This problem is known and there are some customers that do not wait for their orders to be fulfilled so a particular proportion of backordered items become lost sales. This paper considers both mentioned situations simultaneously while the number of defective items follows a uniform distribution and the proportion of backordering is constant. This condition is modelled. The cost function of this inventory model includes order cost, holding cost and two types of shortage costs, one of them is related to backordered items and the other one is related to lost sales. This paper also provides a solution method to obtain optimum values for the decision variables, order quantity and total shortage, then we derive the value of the total cost function according to the optimum values obtained for the decision variables. Finally, some numerical results and diagrams are provided to show how some parameters affect the values of the decision variables and the cost function.

The vendor managed inventory (VMI) is an efficient coordination policy in supply chain management, in which supplier is responsible to manage inventory at the buyer and decides on replenishment policies. This paper presents a VMI model for a supply chain problem with a single supplier, multi-buyer and multi-product, via a production-inventory system in which shortage is allowed and partially backordered. In order to take into account the concern about environmental issues, the total green house gas emission is also considered as a green constraint. The aim is to find the appropriate values of production quantity and the maximum shortage level for all products of all buyers in such a way that the total cost of supply chain is minimized under VMI. The problem is formulated as a non-linear programming model and then a decomposition based analytical approach is proposed to solve it optimally. At the end, a numerical example is discussed to illustrate the proposed approach.

In this paper optimal and near optimal policies are proposed for multi-echelon production/inventory assembly systems under continuous review with constant demand over an infinite planning horizon. Costs at each stage consist of a fixed charge per order or production setup plus a linear holding cost on “echelon” inventory. The objective is minimization of average cost per unit time. The major results of this paper are: a mathematically simple, often optimal, “system myopic” solution, a lower bound on the closeness to optimality of this solution, and a branch and bound algorithm which usually finds the optimal solution quickly.

A two level storage inventory model is constructed in this article. It is well known that the demand for seasonal products (such as fur coats) increases at the beginning of the season until a certain period of time and stabilizes into a fixed amount of time for the rest of the season. To store these extra parts for the buyer arrange additional storage space. This model uses a ramp type demand rate, variable deterioration and shortages are partially backlogged using a variable backordering rate. The entire research is conducted in an inflationary environment. The goal of this model is to reduce the system's total average cost. A numerical assessment and sensitivity analysis are used to verify the suggested model's optimal solution.

This paper presents an integrated single-vendor two-buyer production-inventory model with stochastic demand and service level constraints. Shortage is permitted in the model, and partial backordered partial lost sale. The lead time demand is assumed follows a normal distribution and the lead time can be reduced by adding crashing cost. The lead time and ordering cost reductions are interdependent with logaritmic function relationship. A service level constraint policy corresponding to each buyer is considered in the model in order to limit the level of inventory shortages. The purpose of this research is to minimize joint total cost inventory model by finding the optimal order quantity, safety stock, lead time, and the number of lots delivered in one production run. The optimal production-inventory policy gained by the Lagrange method is shaped to account for the service level restrictions. Finally, a numerical example and effects of the key parameters are performed to illustrate the results of the proposed model.

The stagecoach problem is a special type of network analysis problem in which the cities (nodes) are arranged in stages. By such human or natural arrangement, a journey from City 1 in stage 1 to City n in stage n involves visiting only one city in each intermediate stage. The stagecoach problem involves the determination of the minimum cost flow in which basically two methods have been in use. One of the methods is by listing all possible routes, computing their corresponding costs and selecting the minimum cost route. This procedure entails much computational effort. The second procedure is to determine the suboptimal cost of routes between any two consecutive stages. Though this second approach has less computational effort, its overall optimal policy could be wrong because the choice of a least expensive route at one stage may result in our adopting a more expensive route at a later stage. In this paper we present a recursive dynamic programming algorithm for solving the stagecoach problem. The algorithm is computationally more efficient than the first method as it obtains its minimum total cost using the suboptimal policies of the different stages without computing the cost of all the routes. By the dynamic programming algorithm of this paper, the possibility of missing the minimum cost route is ruled out as could happen in the second approach. The dynamic programming algorithm for obtaining the minimum cost path in a stagecoach problem is numerically illustrated Keywords : Network analysis, stagecoach, dynamic programming algorithm Global Journal of Mathematical Sciences Vol. 7 (1) 2008: pp. 15-20

Due to the great concern with sustainable development, new Economic Production Quantity (EPQ) models are no longer restricted only to traditional economic issues. However, they also focus on sustainability issues, especially regarding environmental aspects. However, the studies addressing the EPQ models together with sustainable issues, remain poorly explored in the literature. Still, these theoretical EPQ models mostly consider essential aspects which simplify the real-life conditions, such as deterministic demand. Thus to fill this gap, the present paper proposes an EPQ model with partial backorder considering environmental issues and stochastic demand, addressing, therefore, more realistic aspects of the real world. To find the optimal solution, we solve the maximisation problem, and we provide a numerical example with sensitivity analysis. We conclude that demand is the parameter that most affects the total profit.KeywordsEPQEnvironmental aspectsStochastic demand

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