A note on the deterministic EPQ with partial backordering

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

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

Emissions trading is a market-based mechanism for curbing emissions, and it has been implemented in Europe, North America, and several other parts of the world. To study its impact on production planning, we develop a dynamic production model, where a manufacturer produces a single product to satisfy random market demands. The manufacturer has access to both a green and a regular production technology, of which the former is more costly but yields fewer emissions. To comply with the emissions regulations, the manufacturer can buy or sell the allowances in each period via forward contracts in an outside market with stochastic trading prices while needing to keep a nonnegative allowance account balance at the end of the planning horizon. We first derive several important structural properties of the model, and based upon them, we characterize the optimal emissions trading and production policies that minimize the manufacturer's expected total discounted cost. In particular, the optimal emissions trading policy is a target interval policy with two thresholds that decrease with the starting inventory level. The optimal production policy is established by first determining the optimal technology choice and then showing the optimality of a base-stock type of production policy. We show that the optimal base-stock level is independent of the starting inventory level and the allowance level when the manufacturer trades the allowance or uses both technologies simultaneously. A numerical study using representative data from the cement industry is conducted to illustrate the analytical results and to examine the value of green technology for the manufacturer.

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.

IntroductionRecent Developments in Monetary Macroeconomics David Altig The contents of this volume hardly require explanation beyond its title: "Recent Developments in Monetary Macroeconomics." Our intent for the conference was to collect a set of papers reflecting the cutting edge of applied monetary macroeconomics. As befitting such an ambitious-sounding goal, the contributions are wide ranging. For purposes of this brief introduction, however, we might organize the papers as answers to three questions. (1) What is the "optimal" Taylor rule? (2) Are "New Keynesian" or "New Neoclassical Synthesis" models the final word on the monetary transmission mechanism? (3) Is there a role for money in the conduct of monetary policy? What is the "Optimal" Taylor Rule The inclusion of the Taylor rule issue is almost a prerequisite for any collection of papers purporting to survey recent developments in monetary macroeconomics. In policy discussions, the Taylor rule is ubiquitous, and it is currently the choice among alternative specifications of central bank behavior. More precisely, perhaps, the general form of the Taylor rule is the choice, as it has come to represent the general class of equations that relate the federal funds rate to some measure of an output gap and inflation rate. There is substantially less unanimity about whether output gaps and inflation rates should be past, present, or (expected) future values, whether past values of the funds rate need to be included, and what are the appropriate magnitudes of the responses to each of these measures. Marc Giannoni and Michael Woodford offer the natural approach to addressing the dispute: find the representation that is optimal within the framework being employed for policy analysis. The model in question here is essentially a derivative of the "New Neoclassical Synthesis" class of models that are currently the workhorses of most monetary policy analyses among academic and central bank staffs alike. Their approach to discovering the optimal policy within this structure has the flavor of [End Page 1039] the "Ramsey problem" familiar from optimal tax policy, although with the strong requirement that the derived policy rule be "robustly optimal": it must support the optimal equilibrium no matter what the distribution of disturbances the model policymakers face. Giannoni and Woodford offer two essential lessons. First, whether optimal policy incorporates forecasts of future price-level growth or output gaps depend critically on the dynamics of the inflation rate. If the adjustment of the price-level to shocks is inertial, then optimal policy necessarily depends on forecasts of future inflation. Second, the response of the funds rate to its own past is inertial. In fact, it is super-inertial, meaning that (all else equal), the contemporaneous funds rate responds more than one-for-one with the lagged value of the funds rate (and lagged changes in the rate). The proposition that monetary authorities ought to aggressively respond to lagged values of the funds rate also arises in the papers by Jess Benhabib, Stephanie Schmitt-Grohé, and Martín Uribe, and George Evans and Seppo Honkapohja. In the latter case, the authors consolidate and expand on their well-known work on learning dynamics. A central contribution of the work presented in this article is the notion that convergence to a unique rational expectations equilibrium under learning, as well as the stability of that equilibrium, serves as basis for the choice (or rejection) of an optimal policy formulation. An apparent lesson from Evans and Honkapohja's analysis is that the learnability criterion prescribes a policy rule that differs in some important ways from the conventional wisdom coming from analyses that invoke the Taylor rule in a pure rational expectations environment. In particular, they conclude that the optimal policy rule in the environment they consider requires the monetary authority to respond directly to private-sector expectations. The environment they consider is essentially the same as in Giannoni and Woodford, with the exception of the central bank's assumed loss function: Giannoni and Woodford assume a preference for interest rate smoothing, Evans and Honkapohja do not. In his comments, John Duffy points out that the introduction of interest-smoothing motive yields an optimal policy rule under learning that is much closer to the conventional view. In particular, it does not require...

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.

In this paper, we develop a dynamic production-inventory control model to determine the optimal production planning for a manufacturer facing stochastic demand and carbon emission allowance price under cap and trade. We derive several structural properties of the model, and characterize the optimal production policies that minimize the expected total discount cost over a finite planning horizon. Specifically, when there is no setup cost for production, the optimal production policy is a hybrid version of two base stock policies; when there has setup cost, the optimal production policy is a hybrid version of a (s, S) and a generalized (s, S) policy.

This paper models a type of vendor-managed inventory (VMI) agreement that occurs in practice called a (z, Z) contract. We investigate the savings due to better coordination of production and delivery facilitated by such an agreement. The optimal behavior of both the supplier and the retailer are characterized. The optimal replenishment and production policies for a supplier are found to be up-to policies, which are shown to be easily computed by decoupling the periods when the supplier outsources from those when the supplier does not outsource. A simple application of the newsvendor relation is used to define the retailer's optimal policy. Numerical analysis is conducted to compare the performance of a single supplier and a single retailer operating under a (z, Z) VMI contract with the performance of those operating under traditional retailer-managed inventory (RMI) with information sharing. Our results verify some observations made in industry about VMI and show that the (z, Z) type of VMI agreement performs significantly better than RMI in many settings, but can perform worse in others.

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

The problem of finding an optimal advertising and production policy in a firm is analysed using a recently proposed model of a marketing‐production system. First, it is shown that the optimal control problem underlying the proposed model is a partially singular control problem. Then, a reverse‐time parametric solution procedure is designed to determine the optimal advertising and production policy for the proposed model. Finally, it is shown that the results derived from the new model are applicable to problems of capacity expansion in a firm.

Problem definition: Production systems deteriorate stochastically due to use and may eventually break down, resulting in high maintenance costs at scheduled maintenance moments. This deterioration behavior is affected by the system’s production rate. Although producing at a higher rate generates more revenue, the system may also deteriorate faster. Production should thus be controlled dynamically to tradeoff deterioration and revenue accumulation in between maintenance moments. We study systems for which the relation between production and deterioration is known and the same for each system and systems for which this relation differs from system to system and needs to be learned on-the-fly. The decision problem is to find the optimal production policy given planned maintenance moments (operational) and the optimal interval length between such maintenance moments (tactical). Methodology/results: For systems with a known production-deterioration relation, we cast the operational decision problem as a continuous time Markov decision process and prove that the optimal policy has intuitive monotonic properties. We also present sufficient conditions for the optimality of bang-bang policies, and we partially characterize the structure of the optimal interval length, thereby enabling efficient joint optimization of the operational and tactical decision problem. For systems that exhibit variability in their production-deterioration relations, we propose a Bayesian procedure to learn the unknown deterioration rate under any production policy. Numerical studies indicate that on average across a wide range of settings (i) condition-based production increases profits by 50% compared with static production, (ii) integrating condition-based production and maintenance decisions increases profits by 21% compared with the state-of-the-art sequential approach, and (iii) our Bayesian approach performs close, especially in the bang-bang regime, to an Oracle policy that knows each system’s production-deterioration relation. Managerial implications: Production should be adjusted dynamically based on real-time condition monitoring and the tactical maintenance planning should anticipate and integrate these operational decisions. Our proposed framework assists managers to do so optimally. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant 439.17.708]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2022.0473 .

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

Summary A new and simple decline formula facilitates the search for an optimal policy of production with or without compressers from a number of gas reservoirs of different characteristics. Optimizing production policy traditionally is carried out by trial and error. The paper leaves this procedure essentially unaltered but reduces the computer time per trial, thus allowing more trials. Results of some sample calculations are presented. Introduction Where numerous wells producing from several as reservoirs feed into a common pipeline system, a great number of alternative production policies present themselves, especially where gas has to be compressed to bring it up to pipeline pressure. To study the alternatives one needs the basic information regarding performance of the individual wells. To describe such performance one may choose among several formulas available in the literature, which are all based on the assumption that, apart from pressures. the physical conditions remain unchanged. A prudent prediction. however, will allow for possible deterioration or even complete loss of one or more wells. On the other hand, one is at liberty, of course, to postulate replacement and infill drilling. Performance of a future producer has to be estimated. Hence the combined performance of all wells in a single gas reservoir can be predicted with only limited accuracy. Once this unavoidable inaccuracy is accepted we must conclude that it is not justified and not desirable to waste computer capacity on excessively accurate well production performances-, computer capacity is employed more usefully on a large number of alternative production policies. This philosophy is worked out in the following chapters. For the time being, we have limited ourselves to reservoirs, without water drive. Performance and Decline Traditionally, inflow performance is defined as the dependence of production rate on bottomhole pressure (BHP). At low rates Darcy's law prevails and the inflow expressed at standard conditions is described by q = C (p2-pw2) (1) This formula is obtained by assuming that the gas has a constant viscosity and follows Boyle's law. At higher rates the flow becomes turbulent in the proximity of the well. There is an upstream section with a linear resistance and a downstream one with a quadratic resistance This leads to (2)Aq+Bq2=p2-pw2 This well-known formula is not exact, because the higher the rate, the longer the section with a quadratic resistance. JPT P. 2285^

Optimal integrated production and inventory control of an assemble-to-order system with multiple non-unitary demand classes

The paper investigates the optimal purchasing and production control policy within a circular production system that processes two returned material types through a shared manufacturing center. Materials are purchased or rejected upon arrival, with differences in procurement cost. At each decision epoch, the system must decide whether to purchase incoming materials and which type to prioritize for production. The problem is formulated as a Markov decision process, and a linear programming approach is developed to compute the optimal purchasing and production policy for the system under both non-preemption and preemption rules. Theoretically, the structure of the optimal policy is analytically characterized under the preemption rule in the case of equal production rates. Numerical experiments reveal that this policy remains optimal or near-optimal for the system under non-preemption and preemption rules with varying production rates. Additionally, comparisons of the optimal policies between non-preemption and preemption rules demonstrate that the system under the preemption rule yields higher average reward than that under the non-preemption rule, to varying degrees depending on system parameters, such as production rate, holding cost, and arrival rate.