Fixed Order Cost Research Articles

Optimal inventory control with cyclic fixed order costs

We consider a periodic review single‐item inventory model under stochastic demand. Every m periods, in the regular order period, fixed order costs are K. In the periods in‐between, the intraperiods, higher fixed order costs of [Formula: see text] apply. The literature on optimal inventory policies under fixed order costs does not account for these time‐dependent fixed order costs. By generalizing existing proofs for optimal inventory policies, we close this gap in inventory theory. The optimal inventory policy is complex in the regular order period and in the intraperiods, a period‐dependent [Formula: see text] policy is optimal. We describe and prove this optimal policy based on the notion of K‐convexity and the optimal ordering behavior in the presence of non– K‐convex cost functions. In a numerical study, we find that a major driver of the optimal policy is a forward‐buying effect that shifts the probability of ordering from the intraperiods to the regular order period. The cost differences between the optimal and a pure period‐dependent [Formula: see text] policy are, however, small.

The aggregate effects of global and local supply chain disruptions: 2020–2022

We study the aggregate effects of supply-chain disruptions in the post-pandemic period in a heterogeneous-firm, general equilibrium model with input-output linkages and a rich set of supply chain frictions: uncertain shipping delays, fixed order costs, and storage costs. Firms optimally hold inventories that depend on the source of supply, domestic or imported. Increases in shipping times are contractionary, raise prices, and increase stockouts, particularly for goods intensive in delayed inputs. These effects are larger when inventories are already at low levels. We fit the model to the U.S. and global economies from 2020 to 2022 and estimate large aggregate effects of supply disruptions. Our model predicts that the boost in output from reducing delays will be smaller than the contraction from the waning effects of stimulus.

Integrating Replenishment Policy and Maintenance Services in a Stochastic Inventory System with Bilateral Movements

We study an inventory control problem with two storage facilities: a primary warehouse (PW) of limited capacity M, and a subsidiary one (SW) of sufficiently large capacity. Two types of customers are considered: individual customers arriving at (positive and negative) linear rates governed by a Markov chain, and retailers arriving according to a Markov arrival process and bringing a (positive and negative) random number of items. The PW is managed according to a triple-parameter band policy (M,S,s),0≤s<S≤M, under a lost sales assumption. Under this policy, as soon as the stock level at the PW falls below s, a refilling to S is performed by a distributor after a random lead-time. However, if the stock exceeds level S when the distributor arrives, no refilling is carried out, and only maintenance services are performed. Items that exceed level M are transferred to the SW at a negligible amount of time for those used in related products. Our cost structure includes a fixed order cost, a variable cost for each item supplied by the distributor, a cost for the additional maintenance, a salvage payment for each transferred item from the PW to the SW, and a loss cost for each unsatisfied item due to demands. We seek to determine the optimal thresholds that minimize the expected overall cost under the discounted criterion. Applying first-passage time results, we present a simple set of equations that provide managers with a useful and an efficient tool to derive the optimal thresholds. Sensitivity analysis and fruitful conclusions along with future scope of research directions are provided.

Periodic review inventory models with multiclass demands and fixed order costs

We consider a periodic review inventory system with multiclass demands and fixed setup cost. Demand arrivals of each class are assumed to be a Poisson process, and a lost-sales setting is adopted. The demand classes are classified by the cost of their unsatisfied demands. We consider two cases: the leftover inventory at the end of a restocking interval is entirely discarded or entirely carried over to the next period. We obtain the optimal rationing policy, the optimal ordering policy and the optimal duration of the periodic review interval that minimize the average cost per unit time. We derive the differential equations satisfied by the value function characterized by the on-hand inventory level and the residual restocking time. This value function does not have the traditional modularity and convexity properties. Hence, the optimal policy is derived directly based on the ordinary differential equations satisfied by the value function. Moreover, some structural properties of the optimal policy such as the monotone property of the optimal rationing policy are obtained.

Replenishment and transshipment in periodic-review systems with a fixed order cost

Small businesses with many stocking locations face high ordering costs and imbalanced inventories. To address those challenges, we consider an inventory system of multiple retailers: each has Poisson demand and backorders are permitted. They order together periodically and there is an opportunity to transship available inventory between retailers in each order cycle. Each order incurs a fixed cost, and there is a variable transshipping cost, a holding cost for inventory, and a penalty cost for backorders. The system aims to determine a feasible ordering and transshipping plan which minimizes the long-run average total cost over time. We provide an analytical approach to evaluating costs, construct bounds to search for the optimal order interval, order-up-to level, and timing of transshipment, and determine the amounts to transship based on a marginal cost. Our numerical studies find that, first, the optimal order interval is between that in the no-transshipment case and an EOQ-like calculation. Second, the optimal timing of transshipments is in the middle of an order cycle or a little later. Third, the optimal order interval and timing of transshipments are insensitive to the introduction of differences between retailers. Fourth, transshipment effectively reduces costs, especially when transshipment costs are low or different between retailers. These findings can simplify the task of cost optimization in complex inventory systems and develop practical measures to further reduce costs.

Optimal Ordering Policy for TwoProduct Inventory Models with Fixed Ordering Costs

Optimal Ordering Policy for TwoProduct Inventory Models with Fixed Ordering Costs

Dynamic Pricing and Inventory Control with Fixed Ordering Cost and Incomplete Demand Information

We consider the periodic review dynamic pricing and inventory control problem with fixed ordering cost. Demand is random and price dependent, and unsatisfied demand is backlogged. With complete demand information, the celebrated [Formula: see text] policy is proved to be optimal, where s and S are the reorder point and order-up-to level for ordering strategy, and [Formula: see text], a function of on-hand inventory level, characterizes the pricing strategy. In this paper, we consider incomplete demand information and develop online learning algorithms whose average profit approaches that of the optimal [Formula: see text] with a tight [Formula: see text] regret rate. A number of salient features differentiate our work from the existing online learning researches in the operations management (OM) literature. First, computing the optimal [Formula: see text] policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in OM that only require solving single-period optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profit-to-go function. The necessity of analyzing action-dependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult than existing bandit learning algorithms. Second, the pricing function [Formula: see text] is of infinite dimension, and approaching it is much more challenging than approaching a finite number of parameters as seen in existing researches. The demand-price relationship is estimated based on upper confidence bound, but the confidence interval cannot be explicitly calculated due to the complexity of the DP recursion. Finally, because of the multiperiod nature of [Formula: see text] policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy [Formula: see text], which is unknown to the learner a priori. In this paper, the demand randomness is approximated by an empirical distribution constructed using dependent samples, and a novel Wasserstein metric-based argument is employed to prove convergence of the empirical distribution. This paper was accepted by J. George Shanthikumar, big data analytics.

End-to-End Deep Learning for Inventory Management with Fixed Ordering Cost and its Theoretical Analysis

We herein investigate the single item, multi-period inventory management problem with fixed ordering cost, holding cost, and lost sales cost. Here, both the demand and vendor lead time (VLT) are uncertain; Additionally, the contextual information that may be useful for predicting the distribution of the demand and VLT is available. We assume that the retailer has access to rich history observations and wishes to decide: (1) when to place an order and (2) how much to order. We solve this problem via an end-to-end (E2E) learning approach, where we develop a deep learning framework that directly outputs the optimal order timing and quantity with the given contextual information as the input. Our numerical experiments, which use the real-world data obtained from an online retailer, demonstrate the advantages of our approach over benchmark methods, such as the (r, Q) policy, predict-then-optimize (PTO) framework, and the E2E method with given order timing. Furthermore, we analyze the convergence of our E2E approach and provide upper bounds for the expected daily average cost of our E2E approach under certain conditions. To compare our approach with the optimal policy, we also provide lower bounds for the cost of the optimal policy under certain assumptions.

Dynamic Pricing and Inventory Control with Fixed Ordering Cost and Incomplete Demand Information

We consider the periodic review dynamic pricing and inventory control problem with fixed ordering cost. Demand is random and price dependent, and unsatisfied demand is backlogged. With complete demand information, the celebrated (s,S,p) policy is proved to be optimal, where s and S are the reorder point and order-up-to level for ordering strategy, and p, a function of on-hand inventory level, characterizes the pricing strategy. In this paper, we consider incomplete demand information and develop online learning algorithms whose average profit approaches that of the optimal (s,S,p) with a tight O (√T) regret rate. A number of salient features differentiate our work from the existing online learning researches in the OM literature. First, computing the optimal (s,S,p) policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in operations management that only require solving single-period optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profit-to-go function. The necessity of analyzing action-dependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult than existing bandit learning algorithms. Second, the pricing function p is of infinite dimension, and approaching it is much more challenging than approaching a finite number of parameters as seen in existing researches. The demand-price relationship is estimated based on upper confidence bound, but the confidence interval cannot be explicitly calculated due to the complexity of the DP recursion. Finally, due to the multi-period nature of (s,S,p) policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy p, which is unknown to the learner a priori. In this paper, the demand randomness is approximated by an empirical distribution constructed using dependent samples, and a novel Wasserstein metric based argument is employed to prove convergence of the empirical distribution.

A Robust Cycle-Based Policy for Inventory Management with Fixed Ordering Costs

We consider a single-installation, single-product inventory management problem with positive fixed ordering costs. We allow the unfulfilled demand to be either backlogged or lost and the lead time to be either zero or positive, and assume that the stochastic demand is correlated and information about the demand distribution is limited. A simple implication of non-negligible fixed ordering costs is that any viable inventory control policy implicitly divides the planning horizon into a sequence of {\em inventory cycles}, where an order is only placed at the beginning of the cycles. Building on this observation, we propose a new policy that manages inventory by directly selecting the order quantity and the cycle length for each cycle using robust optimization. Our cycle-based policy offers a simple and unified approach to solving the inventory problem, and we show that the policy decisions can be computed extremely efficiently by solving a small number of linear programs for all model instances (i.e., backlogging versus lost sales, zero lead time versus positive lead time). To the best of our knowledge, our policy is the first robust policy that can accommodate lost sales, and also the first linear programming-based robust policy that can handle positive fixed ordering costs. Results of extensive numerical experiments show that our policy outperforms several strong benchmark policies.

An optimization model for investment in technology and government regulation

Companies struggle every day to estimate the adequate level of investment in new technologies, and governments lack the tools to determine the impact of their regulations on industry including telecommunications networks. Despite these facts, few studies discuss ways to assess appropriate levels of investment for technological initiatives and government regulations. To fill this gap, this study provides an optimization model for the investment of technology and government regulation, based on efficiencies. Results obtained from surveying northern European companies support the importance of estimating investment in technology and government regulation levels. The survey identified the four most relevant factors for practitioners: quality, cost, technology adoption, and government regulations. Based on the survey’s results, the model evaluates the level of investment for technology adoption and government regulations using cost and quality as target variables. Additional data from a German carrier served to test the model. Results show that technology investment delivers more benefits in cost and quality by increasing technology adoption. However, the model also suggests that diminishing returns make efficiencies stall at a certain level of technology adoption, and shows an investment threshold dependent on the type of benefit, cost, or quality the company seeks to maximize. Regarding government regulation, the model shows a counterintuitive behavior at higher levels of investment for the cost coefficients and at all levels of investment for the quality coefficient. This suggests that government regulation effects could be shifting from fixed-order cost to other types of costs.

Optimization of (${R}$ , ${Q}$ ) Policies for Assembly Inventory Systems With Operating Flexibility

Guaranteed-Service Approach (GSA) was used to set safety stock for multi-echelon inventory systems. This approach assumes that each stock can use operating flexibility measures such as expediting and overtime to fulfill excessive customer demand superior to a bound as a supplement to its safety stock. In this paper, we consider a continuous review assembly inventory system with Poisson final demand and fixed order costs at each stock controlled by a ( $R$ , $Q$ ) policy. We use the GSA to optimize the policy with the consideration of operating flexibility costs and fixed order costs. A deterministic mathematical programming model is established for the problem. And the model is solved by a line search for finding the optimal target cycle service level (CSL) to customer and an iterative procedure for solving the model when the target CSL is given. Moreover, we analyze the optimality conditions for the extended GSA model and obtain some important properties in given conditions. Numerical experiments on randomly generated instances demonstrate the efficiency of the procedure and confirm the solution presented in this paper.

Simultaneous Optimization of Contingent and Advance Purchase Orders with Fixed Ordering Costs

We analyze a finite horizon periodic-review inventory model in the presence of advance purchase discount (APD). In addition to the opportunity to place a contingent order when required, the buyer can also commit to advance orders and receive discounts. Thus the buyer faces the following tradeoff: Contingent orders provide operational flexibility to respond to uncertain demands at a high purchase cost, whereas advance orders incur low purchase cost while sacrificing operational flexibility. We consider a simple and easy-to-implement APD scheme in which the buyer enjoys price discounts by committing at the outset to a fixed size of advanced order for each period along with contingent orders when required. We first show that for any given advance order size, an advance-order dependent (s, S) policy is optimal. We then find the advance order size together with the corresponding (s, S) policy that minimizes the buyer’s total cost.

Coordinating a Semi‐Centralized Global Production Network Through Different Levels of Headquarters Involvement

Motivated by our experience with a global company, we propose and study the concept of a semi‐centralized supply chain and analyze its coordination issues. We focus on a supply chain consisting of a home plant and a foreign branch, both of which are under the same parent company but have considerable autonomy. The role of the home plant is to provide a key component to the foreign branch with guaranteed service (required by the headquarters). Because of a high fixed order cost, the branch orders the component rather infrequently, causing high expediting costs at the home plant. Our purpose is to help the headquarters to improve the supply chain efficiency. We show that under certain conditions, the headquarters can coordinate the supply chain by setting an upper bound on the expediting frequency. If these conditions fail to hold, a simple fixed cost‐sharing contract coordinates the supply chain. When centralized control is too costly or infeasible, the headquarters may delegate the contract design rights to the subunits. If the home plant receives the rights, the supply chain performance can be significantly improved (sometimes to near optimality). These results provide guidance to the headquarters on whether, when, and to whom to delegate the coordination initiatives.

Raw material inventory optimization for MTO enterprises under price fluctuations

Rapidly changing market environment cause raw materials price fluctuations, it is very difficulty for many manufacturing enterprises to make appropriate purchasing strategy aim at management actual. Accordingly, raw materials inventory optimization has become an important research question for MTO enterprises which is one of special manufacturing enterprises. Current literature researches either assumed all product orders can be met or ignored, which is inconsistent with MTO enterprises actual. It is groping direction of the paper that how to introduce customer order rates into raw material inventory optimization model and to make it between an given interval suitable for MTO enterprise’s actual situation, since the shortcoming appeared in previous researches. The model under raw materials price fluctuations for MTO enterprises with the goal of total costs minimal includes decision variables Q, the constraint. Total costs were composed of fixed order costs, purchasing costs and shortage of punishment costs. Then designed approximate algorithm GA, and analyzed approximation ratio, which showed time complexity is o(n4), where n is the number of raw materials. At last take Xi’an & Control Equipment Co. Ltd. as an example, confirmed the model and algorithm effective.

Coordinating a Semi-Centralized Global Production Network Through Different Levels of Headquarters Involvement

Motivated by our experience with a global company, we propose and study the concept of a semi-centralized supply chain and analyze its coordination issues. We focus on a supply chain consisting of a home plant and a foreign branch, both of which are under the same parent company but have considerable autonomy. The role of the home plant is to provide a key component to the foreign branch with guaranteed service (required by the headquarters). Because of a high fixed order cost, the branch orders the component rather infrequently, causing high expediting costs at the home plant. Our purpose is to help the headquarters to improve the supply chain efficiency. We show that under certain conditions, the headquarters can coordinate the supply chain by setting an upper bound on the expediting frequency. If these conditions fail to hold, a simple fixed cost sharing contract coordinates the supply chain. When centralized control is too costly or infeasible, the headquarters may delegate the contract design rights to the subunits. If the home plant receives the rights, the supply chain performance can be significantly improved (sometimes to near optimality). These results provide guidance to the headquarters on whether, when, and to whom to delegate the coordination initiatives.

Periodic review inventory systems with fixed order cost and uniform random yield

We consider periodic review stochastic inventory control problems with both fixed order cost and uniform random yield. Our objective is to characterize some structural properties of the optimal policies so that efficient approaches can then be established towards finding an optimal policy. In particular, we provide a lower-and-upper bound structure for the optimal policy at the beginning of any period, such that it is optimal not to order anything if the initial stock level is above the upper bound, and it is optimal to order a positive quantity if the initial stock level is below the lower bound, where the optimal quantity can be found efficiently. For any initial stock level in-between the upper and lower bounds in each period, a partial characterization of the optimal policy is provided. Furthermore, we show that the optimal order quantity is monotonically decreasing in the initial stock level in any period and this will help the search for the optimal order quantity at any initial stock level. In addition, we illustrate a few interesting phenomena about the behavior of an optimal policy. These phenomena show that the structure of the optimal policy for this problem is in general significantly different from that for the traditional periodic review inventory control problems with certain yield and fixed order cost.

Optimal Policies for a Dual-Sourcing Inventory Problem with Endogenous Stochastic Lead Times

We consider a single-product, two-source inventory system with Poisson demand and backlogging. Inventory can be replenished through a normal supply source, which consists of a two-stage tandem queue with exponential production time at each stage. We can also place an emergency order by skipping the first stage, for a fee. There is no fixed order cost. There are linear order, holding, and back-order costs. Through a new approach, we obtain optimal ordering policies for the discounted or long-run average cost and also characterize near-optimal heuristic policies. The approach consists of four steps. The first step is to establish an equivalent system, in the sense that it has the same optimal policy as the original system. The second step is to construct a tandem queueing system, where costs are charged in accord with the equivalent system’s cost structure. The third step derives an optimal control of the service rate at each server so as to minimize the tandem queue’s system-wide cost. The fourth and final step is to translate the queue’s optimal policy to an optimal policy for the equivalent system and hence the original system.

The Impact of Modeling on Robust Inventory Management Under Demand Uncertainty

This study considers a basic inventory management problem with nonzero fixed order costs under interval demand uncertainty. The existing robust formulations obtained by applying well-known robust optimization methodologies become computationally intractable for large problem instances due to the presence of binary variables. This study resolves this intractability issue by proposing a new robust formulation that is shown to be solvable in polynomial time when the initial inventory is zero or negative. Because of the computational efficiency of the new robust formulation, it is implemented on a folding-horizon basis, leading to a new heuristic for the problem. The computational results reveal that the new heuristic is not only superior to the other formulations regarding the computing time needed, but also outperforms the existing robust formulations in terms of the actual cost savings on the larger instances. They also show that the actual cost savings yielded by the new heuristic are close to a lower bound on the optimal expected cost. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2183 . This paper was accepted by Dimitris Bertsimas, optimization.

(S,s) Inventory Policies in General Equilibrium

We study the aggregate implications of (S,s) inventory policies in a dynamic general equilibrium model with aggregate uncertainty. Firms in the model's retail sector face idiosyncratic demand risk, and (S,s) inventory policies are optimal because of fixed order costs. The distribution of inventory holdings affects the aggregate outcome in two ways: variation in the decision to order and variation in the rate of sale through the pricing decisions of retailers. We find that both mechanisms must operate to reconcile observations that orders are more volatile than, and inventory investment is positively correlated with, sales, while remaining consistent with other salient business cycle characteristics. The model exhibits strong amplification for some shocks, and persistence to a limited extent.