Non-stationary Stochastic Demand Research Articles

The stochastic inventory routing problem on electric roads

In this work we introduce a green inventory routing problem termed the Stochastic Inventory Routing Problem on Electric Roads (S-IRP-ER), in which a hybrid vehicle navigates a road network with charging opportunities in some road sections, to cover the non-stationary stochastic demand of a single product for a set of retailers in the network. We model the problem using isochrone graphs to represent the real road network. In an isochrone graph nodes are located such that the time to travel any arc is constant all over the network. This allows for tracking the battery level of the vehicle serving retailers, as it charges and discharges continuously while travelling. We formulate a mathematical programming heuristics and prove its effectiveness. We use our model on a realistic instance of the problem, showcasing the different strategies that a vehicle may follow depending on fuel costs in relation to the costs of electricity.

Dynamic Inventory Relocation in Disaster Relief

This study investigates dynamic inventory relocation to respond proactively to the changing relief demand forecasts over time. In particular, we examine how to relocate mobile inventory optimally to serve nonstationary stochastic demand at several potential disaster sites. We propose a dynamic relocation model using dynamic programming (DP) and develop both analytical and numerical results regarding optimal relocation policies, the minimum cost‐to‐go function, and the value of inventory mobility over traditional warehouse pre‐positioning. Given the computational complexity of the backwards DP algorithm, we develop a base state heuristic (BSH) for general problems by exploiting the real‐world disaster pattern of occurrence. For problems with temporally independent demand, we propose a polynomial time exact algorithm based on a spatial–temporal graph. For problems with spatially independent demand, we design a speedup technique to implement BSH in polynomial time. The proposed model and algorithms are further extended to consider the impact of transportation uncertainties. Numerical experiments show that the proposed algorithms return high‐quality decisions only in a small fraction of the time required by an exact algorithm and a myopic algorithm. The proposed model and algorithms are applicable to any type of mobile inventory, facility, or server in similar settings.

Optimal Control of a Time-Varying Double-Ended Production Queueing Model

Motivated by production systems with nonstationary stochastic demand, we study a double-ended queueing model having back orders and customer abandonment. One side of our model stores back orders, and the other side represents inventory. We assume first-come-first-served instantaneous fulfillment discipline. Our goal is to determine the optimal (nonstationary) production rate over a finite time horizon to minimize the costs incurred by the system. In addition to the inventory-related (holding and perishment) and demand-related (waiting and abandonment) costs, we consider a cost that penalizes rapid fluctuations of production rates. We develop a deterministic fluid-control problem (FCP) that serves as a performance lower bound for the original queueing-control problem (QCP). We further consider a high-volume system for which an upper bound of the gap between the optimal values of the QCP and FCP is characterized and construct an asymptotically optimal production rate for the QCP, under which the FCP lower bound is achieved asymptotically. Demonstrated by numerical examples, the proposed asymptotically optimal production rate successfully captures the time variability of the nonstationary demand.

Two-stage inventory management with financing under demand updates

Considered is a retailer (she) facing non-stationary stochastic demand. Demand can be fully observed and backlogged, consequently the retailer can update the initial demand information using a Bayesian approach. To alleviate the demand risk, the retailer may use a secondary opportunity to replenish through an option contract. In addition, the retailer also has access to an immediate loan if she faces capital constraints and to a risk-free investment if she has surplus funds. The paper presents a recourse approach to solve the two-stage optimization problem and derive the optimal inventory/financing policies. The results show that the option procurement policy has a two-threshold base-stock structure depending on the first procurement, demand update and also the retailer’s financial state. The initial procurement can be computed subsequently. A sufficiently large initial demand will induce the retailer to seize the secondary procurement opportunity. Finally, a series of numerical examples demonstrate the resulting policy under various inventory/financial situations. This research incorporates the financial and operational decisions into demand updates, and brings new managerial results and insights.

Optimal Control of a Time-Varying Double-Ended Production Queueing Model

Motivated by production systems with nonstationary stochastic demand, we study a double-ended queueing model having backorders and customer abandonment. One side of our model stores backorders, and the other side represents inventory. We assume first-come- first-serve instantaneous ful fillment discipline. Our goal is to determine the optimal (nonstationary) production rate over a finite time horizon to minimize the costs incurred by the system. In addition to the inventory-related (holding, perishment) and demand-related (waiting, abandonment) costs, we consider a cost that penalizes rapid fluctuations of production rates. We develop a deterministic fluid control problem (FCP) that serves as a performance lower bound for the original queueing control problem (QCP). We further consider a high-volume system and construct an asymptotically optimal production rate for the QCP, under which the FCP lower bound is achieved asymptotically. Demonstrated by numerical examples, the proposed asymptotically optimal production rate successfully captures the time variability of the nonstationary demand.

Heuristic methods for the capacitated stochastic lot-sizing problem under the static-dynamic uncertainty strategy

We consider a lot-sizing problem in a single-item single-stage production system facing non-stationary stochastic demand in a finite planning horizon. Motivated by common practice, the set-up times need to be determined and frozen once and for all at the beginning of the horizon while decisions on the exact lot sizes can be deferred until the setup epochs. This operating scheme is referred to as the static-dynamic uncertainty strategy in the literature. It has been shown that a modified base stock policy is optimal for a capacitated system with minimum lot size restrictions under the static-dynamic uncertainty strategy. However, the optimal policy parameters require an exhaustive search, for which the computational time grows exponentially in the number of periods in the planning horizon. In order to alleviate the computational burden for real-life size problems, we developed and tested seven different heuristics for computational efficiency and solution quality. Our extensive numerical experiments showed that average optimality gaps less than 0.1% and maximum optimality gaps below 4% can be attained in reasonable running times by using a combination of these heuristics.

Development and simulation analysis of a new perishable inventory model with a closing days constraint under non-stationary stochastic demand

Abstract The food waste in grocery retail is a worldwide problem. Many mathematical inventory models for perishable items do not have a closing days constraint, although the age of perishable items also increases on closing days in grocery stores. We develop a new age-based inventory model with a closing days constraint. This stochastic multi-item inventory model includes total stock capacity constraints, a positive lead time, a periodic inventory control, a target customer service level and mixed FIFO and LIFO issuing policies for perishable items with a fixed lifetime under a non-stationary random demand. We show in a comparative simulation study under a rolling planning that the closing days constraint improves order decisions and reduces waste quantities and costs in grocery stores.

Dual sourcing in the age of near-shoring: Trading off stochastic capacity limitations and long lead times

We model a periodic review inventory system with non-stationary stochastic demand, in which a manufacturer is procuring a component from two available supply sources. The faster supply source is modeled as stochastic capacitated with immediate delivery, while the slower supply source is modeled as uncapacitated with a longer fixed lead time. The manufacturer’s objective is to choose how the order should be split between the two supply sources in each period, where the slower supply source is used to compensate for the supply capacity unavailability of the faster supply source. This is different from the conventional dual-sourcing problem, and motivated by the new reality of near-shoring options. We derive the optimal dynamic programming formulation that minimizes the total expected inventory holding and backorder costs over a finite planning horizon and show that the optimal policy is relatively complex. We extend our study by developing an extended myopic two-level base-stock policy and we show numerically that it provides a very close estimate of the optimal costs. Numerical results reveal the benefits of dual-sourcing under near-shoring, where we point out that in most cases the manufacturer should develop a hybrid procurement strategy, taking advantage of both supply sources to minimize its expected total cost.

Construction Material Safety-Stock Determination Under Nonstationary Stochastic Demand and Random Supply Yield

Nonstationary stochastic demand and random supply yield have led to a great challenge in construction material management. Appropriate safety stocks can cope with uncertainty in both demand and supply yield, and improve the construction cost and productivity significantly. But the safety stock is usually determined by planner's experience or rough evaluation in management practice, which results in overstock or understock situations frequently. For changing this situation, this paper focuses on analysis of the stochastic inventory control problem of construction material, and develops a general safety-stock determination approach under nonstationary stochastic demand and random supply yield. Based on the linear-order release rule and modified base-stock definition, the inventory balance equation is developed. Then, a coverage random variable, which incorporates the complete information of outstanding orders, is defined to represent the outflows and inflows to stock over risk periods. Meanwhile, the fixed-point iteration method is adopted to calculate the time-varying safety-stock level under predefined service-level (nonstockout probability) constraint. Finally, a numerical analysis is conducted to investigate the relative performance of the proposed safety-stock approach and days of supply rule. The result demonstrates a considerable service improvement and inventory decrease of the safety-stock approach.

Non-stationary stochastic inventory lot-sizing with emission and service level constraints in a carbon cap-and-trade system

Firms worldwide are taking major initiatives to reduce the carbon footprint of their supply chains in response to the growing governmental and consumer pressures. In real life, these supply chains face stochastic and non-stationary demand but most of the studies on inventory lot-sizing problem with emission concerns consider deterministic demand. In this paper, we study the inventory lot-sizing problem under non-stationary stochastic demand condition with emission and cycle service level constraints considering carbon cap-and-trade regulatory mechanism. Using a mixed integer linear programming model, this paper aims to investigate the effects of emission parameters, product- and system-related features on the supply chain performance through extensive computational experiments to cover general type business settings and not a specific scenario. Results show that cycle service level and demand coefficient of variation have significant impacts on total cost and emission irrespective of level of demand variability while the impact of product's demand pattern is significant only at lower level of demand variability. Finally, results also show that increasing value of carbon price reduces total cost, total emission and total inventory and the scope of emission reduction by increasing carbon price is greater at higher levels of cycle service level and demand coefficient of variation. The analysis of results helps supply chain managers to take right decision in different demand and service level situations.

Inventory lot-sizing with supplier selection under non-stationary stochastic demand

Inventory lot-sizing and supplier selection problem has been studied in the literature considering mainly time-varying deterministic demand. However, in real life, most of the products exhibit non-stationary stochastic demand. In this context, we propose an integer linear programming model for inventory lot-sizing and supplier selection problem under non-stationary stochastic demand with all-units quantity discounts and fill rate constraints. Through detailed analysis of experimental results, we show the impacts of fill rate requirements and demand coefficient of variation on costs, inventory levels and order allocations.

A branch-and-bound algorithm for shift scheduling with stochastic nonstationary demand

Many shift scheduling algorithms presume that the staffing levels, required to ensure a target customer service, are known in advance. Determining these staffing requirements is often not straightforward, particularly in systems where the arrival rate fluctuates over the day. We present a branch-and-bound approach to estimate optimal shift schedules in systems with nonstationary stochastic demand and service level constraints. The algorithm is intended for personnel planning in service systems with limited opening hours (such as small call centers, banks, and retail stores). Our computational experiments show that the algorithm is efficient in avoiding regions of the solution space that cannot contain the optimum; moreover, it requires only a limited number of evaluations to encounter the estimated optimum. The quality of the starting solution is not a decisive factor for the algorithm׳s performance. Finally, by benchmarking our algorithm against two state-of-the-art algorithms, we show that our algorithm is very competitive, as it succeeds in finding a high-quality solution fast (i.e., with a limited number of simulations required in the search phase).

The value of VMI beyond information sharing in a single supplier multiple retailers supply chain under a non-stationary (Rn, Sn) policy

This study aims to determine the value of vendor-managed inventory (VMI) over independent decision making with information sharing (IS) under non-stationary stochastic demand with service-level constraints. For this purpose, we utilize mixed-integer linear programming formulations to quantify the benefits that can be accrued by a supplier, multiple retailers and the system as a whole by switching from IS to VMI. More specifically, we investigate the incremental value that VMI provides beyond IS in terms of expected cost savings, inventory reductions, and decrease in shipment sizes from the supplier to the retailers by conducting a large number of computational experiments. Results reveal that the decision transfer component of VMI improves these performance measures significantly when the supplier׳s setup cost is low and order issuing efficiency is high. The benefits offered by VMI are negligible under the problem settings where the supplier׳s order issuing efficiency is low and the production setup serves solely a single replenishment under IS.

Exploring the economic consequences of letting a supplier hold reserve storage

We consider a single-item, periodic review inventory control problem with discrete non-stationary stochastic demand. The time horizon is finite and all shortages at the downstream level are backordered. There are two modes of supply: a normal supplier and a reserve storage supply. The reserve storage is capacitated and the downstream buyer can only order the entire inventory in the reserve storage or nothing. If the reserve storage is empty, it takes a fixed time interval before it is replenished again. Provided that the reserve storage is fully replenished it can be used at any time period, whereas orders to the normal supplier can only be issued at specific time periods. The lead time from the reserve storage is shorter than from the normal supplier, but using the reserve storage is more expensive than using the normal supplier. We use stochastic dynamic programming to specify an exact model of the problem. We also develop an approximate model which is computationally faster than the first model. The models are then used to analyze numerically the sensitivity with respect to key parameters like the reserve storage size and unit purchase cost. This paper is motivated by a problem presented during contacts with a leading Danish provider of communications solutions.

Constraint-Based Local Search for Inventory Control Under Stochastic Demand and Lead Time

In this paper, we address the general multiperiod production/inventory problem with nonstationary stochastic demand and supplier lead time under service-level constraints. A replenishment cycle policy is modeled. We propose two hybrid algorithms that blend constraint programming and local search for computing near-optimal policy parameters. Both algorithms rely on a coordinate descent local search strategy; what differs is the way this strategy interacts with the constraint programming solver. These two heuristics are first, compared for small instances against an existing optimal solution method. Second, they are tested and compared with each other in terms of solution quality and run time on a set of larger instances that are intractable for the exact approach. Our numerical experiments show the effectiveness of our methods.

Static-dynamic uncertainty strategy for a single-item stochastic inventory control problem

We consider a single-stage inventory system facing non-stationary stochastic demand of the customers in a finite planning horizon. Motivated by the practice, the replenishment times need to be determined and frozen once and for all at the beginning of the horizon while decisions on the exact replenishment quantities can be deferred until the replenishment time. This operating scheme is refereed to as a “static-dynamic uncertainty” strategy in the literature [3]. We consider dynamic fixed-ordering and linear end-of-period holding costs, as well as dynamic penalty costs, or service levels. We prove that the optimal ordering policy is a base stock policy for both penalty cost and service level constrained models. Since an exponential exhaustive search based on dynamic programming yields the optimal ordering periods and the associated base stock levels, it is not possible to compute the optimal policy parameters for longer planning horizons. Thus, we develop two heuristics. Numerical experiments show that both heuristics perform well in terms of solution quality and scale-up efficiently; hence, any practically relevant large instance can be solved in reasonable time. Finally, we discuss how our results and heuristics can be extended to handle capacity limitations and minimum order quantity considerations.

Constraint programming for stochastic inventory systems under shortage cost

One of the most important policies adopted in inventory control is the replenishment cycle policy. Such a policy provides an effective means of damping planning instability and coping with demand uncertainty. In this paper we develop a constraint programming approach able to compute optimal replenishment cycle policy parameters under non-stationary stochastic demand, ordering, holding and shortage costs. We show how in our model it is possible to exploit the convexity of the cost-function during the search to dynamically compute bounds and perform cost-based filtering. Our computational experience show the effectiveness of our approach. Furthermore, we use the optimal solutions to analyze the quality of the solutions provided by an existing approximate mixed integer programming approach that exploits a piecewise linear approximation for the cost function.

An Inventory Optimization Model To Support Operating Room Schedules

Health care costs continue to rise at unprecedented rates in the United States. While past research has addressed optimal patient scheduling to improve the utilization of operating theater resources (physicians, specialized equipment, and operating rooms), little research has attempted to optimize material requirements and safety stocks to support these optimized schedules. Although classical inventory models can and should be used, additional considerations in hospital environments remain unaddressed. Based on our empirical observations, significant opportunities exist to reduce waste by improving the planning and coordination of materials and information in hospital operating theaters. To quantify these opportunities, we develop an analytic optimization framework for planning and executing material quantity preparation and placement decisions to support hospital operating theaters. Our model considers the non-stationary stochastic demand associated with block scheduling of surgical departments and a stochastic bill of materials, common in surgical treatments. Our approach is novel in that we formulate the stochastic decision problem exactly as a linear program by leveraging the convexity of the expected cost functions. The model is computationally efficient and flexible, allowing fast solutions and unique considerations for individual hospitals while considering uncertainty explicitly. We conduct a numerical study and sensitivity analysis to pinpoint the leverage points in the system for reducing inventory carrying and materials management costs. Surprisingly, reducing the uncertainty in the physician-determined bill of materials through standardization exhibited the greatest potential for savings.

Computing the non-stationary replenishment cycle inventory policy under stochastic supplier lead-times

In this paper we address the general multi-period production/inventory problem with non-stationary stochastic demand and supplier lead-time under service level constraints. A replenishment cycle policy ( R n , S n ) is modeled, where R n is the nth replenishment cycle length and S n is the respective order-up-to-level. We propose a stochastic constraint programming approach for computing the optimal policy parameters. In order to do so, a dedicated global chance-constraint and the respective filtering algorithm that enforce the required service level are presented. Our numerical examples show that a stochastic supplier lead-time significantly affects policy parameters with respect to the case in which the lead-time is assumed to be deterministic or absent.

A state space augmentation algorithm for the replenishment cycle inventory policy

In this work we propose an efficient dynamic programming approach for computing replenishment cycle policy parameters under non-stationary stochastic demand and service level constraints. The replenishment cycle policy is a popular inventory control policy typically employed for dampening planning instability. The approach proposed in this work achieves a significant computational efficiency and it can solve any relevant size instance in trivial time. Our method exploits the well known concept of state space relaxation. A filtering procedure and an augmenting procedure for the state space graph are proposed. Starting from a relaxed state space graph our method tries to remove provably suboptimal arcs and states (filtering) and then it tries to efficiently build up (augmenting) a reduced state space graph representing the original problem. Our experimental results show that the filtering procedure and the augmenting procedure often generate a small filtered state space graph, which can be easily processed using dynamic programming in order to produce a solution for the original problem.