Safety Stock Placement Research Articles

Shipment planning and safety stock placement in maritime supply chains with stochastic demand and transportation times

This paper introduces and studies shipment planning and safety stock placement in bulk and semi-bulk maritime supply chains. This problem jointly optimizes the assignment of destinations to origins, shipment quantities of multiple products by determining the share of the contract by sea transport options, sailing speed, and multi-echelon safety stocks with the objective to minimize the expected total transportation and inventory cost. In the case of a serial system supplying a single product, we analytically characterize the effect of sailing speed and different time and cost parameters on optimal safety stock placement. For the general case, the integrated problem can be solved in four different ways. We present a mixed integer nonlinear program (MINLP) and the superior reformulation of the mixed integer conic quadratic integer program (MICQP) that can be solved using state-of-the-art solvers. Then, we propose two exact solution procedures that iteratively approximate the nonlinear terms in the objective function and constraints. Our numerical experiments illustrate the interdependence between safety stock placement, assignment decisions, and sailing speed. Results also show that it is important to adopt multi-stage inventory optimization and there is an added value in integrating shipment planning and safety stock placement. Furthermore, computational experiments demonstrate that all the solution approaches are alternatives to solving small instance sizes. For medium and large instances, the solvers are unable to provide quality solutions within one hour while the iterative procedures find optimal or near-optimal solutions in reasonable CPU times.

Safety stock placement in supply chains with expediting

Numerous documented implementations demonstrate the guaranteed-service (GS) model of safety stock optimization has applicability to practice. The standard GS (S-GS) model employs safety stock as the only countermeasure to satisfy a specified demand bound, assuming any demand in excess of the bound will be addressed by countermeasures which are not modeled. We extend the S-GS model to explicitly include expediting as a second countermeasure. Thus, our two-countermeasure GS (2C-GS) formulation addresses more demand variability explicitly than the S-GS model. We characterize a stage’s optimal base-stock level as the maximum of two base-stock calculations, and show how the resulting optimization problem can be efficiently solved using existing GS solution techniques. For normally distributed demand, we analytically characterize when the S- and 2C-GS approaches deploy different configurations of safety stock. For a series of numerical experiments, we find that the 2C-GS model places safety stocks at more stages than the S-GS model. When compared to the 2C-GS solution, the suggested safety stock positioning of the S-GS approach results in an average cost increase of 7%, and a maximum of 42%, in real-world supply chain settings. Furthermore, the benefits from expediting increase when downstream stages have greater expediting capability.

Safety stock placement with market selection under load-dependent lead times

We study the problem of safety stock placement in a supply chain with market selection decisions. A manufacturer with deterministic, load-dependent lead time supplies multiple warehouses, each serving multiple retailers. Each retailer has access to a set of potential markets with different characteristics. Serving more markets increases revenues, but also increases the manufacturer’s lead time, resulting in higher inventory costs. Adopting the Guaranteed Service Approach, we present a nonlinear mixed-integer programming model and reformulate it to eliminate integer variables related to service times at warehouses. We then propose a successive piecewise linearization algorithm and a mixed-integer conic quadratic formulation to solve the resulting nonlinear binary formulation. Computational experiments show that the successive piecewise linearization algorithm outperforms two state-of-the-art solvers, BARON and CPLEX, which are used to solve instances of the original formulation and the mixed-integer conic quadratic reformulation, respectively. The value of incorporating load-dependent lead times is greatest when capacity is limited relative to available demand. The benefit of integrating market selection and safety stock decisions is greatest when capacity is limited and marginal revenue is relatively low.

Joint optimization of safety stock placement and supplier selection in a multi-layered distribution network

This paper discusses the challenging problem of jointly optimizing safety stock placement and supplier selection in a multi-layered distribution network, under the guaranteed service model (GSM) approach and single-supplier strategy for the customers. The original GSM-based model is extended to include optimal echelon (R, Q) policies. We propose an approximation model, which determines the optimal configuration flow of orders, order quantities, reorder points, safety stock amounts, inbound and outbound service times at each node. A two-stage heuristic is developed for this complex model that optimizes the flow configuration and then determines the other variables. Current experimental results on a few test networks indicate a promising computational time performance.

Optimizing inventory management costs in supply chains by determining safety stock placement

For a production business, the measurement of safety inventory to be maintained during each step of a supply chain is a key concern and requires providing the clients an irregular state of management. The stock held should be small to reduce holding costs and capacity while maintaining the capacity to service customers in time to satisfy much, if not all, the demand. This paper discusses this issue by using a deterministic time structure and provides a measure of the security position of stock in supply chains for the overall network. First, prove that the overall problem is NP-hard. Then set up a couple of parameters that characterize an optimal overall network structure. To take care of these problems, a polynomial approximation is considered. An arrangement of computational tests to survey the execution of the general-network calculation and to decide how to set different parameters for the calculation is selected. In addition to the general network case, the two-layer network issues are considered. Also, a nonlinear model for determining the level of safety stock in different components of the supply chain to minimize the related safety stock costs is developed.

Finite inventory budgets in production capacity and safety stock placement under the guaranteed service approach

Following the guaranteed service approach (GSA), several studies have investigated the interaction between production and inventory decisions in multistage supply systems and developed efficient algorithms to solve the large-scale optimization problems that emerge. These works have largely ignored limited inventory budgets and their effect on production and inventory decisions. In reality, however, firms tend to limit the amount of capital tied-up in inventories, and there is usually a limited amount of storage space at a warehouse or a retailer. This study investigates the problem of production capacity and safety stock placement with both limited capacity and inventory budgets in general acyclic supply chains. This problem allocates capacity to production stages and sets inventory targets at logistics stages, with the objective to minimize the expected total supply chain cost while meeting a target customer service level.The present paper contributes to the GSA literature in several ways. First, we compare guaranteed and chance-constraint approaches to model limited inventory budgets and analyze their effect on safety stock placement. Second, we present two new equivalent formulations of the production capacity and safety stock placement problem. Solving these new models using a successive piecewise linear approximation algorithm outperforms existing solution procedures in terms of solution quality and time, in both cases of infinite and finite inventory budgets. Third, taking advantage of the structure of these formulations, we develop a two-phase heuristic, which finds optimal or near-optimal solutions and greatly improves CPU times. Finally, our numerical experiments show that inventory budgets affect production capacity and safety stock placement decisions, increasing both work in process (WIP) and safety stock levels.

Planning capacity and safety stocks in a serial production–distribution system with multiple products

This study jointly optimizes the production capacity and safety stocks in a serial production–distribution system supplying multiple products under a guaranteed service approach (GSA). The network comprises one manufacturer operating multiple workcenters, one warehouse with limited storage capacity, and one retailer. The manufacturer must efficiently allocate capacity to the workcenters under a limited budget, while the warehouse and retailer need to maintain safety stocks to achieve a target service level. For a single workcenter processing a single product, the interaction between the production lead time, storage capacity, inventory costs, and safety stock placement is characterized. When the manufacturer has multiple workcenters, the integrated problem is formulated as a non-convex program and is solved using a nested Lagrangian relaxation heuristic. The algorithm dualizes the storage constraint in the first phase and the budget constraint in the second phase. A simulation study is conducted to assess the value of the integration, and computational experiments demonstrate that the nested Lagrangian relaxation heuristic can identify optimal or near-optimal solutions in reasonable CPU times.

Optimizing production capacity and safety stocks in general acyclic supply chains

This paper addresses the joint optimization of production capacity and safety stocks in supply chains under the guaranteed service approach (GSA). The integrated problem is formulated as a mixed integer nonlinear program (MINLP) and solution procedures are proposed in the cases of general acyclic and spanning tree networks. For general acyclic supply chains, the integrated problem is solved using a Lagrangian decomposition method which iteratively solves capacity planning and safety stock placement subproblems, and adds budget feasibility constraints to strengthen the Lagrangian decomposition lower bound. When the supply chain has a spanning tree structure, an efficient Lagrangian relaxation heuristic dualizes the budget constraint and solves the relaxed problem using a dynamic programming algorithm. Computational experiments on real-world instances show that the Lagrangian decomposition method is able to solve all instances within 0.1% optimality, while a state-of-the-art solver is unable to provide feasible solutions for large instances. In the case of spanning tree networks, the proposed Lagrangian relaxation heuristic finds optimal or near-optimal solutions and greatly improves running time in comparison to the Lagrangian decomposition method. In addition, numerical experiments show that savings can be achieved through joint optimization of capacity and safety stocks.

Safety stock placement for serial systems under supply process uncertainty

In this study, we address safety stock positioning when demand per period is a known constant but supply is uncertain. The supply is either available or not available, while the setting is that of a periodically reviewed, serial system following a base stock policy. Each stage is allowed to operate according to the guaranteed or stochastic service model. We use a Discrete Time Markov Chain model for expressing the expected on-hand inventories for each stage, along with other terms of interest, as a function of policy parameters determined by a given service level requirement for the end product. Exact models are constructed for single-stage and two-stage systems. As the number of states for a two-stage system grows exponentially, we propose an approximation for expressing the effect of the input stage using a single parameter. A generalization for the approximation is provided for a multi-stage problem. Computational evaluations of the approximation, as well as numerical comparisons of different cases, are presented.

Extending the strategic safety stock placement model to consider tactical production smoothing

This paper extends strategic safety stock placement models under the guaranteed service approach (GSA) to incorporate tactical production smoothing. We propose a model to jointly optimize production capacity, production smoothing, and service times between all stages in the supply chain. Analysis of the model leads to several interesting findings. First, for certain service times, production smoothing is desirable to reduce both capacity and inventory costs. Second, inventory cost at a production stage is non-monotonous in its net service time and consequently quoting a large service time may increase costs at both the production stage and its customers. Third, safety stocks can be pooled at downstream stages only when production is not smoothed, while production smoothing necessitates holding safety stocks. The formulated problem is solved using a dynamic program and numerical experiments are conducted based on a real-world instance from the literature. Through these experiments, we show that integrating production smoothing in GSA models results in considerable savings, especially when capacity costs are neither too high nor too low relative to holding costs.

On the effect of overtime and subcontracting on supply chain safety stocks

In supply chain safety stock placement models, production lead times are typically considered exogenous; however, in many supply networks, lead times are controllable and should be considered as endogenous variables. This work integrates strategic safety stock placement with subcontracting and overtime planning that can be used to optimize lead times. We formulate the problem of jointly optimizing safety stocks and lead times to minimize production and inventory holding costs, while guaranteeing a customer service level.Using a two-stage supply chain, we analyze the impact of subcontracting and overtime on optimal safety stock placement both analytically and numerically. Our analysis suggests that high usage of subcontracting and overtime tend to decouple safety stocks such that both raw materials and finished goods safety stocks are held. For multistage networks, we derive bounds on decision variables, which are used in a dynamic programming algorithm to solve the integrated problem. Experiments on selected supply chains from publicly available data sets show the value resulting from joint optimization of lead times and safety stocks. The results indicate that the integrated problem leads to greater usage of operational flexibility and shorter lead times as compared to the sequential problem, where lead times are optimized locally. In addition, the value of integration depends on the subcontracting and overtime costs along with the value added at a stage.

Effect of setup time reduction on supply chain safety stocks

In most safety stock placement models, manufacturing lead times are treated as constant and exogenous quantities. In reality, however, manufacturing lead times depend on operating policies such as batch sizing and can be decreased by systematic investments in setup time reduction. This paper considers a manufacturer supplying a distribution network and integrates strategic safety stock placement with setup time reduction and batch sizing. We formulate the integrated problem with the objective of minimizing work-in-process and inventory holding costs and setup time reduction investment. An existing dynamic programming algorithm is extended to solve this problem efficiently. Using a two-stage distribution network, we characterize analytically the impact of setup time, batch sizes and integration on optimal positioning of safety stocks. Numerical studies on a three-stage network highlight the economic value of integration and setup time reduction. Setup time reduction is shown to be highly effective when utilization is high or when it can be accomplished with small investments. The savings from integration are high when value added at the manufacturer is high, and low when the marginal cost of setup reduction is either very high or very low.

Two-stage supply chain design with safety stock placement decisions

In this paper, we propose a supply chain design model that integrates facility location with safety stock placement and delivery strategy decisions, to reflect their interdependence and ultimately improve the resulting supply chain design. Safety stock placement decisions are modelled using the guaranteed-service approach. We also consider two customer classes that differ with respect to their delivery time preferences. The resulting non-linear model is formulated as a conic quadratic mixed-integer program that can be solved to optimality. We show that the model can be extended to include demand correlation and stochastic lead times. The computational experiments capture the trade-offs between location and safety stock placement decisions, between demand variability pooling and proximity to retailers, and between lead times and service times. We show that, when the distribution centres have short lead times to the retailers, distribution centres tend to hold safety stocks, and the retailers offer express deliveries. Conversely, when lead times are long, only the retailers hold safety stocks and they do not offer express deliveries.

Optimising safety stocks and reorder points when the demand and the lead-time are probabilistic in cement manufacturing

Inventory control is an extremely challenging task, complicated by a probability of demand and lead-time. An inventory system attempts to balance between overstock and understock to reduce the total cost and achieve customer demand in a timely manner. The optimality of inventory policies for a cement industry is still unknown for many types of inventory systems. In this paper, probability distribution of demand during lead-time is considered when the demand and lead-time are probabilistic while extracting the optimal safety stock placement and the reorder point in cement manufacturing. Probability distributions of demand during lead-time include Pearson type 6 four-parameter, log-Pearson 3, fatigue life (Birnbaum-Saunders), and inverse Gaussian three-parameter distributions. The probability distribution of demand during lead-time is established when the demand and the lead-time are probabilistic; each one has a probability distribution function. The safety stock depends on the safety factor under certain service level and the standard deviation of demand during lead-time which can be obtained from the distribution. The quantities of the safety stock and the reorder point represent an optimal value at each position to avoid over or under stock, the most controversial issue in the inventory system.

Integrated lot sizing and safety stock placement in a network of production facilities

In a network of production facilities, safety stock placement is determined at the network level to hedge against uncertainty in demand, while considering the production cycle time (PCT) of each stage as an exogenous parameter. The lot sizing policy at each stage affects setup time, capacity, and is a major determinant of queuing behavior and PCT. This paper integrates strategic safety stock placement and tactical lot sizing, which are typically optimized independently. We model the integrated problem to minimize system-wide production and inventory costs subject to meeting service-level requirements of external customers. We establish bounds for optimal lot sizes and extend existing safety stock placement dynamic programming algorithm to solve the integrated problem. Based on numerical experiments, we assess the value of integrating the two decisions and find that integration leads to smaller lot sizes, shorter lead times, and favors pooling of safety stocks. The setup cost, setup time, and the added value at a production stage are the most significant factors in determining the amount of savings generated through integration. In addition, simulation studies show that integration reduces PCT variability and decreases the fraction of delayed items.

Genetic algorithm for inventory positioning problem with general acyclic supply chain networks

Inventory positioning, also known as safety stock placement, in supply chain networks is an important optimisation problem that has various applications in supply chain design and configuration. In this paper, we develop a new genetic algorithm (GA) for this NP-hard problem. Our new GA features custom designed procedure to generate feasible individuals by exploiting the problem structure. It also implements a multi-start strategy to enhance solution quality. Computational results show that our GA is able to offer near optimal solutions in reasonable computational time. [Received 4 October 2014; Revised 19 October 2015; Revised 14 January 2016; Accepted 18 January 2016]

Safety stock placement in supply chains with demand forecast updates

Supply chains are exposed to many types of risks and it may not be obvious where to keep safety stocks in the supply chain to hedge against those risks, while maintaining a high customer service level. In this paper, we develop an approach to determine the safety stock levels in supply chain systems that face demand uncertainty. We model customer demand following the Martingale Model of Forecast Evolution (MMFE). An extensive body of literature discusses the safety stock placement problem in supply chains, but most studies assume independent and identically distributed demand. Our approach is based on a simulation study in which mathematical models are solved in a rolling horizon setting. It allows determining the safety stock levels at each stage of the supply chain. Based on a numerical study, we find that a big portion of the safety stocks should be placed downstream in the supply chain to achieve a high customer service level.

Trade-off between service time and carbon emissions for safety stock placement in multi-echelon supply chains

With the low-carbon revolution, many firms face the continuous challenge for reducing carbon emissions in their supply chain activities. Built upon the strategic safety stock placement model, in this paper, we model the carbon emissions for each stage of a multi-echelon supply chain as a function of the service time guaranteed by this stage to its immediate downstream stages. We establish that there is a negative correlation between service time and carbon emissions at each stage. Based on this, we develop two models to study the trade-off between service time and carbon emissions for safety stock placement in multi-echelon supply chains by considering carbon cap and carbon tax, respectively. Both of them are solved using iterative piecewise linear approximations. We implement these two models to study how carbon cap, carbon emission cost rate and guaranteed service time affect the optimal safety stock placement using a real chain.

Meta-heuristics for placing strategic safety stock in multi-echelon inventory with differentiated service times

The computational resolution of multi-echelon safety stock placement problems has attracted ample attention in recent years. Practitioners can obtain good solutions for large supply networks with general structure using the guaranteed service model. The mainstream assumption in this model is that a stock point quotes identical service times to its successors. In business, it is common to assign customers to different customer- and service classes, as it can yield significant cost improvements. Nevertheless, differentiated service times have rarely been considered in computational methods of safety stock placement. We relax the assumption of identical service times in the guaranteed service approach and allow stock points to prioritize between their successors. This increases the complexity of the problem considerably, so that meta-heuristics become the methods of choice. Meta-heuristics need a mapping between safety stock levels in the supply network (from which they compute the holding cost) and an internal representation of stocking decisions (from which they generate new solutions). The design of a representation is non-trivial because of complex interactions between stocking decisions and stock levels. We propose a representation for the safety stock allocation problem with differentiated service times that can be used in general-acyclic supply networks. We apply a local search, a simple genetic algorithm and a problem-adjusted simulated annealing to 38 general-acyclic real-world instances. Results suggest that service time differentiation indeed decreases total holding cost in the network. Simulated annealing outperforms the other meta-heuristics within the set of tested methods with respect to speed and solution quality.

Strategic Safety Stock Placement in Supply Networks with Static Dual Supply

Many real-world supply networks source required materials from multiple suppliers. Existing multiechelon inventory optimization approaches either restrict their scope to multiple supply sources in two-echelon systems or single suppliers in multiechelon systems. We develop an exact mathematical model for static dual supply in a general acyclic N-echelon network structure, which builds on the guaranteed-service framework for safety stock optimization. It is assumed that the suppliers are allocated static fractions of demand. We prove that for normally distributed demand an extreme point property holds. We present a real example from the industrial electronics industry consisting of five echelons and three dual-sourced materials. This example forms the basis for a numerical analysis. Compared with the only previously published approximate solution, our exact approach results in considerable cost savings because the exact model captures inventory pooling in a way that the approximation is unable to do. For a set of test problems, total safety stock cost savings are 9.1%, on average.