Guaranteed Service Approach Research Articles

Efficient computational strategies for a mathematical programming model for multi-echelon inventory optimization based on the guaranteed-service approach

This paper presents a Multi-Echelon Inventory Optimization (MEIO) framework, based on the Guaranteed-Service Model (GSM), to allocate safety stocks across a supply chain with several locations and products, minimizing costs while meeting service level objectives. Extending previous work by Achkar et al. (2023), this paper enhances the Mixed-Integer Quadratically Constrained Program (MIQCP) with a highly efficient solution approach. The model introduces a piecewise linear approximation, significantly improving computational efficiency and the accuracy of the approximation for the fill rate function. It also introduces a different and more efficient approach to account for stochastic lead times using a discrete function. Moreover, an extension of the approach to account for non-normally distributed demands is proposed. The model is applied to several instances of a real-world case study from a pharmaceutical company, with more than 7300 product-location combinations, showing that optimal solutions can be obtained within few seconds of computational time.

Extensions to the guaranteed service model for industrial applications of multi-echelon inventory optimization

Multi-echelon inventory optimization (MEIO) plays a key role in a supply chain seeking to achieve specified customer service levels with a minimum capital in inventory. In this work, we propose a generalized MEIO model based on the Guaranteed Service approach to allocate safety stock levels across the network at the lowest holding cost. This model integrates several existing and some novel features that are usually present in pharmaceutical multi-echelon supply chains into a single model: review periods, manufacturing facilities, hybrid nodes (nodes with both internal and external demand), minimum order quantities (MOQ), and different service level performance indicators (fill rate and cycle service levels). We include a polynomial regression to approximate fill rates as a possible target measure to set safety stocks. To improve efficiency, we propose a nonlinear programming model to support decision making, which can be reformulated as a Quadratically Constrained Program (QCP), which leads to order of magnitude reductions in computational time. The performance of the model is evaluated by solving illustrative and real-world cases, and is validated with simulation.

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.

Inventory Positioning in Supply Chain Network: A Service-Oriented Approach

This study develops a mixed-integer linear programming model based on a guaranteed service approach for an inventory positioning problem in a supply chain under the base stock inventory policy. Our proposed model aims to determine appropriate inventory positions and amounts and the optimal service level for the supply chain to minimize the total cost of safety inventory holding and shortage. Two demand scenarios, based on normal and empirical distributions, are investigated. An extensive numerical experiment is conducted to illustrate the applicability and effectiveness of our model, especially under empirical distribution. The experiment features a practical network structure and demand data from an industrial user. Moreover, to further validate the experimental results from the mathematical model, they are compared with the result from a simulation model, which is constructed to imitate the operations of the supply chain. The comparison result indicates that the model solution under the empirical demand distribution is close to the simulation regarding the difference in the total cost (less than 1%). This solution significantly outperforms the model solution under the normal demand, which results in a significant difference in total cost (more than 25%) compared to the simulation.

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.

An Optimization Model for Multi-Echelon Multi-Commodity Supply Chain Design with Side Constraints

A proper strategies in designing and optimizing supply chains can help enterprises to stay competitive in this volatile economy. Identifying various type of products that run through the supply chain and aligning several parts of the supply chain to these product types are considered to be effective strategies adapted by successful companies. In this paper we address an optimal design of a multi-echelon supply chain problem with various commodity that is of significance to the business enterprise, along with the associated inventory systems. By using the guaranteed service approach to model the multi-echelon inventory system, we develop an MINLP model to simultaneously optimize the transportation, inventory and network structure of a multi-echelon supply chain which involve multi commodity. The optimization model is then solved using a strategy of releasing nonbasic variables from their bounds, combined with the “active constraint” method.

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.

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.

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.

A Comparison Study of Stochastic- and Guaranteed- Service Approaches on Safety Stock Optimization for Multi Serial Systems

Two competing approaches have been developed over the years for multi-echelon inventory system optimization, stochastic-service approach (SSA) and guaranteed-service approach (GSA). Although they solve the same inventory policy optimization problem in their core, they make different assumptions with regard to the role of safety stock. This paper provides a detailed comparison of the two approaches by considering operating flexibility costs in the optimization of (R, Q) policies for a continuous review serial inventory system. The results indicate the GSA model is more efficiency in solving the complicated inventory problem in terms of the computation time, and the cost difference of the two approaches is quite small.

An Integer Programming Model for Multi-Echelon Supply Chain Decision Problem Considering Inventories

In this paper we address a problem that is of significance to the industry, namely the optimal decision of a multi-echelon supply chain and the associated inventory systems. By using the guaranteed service approach to model the multi-echelon inventory system, we develop a mixed integer; programming model to simultaneously optimize the transportation, inventory and network structure of a multi-echelon supply chain. To solve the model we develop a direct search approach using a strategy of releasing nonbasic variables from their bounds, combined with the “active constraint” method. This strategy is used to force the appropriate non-integer basic variables to move to their neighbourhood integer points.

Performance Measurement of a Rotavirus Vaccine Supply Chain Design by the Integration of Production Capacity into the Guaranteed Service Approach

Previous research has integrated multi-echelon inventory management into the design of a responsive supply chain by the use of the guaranteed service approach. We build further upon this work by integrating the production capacity and product flow to minimize the supply chain’s inventories. The production capacity is modeled with a queuing network to handle the variability of the batch production processes as well as the demand variability. We test and validate our model with adapted instances from literature and apply it to the rotavirus vaccine supply chain. This vaccine supply chain is seen as complex on the manufacturing side as well as on the distribution side. For our industrial application we show how this work is embedded in a scenario approach and the contribution of our model to evaluate a single scenario according to multiple performance indicators. For this paper, our scenarios consist of different lead time reduction programmes and varying demand levels. We demonstrate how to extract the best performing scenario.

Optimization of (R, Q) policies for serial inventory systems using the guaranteed service approach

The guaranteed-service approach (GSA) was recently used to optimally place strategic safety stocks in multi-echelon inventory systems without fixed order costs. Under the GSA, excessive customer demand superior to a bound is fulfilled by using operating flexibility measures such as expediting and overtime. In this paper, the GSA is used to optimize the (R, Q) policy for a continuous review serial inventory system with Poisson demand and fixed order costs at each stocking location. Except for inventory holding costs, we also consider fixed order costs and operating flexibility costs in the system. Firstly, a deterministic mathematical programming model that considers the effects of using operating flexibility measures on the physical flow and the total cost of the system is established for optimizing the inventory policy under a given cycle service level. The model is solved by using an iterative procedure. In each iteration of the procedure, the fill rate of the system is estimated, and two sub-problems of the model, the order size decision sub-problem and the reorder point decision sub-problem, are solved by using dynamic programming algorithms for a given fill rate. With this procedure, the original optimization problem is solved by finding the optimal cycle service level of the system using a line search. Numerical results on randomly generated instances demonstrate the efficiency of the algorithms and the procedure.

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.

Optimal batch ordering policies for assembly systems with guaranteed service

For a supply chain modelled as a multi-echelon inventory system, effective management of its inventory at each stock is critical to reduce inventory costs while assuring a given service level to customers. In our previous work, we used the guaranteed-service approach (GSA) to design optimal echelon batch ordering policies for continuous-review serial systems with Poisson customer demand and fixed order costs. The approach assumes that the final customer demand is bounded and each stock has a guaranteed service time in the sense that the demand of its downstream stock can always be satisfied in the service time. This paper extends this work by considering more general assembly systems. We first derive an analytical expression for the total cost of the system in the long run. The problem of finding optimal echelon batch ordering policies for the system can then be decomposed into two independent sub-problems: order size decision sub-problem and reorder point decision sub-problem. We develop efficient dynamic programming algorithms for the two sub-problems. Numerical experiments on randomly generated instances show the effectiveness of the proposed approach.

Optimising reorder intervals and order-up-to levels in guaranteed service supply chains

We consider the problem of determining the optimal reorder intervals R and order-up-to levels S in a multi-echelon supply chain system where all echelons are assumed to have fixed ordering costs and to operate with a (R, S) policy with stationary nested power-of-two reorder intervals. By using the guaranteed service approach to model the multi-echelon system facing a stochastic demand, we formulate the problem as a deterministic optimisation model in order to simultaneously determine the optimal R and S parameters as well as the guaranteed service times. The model is a non-linear integer programming (NLIP) problem with a non-convex and non-concave objective function including rational and square root terms. Then, we propose a sequential optimisation procedure (SOP) to obtain near-optimal solutions with reasonable computational time. The numerical study demonstrates that for a general acyclic multi-echelon system with randomly generated parameters, the SOP is able to obtain near-optimal solutions of about 0.46% optimality gap in average in a few seconds. Moreover, we propose an improved direct approach using a global optimiser, bounding the decision variables in the NLIP model and considering the SOP solution as an initial solution. Numerical examples illustrate that this reduces significantly the computational time.

An integrated guaranteed- and stochastic-service approach to inventory optimization in supply chains

Multi-echelon inventory optimization literature distinguishes stochastic- (SS) and guaranteed-service (GS) approaches as mutually exclusive frameworks. While the GS approach considers flexibility measures at the stages to deal with stockouts, the SS approach only relies on safety stock. Within a supply chain, flexibility levels might differ between stages rendering them appropriate candidates for one approach or the other. The existing approaches, however, require the selection of a single framework for the entire supply chain instead of a stage-wise choice. We develop an integrated hybrid-service (HS) approach which endogenously determines the overall cost-optimal approach for each stage and computes the required inventory levels. We present a dynamic programming optimization algorithm for serial supply chains that partitions the entire system into subchains of different types. From a numerical study we find that, besides implicitly choosing the better of the two pure frameworks, whose cost differences can be considerable, the HS approach enables additional pipeline and on-hand stock cost savings. We further identify drivers for the preferability of the HS approach.

Stochastic inventory management for tactical process planning under uncertainties: MINLP models and algorithms

AbstractWe address in this article the mid‐term planning of chemical complexes with integration of stochastic inventory management under supply and demand uncertainty. By using the guaranteed service approach to model time delays in the flows inside the network, we capture the stochastic nature of the supply and demand variations, and develop an equivalent deterministic optimization model to minimize the production, feedstock purchase, cycle inventory, and safety stock costs. The model determines the optimal purchases of the feedstocks, production levels of the processes, sales of final products, and safety stock levels of all the chemicals. We formulate the model as a mixed‐integer nonlinear program with a nonconvex objective function and nonconvex constraints. To solve the global optimization problem with modest computational times, we exploit some model properties and develop a tailored branch‐and‐refine algorithm based on successive piecewise linear approximation. Five industrial‐scale examples with up to 38 processes and 28 chemicals are presented to illustrate the application of the model and the performance of the proposed algorithm. © 2010 American Institute of Chemical Engineers AIChE J, 2011