Ship-pack optimization in a two-echelon distribution system

In large distribution systems, distribution centers (DC) deliver some merchandize to their retail stores in size-specific packages, also called ship-packs. These ship-packs include cases (e.g., cartons containing 24 or 48 units), inners (packages of 6 or 8 units) or eaches (individual units). For each Stock Keeping Unit (SKU), a retailer can decide which of these ship-pack options to use when replenishing its retail stores. Working with a major US retailer, we have developed a cost model that balances DC handling costs, store handling costs and inventory-related costs at both the DC and the stores, and therefore can help to determine the optimum warehouse ship-pack for each SKU. We implement our model for a sample of 529 SKUs, and show that by changing ship-pack size for about 30 SKUs, the retailer can reduce its total cost by 0.3% - 0.4%. Interestingly, we find that most of the cost savings occurs at the DC level.     

A simulation optimization approach for a two-echelon inventory system with service level constraints

In this paper, we present a simulation optimization algorithm for solving the two-echelon constrained inventory problem. The goal is to determine the optimal setting of stocking levels to minimize the total inventory investment costs while satisfying the expected response time targets for each field depot. The proposed algorithm is more adaptive than ordinary optimization algorithms, and can be applied to any multi-item multi-echelon inventory system, where the cost structure and service level function resemble what we assume. Empirical studies are performed to compare the efficiency of the proposed algorithms with other existing simulation algorithms.     

Determining the Pareto set in a bicriteria two-echelon inventory/distribution system

This article concerns two-echelon inventory/distribution system, consisting of a warehouse and a retailer. We assume that the demand is deterministic and stockouts are not permitted. Two criteria are considered: to minimize the annual inventory cost and the annual total number of damaged items by improper shipment handling. The problem consists of determining the non-dominated inventory policies in such a way that the trade-off between both criteria is achieved. We present the characterization of the non-dominated optimal solution set and we use this result to correct the solution method previously proposed by other authors for a problem with identical cost structure. An efficient algorithm to calculate the non-dominated solution set is introduced. Computational results on several randomly generated problems are reported.     

A cooperative inventory policy with deteriorating items for a two-echelon model

A cooperative inventory policy between supplier and buyer is proposed. Unlike other studies, we consider the case of deteriorating items and permit the completed back-order in the problem. We solve the problem without the condition of equal replenishments periods during a specified planning horizon and present a procedure to find the optimal solution. A case is presented to demonstrate the application of the proposed approach. The sensitivity analysis for a cooperation policy between supplier and buyer also are explored.     

A multi-criteria optimization model for humanitarian aid distribution

Natural disasters are phenomenons which strike countries all around the world. Sometimes, either by the intensity of the phenomenon or the vulnerability of the country, help is requested from the rest of the world and relief organizations respond by delivering basic aid to those in need. Humanitarian logistics is a critical factor in managing relief operations and, in general, there is a lack of attention on the development of mathematical models and solution algorithms for strategic and tactical decisions in this area. We acknowledge that in humanitarian logistics traditional cost minimizing measures are not central, and postulate that other performance measures such as time of response, equity of the distribution or reliability and security of the operation routes become more relevant. In this paper several criteria for an aid distribution problem are proposed and a multi-criteria optimization model dealing with all these aspects is developed. This model is the core of a decision support system under development to assist organizations in charge of the distribution of humanitarian aid. Once the proposed criteria and the model are described, an illustrative case study based on the 2010 Haiti catastrophic earthquake is presented, showing the usefulness of the proposal.     

Designing a two-echelon distribution network under demand uncertainty

This paper proposes a comprehensive methodology for the stochastic multi-period two-echelon distribution network design problem (2E-DDP) where product flows to ship-to-points are directed from an upper layer of primary warehouses to distribution platforms (DPs) before being transported to the ship-to-points. A temporal hierarchy characterizes the design level dealing with DP location and capacity decisions, as well as the operational level involving transportation decisions as origin-destination flows. These design decisions must be calibrated to minimize the expected distribution cost associated with the two-echelon transportation schema on this network under stochastic demand. We consider a multi-period planning horizon where demand varies dynamically from one planning period to the next. Thus, the design of the two-echelon distribution network under uncertain customer demand gives rise to a complex multi-stage decisional problem. Given the strategic structure of the problem, we introduce alternative modeling approaches based on two-stage stochastic programming with recourse. We solve the resulting models using a Benders decomposition approach. The size of the scenario set is tuned using the sample average approximation (SAA) approach. Then, a scenario-based evaluation procedure is introduced to post-evaluate the design solutions obtained. We conduct extensive computational experiments based on several types of instances to validate the proposed models and assess the efficiency of the solution approaches. The evaluation of the quality of the stochastic solution underlines the impact of uncertainty in the two-echelon distribution network design problem (2E-DDP).     

A two-echelon inventory model with lost sales

This paper considers a single-item, two-echelon, continuous-review inventory model. A number of retailers have their stock replenished from a central warehouse. The warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. Demand not met at a retailer is lost. The order quantity from each retailer on the warehouse and from the warehouse on the supplier takes the same fixed value Q, an exogenous variable determined by packaging and handling constraints. Retailer i follows a (Q, R_i) control policy. The warehouse operates an (SQ, (S − 1)Q) policy, with non-negative integer S. If the warehouse is in stock then the lead time for retailer i is the fixed transportation time L_i from the warehouse to that retailer. Otherwise retailer orders are met, after a delay, on a first-come first-served basis. The lead time on a warehouse order is fixed. Two further assumptions are made: that each retailer may only have one order outstanding at any time and that the transportation time from the warehouse to a retailer is not less than the warehouse lead time. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Procedures for determining these performance measures and optimising the behaviour of the system are developed.     

Towards a multi-objective performance assessment and optimization model of a two-echelon supply chain using SCOR metrics

This paper aims at multi-objective performance assessment and optimization of a multi-period two-echelon supply chain consisting of a supplier and a manufacturer. On the basis of the assessment system of the supply-chain operations reference model, the supply chain’s performance is investigated with respect to costs, assets, agility, reliability and responsiveness. First, methods to quantify these five performance attributes are put forward. Then a multi-objective mathematical programming model is developed for production decision making of components and products so that the supply chain’s performance frontier formed with Pareto efficient performance values can be achieved. Thereafter a simple augmented \(\epsilon \) -constraint method is proposed for searching for all Pareto efficient solutions of the multi-objective mathematical programming problem. Finally, efficiency of the method is demonstrated with a numerical example and a sensitivity analysis is implemented to reveal effects of capacity expansion on supply chains’ performance.     

A two-echelon stochastic facility location model for humanitarian relief logistics

We develop a two-stage stochastic programming model for a humanitarian relief logistics problem where decisions are made for pre- and post-disaster rescue centers, the amount of relief items to be stocked at the pre-disaster rescue centers, the amount of relief item flows at each echelon, and the amount of relief item shortage. The objective is to minimize the total cost of facility location, inventory holding, transportation and shortage. The deterministic equivalent of the model is formulated as a mixed-integer linear programming model and solved by a heuristic method based on Lagrangean relaxation. Results on randomly generated test instances show that the proposed solution method exhibits good performance up to 25 scenarios. We also validate our model by calculating the value of the stochastic solution and the expected value of perfect information.     

Finite element approximation of a shape optimization problem for von karman system

An optimal shape design problem of an elastic body described by a system of two nonlinear elliptic equations is considered. The problem is to find the boundary of the domain occupied by the body in such a way that the stiffnes of the system in the equilibrium state is minimized.

It is assumed that the volume of the body is constant. Moreover, the function describing the boundary of the domain and its gradient are bounded.     

A genetic algorithm for a single product network design model with lead time and safety stock considerations

We consider a two-stage supply chain with a production facility that replenishes a single product at retailers. The objective is to locate distribution centers in the network such that the sum of facility location, pipeline inventory, and safety stock costs is minimized. We explicitly model the relationship between the flows in the network, lead times, and safety stock levels. We use genetic algorithms to solve the model and compare their performance to that of a Lagrangian heuristic developed in earlier work. A novel chromosome representation that combines binary vectors with random keys provides solutions of similar quality to those from the Lagrangian heuristic. The model is then extended to incorporate arbitrary demand variance at the retailers. This modification destroys the structure upon which the Lagrangian heuristic is based, but is easily incorporated into the genetic algorithm. The genetic algorithm yields significantly better solutions than a greedy heuristic for this modification and has reasonable computational requirements.     

A genetic algorithm for optimal operating parameters of VMI system in a two-echelon supply chain

This paper deals with the operational issues of a two-echelon single vendor–multiple buyers supply chain (TSVMBSC) model under vendor managed inventory (VMI) mode of operation. The operational parameters to the above model are: sales quantity and sales price that determine the channel profit of the supply chain, and contract price between the vendor and the buyer, which depends upon the understanding between the partners on their revenue sharing. In order to find out the optimal sales quantity for each buyer in TSVMBSC problem, a mathematical model is formulated. Optimal sales price and acceptable contract price at different revenue share are subsequently derived with the optimal sales quantity. A genetic algorithm (GA) based heuristic is proposed to solve this TSVMBSC problem, which belongs to nonlinear integer programming problem (NIP). The proposed methodology is evaluated for its solution quality. Furthermore, the robustness of the model with its parameters, which fluctuate frequently and are sensitive to operational features, is analysed.     

Hybrid algorithm for a vendor managed inventory system in a two-echelon supply chain

In this paper we address the issue of vendor managed inventory (VMI) by considering a two-echelon single vendor/multiple buyer supply chain network. We try to find the optimal sales quantity by maximizing profit, given as a nonlinear and non-convex objective function. For such complicated combinatorial optimization problems, exact algorithms and optimization commercial software such as LINGO are inefficient, especially on practical-size problems. In this paper we develop a hybrid genetic/simulated annealing algorithm to deal with this nonlinear problem. Our results demonstrate that the proposed hybrid algorithm outperforms previous methodologies and achieves more robust solutions.     

A two-echelon inventory system with transportation capacity constraint

Consider a one-warehouse multi-retailer system under constant and deterministic demand, which is subjected to transportation capacity for every delivery period. To search for the best stationary zero inventory ordering (ZIO) policy, or the best power-of-two policy, or the best nested policy, the problem is formulated as a 0–1 integer linear program in which the objective function comprises of a fixed transportation cost whenever a delivery is made and the inventory costs for both the warehouse and retailers. To overcome the transportation capacity limitation, we extend the policies to allow for staggering deliveries. It is shown that with transportation capacity constraint the non-staggering policy can have its effectiveness close to 0% from the best staggering policy and the power-of-two policy with staggering allowed can have its effectiveness close to 0% from the optimal policy. Nevertheless in general, the power-of-two policy fairs well on a number of randomly generated problems. To solve the large distribution network problem, an efficient heuristic based on the power-of-two policy with staggering of deliveries is suggested.     

Valuing executive stock options: A quadratic approximation

This paper develops a continuous-time model for valuing executive stock options (ESOs) with features of early exercise, delayed vesting and forfeiture. Applying the quadratic approximation established for valuing American options into ESOs, we obtain an explicit formula for the fair ESO value at its grant date. We show that the approximation formula is consistent with the exact results for two special cases either with no dividend or infinite maturity, and also that the perpetual value for the latter case gives an upper bound of the ESO value. To see the performance of the formula, we numerically examine it with benchmark results generated by a binomial-tree model for some particular cases. Numerical experiments show that there is a complementary relation between the vesting and trading periods with respect to exit rate of ESO holders.     

A Wiener process model with truncated normal distribution for reliability analysis

The use of degradation model to perform the reliability analysis has drawn much attention due to the fact that the performance of numerous highly reliable systems degrades over time. To describe the unit-to-unit variability for a population of systems, the random effect has been incorporated into the degradation model that plays an important part in assessing the reliability of deteriorating systems. In the existing literature, the normal distribution is commonly adopted to represent the random effect, but the assumption can be unsuitable for some practical applications, such as the degradation process of train wheels. In this paper, we present a degradation modeling and reliability estimation approach by using truncated normal distribution to characterize the unit-to-unit variability. A Wiener process with truncated normal distribution is firstly applied to model the degradation process of the deteriorating system, and the analytical expressions of probability density function and reliability function are derived. Expectation maximization algorithm is then used to estimate the model parameters. The effectiveness and feasibility of the presented approach are illustrated through a numerical example and practical case studies for laser devices and train wheels. The results indicate that the presented approach can obtain better reliability estimation results by considering the truncated normal distribution when the unit-to-unit variability has significant difference.     

A production-inventory model with probabilistic deterioration in two-echelon supply chain management

In this study, a production-inventory model is developed for a deteriorating item in a two-echelon supply chain management (SCM). An algebraical approach is applied to find the minimum cost related to this entire SCM. We consider three types of continuous probabilistic deterioration function to find the associated cost. The purpose of this study is to obtain the minimum cost with integer number of deliveries and optimum lotsize for the three different models. Some numerical examples, sensitivity analysis and graphical representation are given to illustrate the model. A numerical comparison between the three models is also given.     

A dynamic system model for solving convex nonlinear optimization problems

This paper proposes a feedback neural network model for solving convex nonlinear programming (CNLP) problems. Under the condition that the objective function is convex and all constraint functions are strictly convex or that the objective function is strictly convex and the constraint function is convex, the proposed neural network is proved to be stable in the sense of Lyapunov and globally convergent to an exact optimal solution of the original problem. The validity and transient behavior of the neural network are demonstrated by using some examples.