Evolutionary multiobjective optimization of the multi-location transshipment problem

We consider a multi-location inventory system where inventory choices at each location are centrally coordinated. Lateral transshipments are allowed as recourse actions within the same echelon in the inventory system to reduce costs and improve service level. However, this transshipment process usually causes undesirable lead times. In this paper, we propose a multiobjective model of the multi-location transshipment problem which addresses optimizing three conflicting objectives: (1) minimizing the aggregate expected cost, (2) maximizing the expected fill rate, and (3) minimizing the expected transshipment lead times. We apply an evolutionary multiobjective optimization approach using the strength Pareto evolutionary algorithm (SPEA2), to approximate the optimal Pareto front. Simulation with a wide choice of model parameters shows the different trades-off between the conflicting objectives.

💡 Research Summary

The paper addresses a multi‑location inventory system in which a central decision maker determines the order quantities for each of n retail stores that sell a single product. After the replenishment phase, stochastic demand is realized at each store. Unsatisfied demand at a store can be met by lateral transshipments from other stores that have surplus inventory. Each unit shipped from store i to store j incurs a deterministic cost τij and a deterministic lead time Lij; remaining surplus inventory is penalized by a holding cost hi, while any residual shortage after transshipment is penalized by a shortage cost pj.

Three conflicting objectives are formulated: (1) minimize the expected total system cost C(S), which aggregates holding, shortage, and transshipment costs; (2) maximize the expected fill rate F(S), defined as the proportion of total demand satisfied after transshipment; and (3) minimize the expected total transshipment lead time L(S), calculated as the sum over all shipments of the product of shipped quantity and its associated lead time. The decision variables are the order‑up‑to levels S = (S1,…,Sn). Because the transshipment quantities Tij are determined by a complete‑pooling policy (the minimum of available surplus at the source and unmet demand at the destination), they are obtained by solving a linear program K for each demand realization. Closed‑form expressions for K exist only for highly symmetric cases; therefore, the authors solve K numerically (Simplex) and estimate the expectations of C, F, and L by Monte‑Carlo resampling.

The resulting multi‑objective stochastic programming problem (P) seeks a Pareto‑optimal set of order‑quantity vectors that simultaneously achieve low cost, high service, and short lead times. Classical scalarization methods (weighted sums) are deemed inadequate because they obscure the trade‑offs and provide only a single solution. Consequently, the authors employ the Strength Pareto Evolutionary Algorithm 2 (SPEA2), an elitist multi‑objective evolutionary algorithm that maintains an external archive of non‑dominated solutions, assigns strength values based on domination count, and uses binary tournament selection, crossover, and mutation to evolve a population over successive generations.

A comprehensive experimental study is conducted. The authors generate a variety of test instances by varying the number of stores, cost parameters (hi, pj, τij), and lead‑time matrices Lij. For each instance, 2,000 demand scenarios are sampled to evaluate candidate solutions, and SPEA2 is run for a sufficient number of generations to converge to a stable Pareto front. The empirical results illustrate clear trade‑offs: reducing expected cost typically lowers the fill rate, while minimizing lead time often requires increased transshipment activity, which raises total cost. When transshipment costs are low (τij < hi + pj), the Pareto front expands, indicating many viable compromises; when transshipment costs are high, the front collapses toward a cost‑only solution because shipments become uneconomical.

The paper’s contributions are threefold. First, it introduces a novel multi‑objective formulation that simultaneously captures cost, service level, and time dimensions—an extension beyond most prior work that considered only cost or fill rate. Second, it integrates deterministic lead times into the objective function, thereby acknowledging the practical impact of shipment delays on customer satisfaction. Third, it demonstrates that SPEA2 can effectively approximate the Pareto set for this stochastic, combinatorial problem, providing decision makers with a rich set of alternatives rather than a single point solution.

Future research directions suggested include incorporating fixed transshipment setup costs, extending the model to multi‑echelon or multi‑product settings, exploring alternative transshipment policies (e.g., threshold‑based or dynamic pooling), and benchmarking SPEA2 against other state‑of‑the‑art evolutionary multi‑objective algorithms such as NSGA‑III or MOEA/D. Additionally, the authors propose investigating online or adaptive versions of the approach that could update order quantities in real time as demand information becomes available.