Retailers repeatedly decide how much stock to order, how to split it across stores and how to ship it, under uncertain demand and with limited stock, labour and vehicle capacity. This study examines how Operations Research (OR) has been applied to these inventory replenishment and distribution decisions. Five published studies are reviewed, two in full and three as supporting papers, covering four stages of the retail supply chain: ordering, allocation, shipping and the store shelf record. For each study, the decision variables, objective function, constraints, assumptions, solution method, software and validation are set out and assessed, and the integer programme of one study is re-solved with its own published data. In four of the five studies, the model outperformed the benchmark its authors chose, from a profit gain of 2.23% to 4.92% of sales at a catalogue retailer to an estimated 3% to 4% more sales at Zara, using existing stock, staff and vehicles. None of the problems was solved exactly in daily use; each relied on a tested simplification, usually the newsvendor trade-off or a linear approximation. The re-solved model reproduces its reported optimum of 342 pallets, but shows that the paper's claimed gain over past dispatch rests on the wrong figure, and that relaxing the full-load rule alone would raise the optimum to 379 pallets. The quality of validation is what separates the strongest studies from the weakest.

Keywords— Operations research; inventory replenishment; retail supply chain; integer programming; newsvendor model; stock allocation