This paper applies the simplex algorithm of linear programming to the product-mix decision of an anonymised pure-water bottling factory in Agbor, Delta State, which produces three product lines [sachet water (bagged, 20 × 50cl sachets per bag), 50cl bottled water (24 bottles per carton) and 75cl can water (24 cans per carton)] from a shared filling-and-sealing line. Per-unit profit contributions (short-run contribution margins, before fixed overhead) and resource requirements (treated water, machine time, and a packaging working-capital budget) are derived explicitly from estimated unit prices, material costs and processing times, and combined with market-survey demand ceilings for each product to formulate a three-variable, six-constraint linear program. Starting from the all-slack basic feasible solution, the simplex method is solved to optimality in three non-degenerate pivot iterations, and the resulting basis is confirmed unique. The optimal continuous solution recommends approximately 614 bags of sachet water, 150 cartons of 50cl bottled water and 35 cartons of 75cl can water per week, for a maximum weekly contribution margin of ₦97,225; a feasible integer plan of 608, 150 and 35 units respectively attains ₦97,012, within 0.2% of the continuous optimum. Only the treated-water supply, the packaging budget, and the bottled-water demand ceiling bind at the optimum; the shadow price of the binding demand ceiling (₦221.58 per additional carton) is far larger than that of either binding resource constraint, and right-hand-side ranging shows this shadow price holds only while the ceiling is relaxed up to 220.46 cartons per week, beyond which can-water production is crowded out entirely and the basis changes. The paper illustrates, on a small and fully worked application, both the mechanics of the simplex algorithm and the economic information (optimal quantities, shadow prices, and their valid ranges) that a completed tableau yields.

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