Abstract
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.