13.2 Variations

Planned shortages. If unmet demand can be backordered at cost \(p\) per unit per year, it pays to let the stock run negative for part of each cycle. The optimal order size and maximum backlog are \[ Q^* = \sqrt {\frac {2KD}h\cdot \frac {h + p}p}, \qquad B^* = Q^*\frac h{h + p} , \] and as \(p \to \infty \) these reduce to the EOQ with no backlog.

Finite production rate. If the items are made at rate \(P > D\) rather than bought, the stock builds up at rate \(P - D\) during production, the maximum stock is \(Q(1 - D/P)\), and the economic production quantity is \(Q^* = \sqrt {2KD/h(1 - D/P)}\).

Quantity discounts. If the unit price falls with the order size, as \(c_1 > c_2 > \cdots \) for \(Q\) in successive ranges, the total cost \(C_k(Q) = KD/Q + ic_kQ/2 + c_kD\) includes the purchase cost and is a different convex function on each range. For each range, the best feasible \(Q\) is the EOQ for that price if it lies in the range, and otherwise the nearest end of the range, which in practice is the lower end: the EOQs differ little between prices, while the cheaper ranges start at larger orders. The best of these candidates is optimal. In Example 13.1.2 with prices Rs. 12, 11.5 and 11 for orders below \(2000\), from \(2000\) to \(4999\), and of \(5000\) or more, the candidates cost Rs. 149 367, 143 254 and 138 700, so the firm should order \(5000\) at a time, more than twice the EOQ, to earn the lowest price (Software 13.5.1).

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