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# 数学代写|运筹学代写Operations Research代考|MAST30013 The Economic Order Quantity with Quantity Discounts

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## 数学代写|运筹学代写Operations Research代考|The Economic Order Quantity with Quantity Discounts

So far we have assumed that the unit purchasing cost $p$ is constant and independent of the order size $Q$. Recall that in the original economic order quantity model, the costs actually were the sum of ordering, holding, and purchasing costs, viz.,
$$T C(Q, p)=c_o D / Q+1 / 2 c_h Q+p D .$$
However, we argued, for a fixed price $p$, the purchasing costs during the period under consideration are $p D$, which is a constant, which does not influence the solution, so that we could (and did) ignore the purchasing costs. In practice, however, many suppliers will offer incentives for purchases of larger quantities in the form of lower unit costs. The basic economic order quantity model described in Sect. $12.2$ can easily be modified to take such quantity discounts into consideration. For simplicity, we will restrict ourselves to the standard model with no shortages allowed and three price levels: the original non-discounted price and two discount levels. It is straightforward to extend the model to any number of discount levels.
Before we proceed, we have to make a minor modification. Recall that the main component of the holding costs is the cost for tied-up capital. Given that we paid a fixed price for the good so far, this cost could simply be expressed as a dollar amount for each unit in stock. Since we are now paying a price that is no longer fixed but does depend on the discount level that we choose, the unit holding costs have to be redefined. This is most easily done by defining $c_h$ as a proportion of the unit purchasing price $p$. Given that, the economic order quantity is redefined as
$$Q^*=\sqrt{\frac{2 D c_o}{c_h p}}$$

## 数学代写|运筹学代写Operations Research代考|The Production Lot Size Model

As an alternative to ordering models of the type discussed so far in this chapter, we may produce the desired items ourselves. In such a case, the items will not arrive in one bulk as they do in case of orders, but they arrive one piece at a time from our machines. Assume for the time being that we cannot regulate the speed with which our machines produce the units: we either turn the machine on, in which they churn out $r$ units per day (this is our production rate), or we turn off the machine, in which case we make nothing. Recall that our annual demand was assumed to be of magnitude $D$, from which we can easily compute the daily demand $d$, which is $D$ / 365 , $D / 360$, or $D / 250$ (working days), depending on the decision maker’s specifications. Before engaging in any computations, it is necessary to determine whether or not the system has any feasible solutions. The simple regularity condition is that $r \geq d$. If this condition is not satisfied, then it will not be possible to satisfy the total demand, and we have to find ways that allow us to do so. The following arguments assume that the regularity condition is satisfied.

Batch or intermittent production as described above occurs in many vertically integrated companies, where the ordered items are produced internally. A production run can then be considered an order, with the production run size corresponding to the order size $Q$, and the production setup costs corresponding to the ordering costs $c_o$

## 数学代写|运筹学代写Operations Research代考|The Economic Order Quantity with Quantity Discounts

$$T C(Q, p)=c_0 D / Q+1 / 2 c_h Q+p D .$$

$$Q^*=\sqrt{\frac{2 D c_o}{c_h p}}$$

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