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This article deals with the situation where the producer manufactures several products with given production rates on a single machine. Only one product can be produced at a time on the machine. As an example, the producer may use higher or lower quality components in the production. Also, the products may have different holding costs. We present a model, where the demand appears at discrete points in time, and all demand must be met. We assume a deterministic, varying demand. By a production batch we mean a sequence of individual demands that is possible to manufacture in a continuous run to satisfy all the demands on time. A production run consists of a sequence of batches. There is a fixed set-up cost associated with each production run. By a production schedule we mean a sequence of “run” intervals such that each of them is partitioned into parts – for each product. This paper is concerned with the problem of how to determine the production schedule that minimizes the total production and holding cost over a finite time horizon.

We consider the problem of finding the optimal schedule of production runs of single machine to meet all discrete multiproduct demands which are required at discrete points of time. There is a time-invariant fixed joint setup cost. We assume there is no costs associated with switching the production rate. Each product has its own time-proportional stock holding cost and production rate. This type of finite time horizon problem is related to the multiproduct capacitated lot-sizing problem in which the production rate determines the period's production capacity. A schedule is described by the collection of production amount functions – each one for an individual product. The sum of these functions is equal to zero on the intervals where no product is manufactured. The sequential segments (where it is positive) determine the sequence of production batches. The objective is to determine the (joint) production amount function that minimizes the total cost over a finite time horizon. Hill [1], in the case of a single product, shows how the finite production rate problem can be transformed into a discrete time uncapacitated lot-sizing problem of the Wagner–Whitin type. We refer to Hill [1] for a detailed literature review of these relations. The objective of the present paper is to generalize the approach suggested by Hill to the multiproduct case. We do it by a natural generalization of his methodology. We assume that products are similar enough to be aggregated (using the same units). To find the joint batch production time, we decompose it for each product to solve an auxiliary allocation problem which has received considerable attention also in our paper [2]. Batching is clustering the products for manufacturing processing in the same production run. In this way, batching typically generates cycle inventory which can be advantageous for economic reasons; see Kuik and Salomon [3] for a discussion of batching analysis. Production in larger quantities reduces the number of setups.