In contemporary market, people prefer to products with best quality at the lowest prices, regardless of where they are produced. Hence, most companies can no longer afford to produce their products in a single plant to provide the products needed with the lowest production cost in the market [5]. To get the edge of competitive strategy, several studies ([9], [20], [37]) suggest that firms should transfer production to locations for lowing production cost and maximizing total production quantities in a whole while maintaining quality and reliability.
The term production is typically used to denote the process of transforming input resources into output products. The allocation problem of input resources such that the output is maximized has been a critical issue since early 1900. The neoclassical authors around early 1900 used production function to relate outputs to its underlying input factors to manifest production-related phenomena such as diminishing returns [26], [34]. Although Wicksell contributed the initiative work of production function ([14], [26], [31], [36]), the term Cobb-Douglas function stands for production function in honor of Cobb and Douglas’ articles published in the American Economic Review ([8]).
In this article we consider an allocation problem of production resources in a multi-plant firm and formulate it as a mathematical programming model. The objective function of interest is modeled as a Cobb-Douglas-like function. Based on the Kuhn-Tucker conditions, two algorithms are proposed to obtain the solution to the mathematical programming model. To be specific, the problem of interest can be described as follows.
A multi-plant firm contains n plants, each employs production factors to produce the same product. In a production horizon, the amount of a specific production factor x is bound up to a fixed number H. Meanwhile, other corresponding input factors for each plant are held to be fixed. Since the production efficiency and other corresponding production factors for each plant are not quite the same, the output quantities among various plants are different. The problem is to allocate the amount of the specific production factor x employed among n plants such that the total product quantity is maximized. This is indeed a multiple-input, single-output (MISO) case. Intuitively, a multiple-input, multiple-output (MIMO) case could be more applicable than a MISO one. However, a MIMO case needs additional assumptions to accommodate such a setting in various analyses; it is not commonly seen in practice [3]. Therefore, a MISO case is considered in this article instead of MIMO one.
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