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行业空间配置的线性对偶模式

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A LINEAR MODEL WITH ITS DUAL OF SPATIAL ALLOCATION FOR SOME SPECIFIED INDUSTRIES

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- Department of geography, Beijing Narmal University, 100875

**Received date: **1993-02-01

**Revised date: **1993-06-01

**Online published: **1994-03-15

梁进社
. 行业空间配置的线性对偶模式[J]. *地理学报*, 1994
, 49(2)
: 128
-138
.
DOI: 10.11821/xb199402004

Locational researchers are interested in not only the principles of location choice for an individual firm, but also the spatial allocation of an industry. This paper attends to study the production allocation in space for such industries as iron and steel, nonferrous metal, and organic synthesis etc, in each of which firms are quite big in scale, but the total number of them is small, thereby locational choice for each of them has a quite great effect on the others. Otherwise, the demands of these industries distribute unequally in space. For this kind of industries the analytical methods used by Weber and Losch appear to be invalid because their presumptions do not exist here. Weber’s theory needs to be given the material and product sale sites for each one, therefore it does not consider the spatial competition in the same industry. Losch’s theory ignores the unequality of resource and product demand distribution in space. The author of this paper proposed a linear model with its dual for explaining the spatial allocation of this kind of industries.First, the author assumes that there are many production sites available for an industry which has given materials supply and markets. Also he makes capacity b^{h}_{i} and unit cost f^{h}_{i} of material h in its site i as well as demand amount q_{j}; of the product in its sale place j presumptively. Simultaneously there are some exogenous variables such as unit product processing cost c_{1}; of production site i, transport cost rate t^{h}_{ij}; of material h from its site i to production site j,transport cost rate t_{ij} of the product from production site i to its sale place j. Then the spatial allocation of the industry can be reduced to a linear programming problem as follow: Where rh =technique coefficient related to material h;xij=transport amounts of material h from its site i;to production site j;yi =output of the product in production site i;zij=transport amounts of the product from production site i to sale place j.The model is to minimize the total cost including the production and transport both for material and product under the capacity of every material site and the demand quantites of every market. Its solution is composed of the output from every material and product site, as well as transport amounts from material sites to product processing sites, and then to the markets. Based upon spatial price equilibrium the author sets up the dual model of the linear programming. Its objective function is to maximize the industrial economic income obtained from the production and transport of materials and products: Its constraints are as follows:<1> The difference between the product shadow prices in a market and production site is not in excess of the transport cost rate conneected the two places:pj-ωi≤tij <2> The difference between the product shadow prices and its material cost is not in excess of its production cost per unit at every producing site: <3> The difference between shadow price of a material at production site and the economic rent at its site is not in excess of the sum of its unit cost added transport cost rate connected the two places: where pj= shadow price of the product at sale place j wi=shadow price of the product at production place i vjh=shadow price of material h at product processing site j uih=shadow price or economic rent of material h at its site i From solving the dual model, what can be got are the shadow prices of the materials and product at respective places. Thus, the solutions of the primary and its dual model give out the industrial distribution in space with the minimum-total-cost and shadow prices of the materials and product at respective locations.Afterwards, the author seeks the relationships between solutions of the two models from spatial price equilibrium, and gets following results: <1> The product is transported to a sale place from a production site only when the difference of the shadow prices offsets the transport cost rate between the two places:(pj-ωi-tij)zij = 0<2> The product is produced only when the shadow price blances its per unit cos

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