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Figure A.1: Structure of the interregional trade flows optimisation problem
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The capital letters R and S are used to respectively denote regions and sectors (ie. R1S1 denotes sector 1 in region 1. The letter n represents the number of regions, while the letter m represents the number of sectors. A total of 16 regions (Auckland inclusive), covering all 16 regional councils in New Zealand, were used in the Auckland region empirical analysis. This analysis was performed for all 23 sectors.
Matrix X (n x m x 2, (n x n - n) x m). This matrix describes the flow of trade between regions. A negative sign (-) denotes exports, while a positive sign (+) denotes imports. Column 1, for example, describes the trade flows between region 1, R1, and region 2, R2.
Vector y (n x m x 2, 1). This column vector describes imports to, and exports from, each region by sector. The element in row 1, for example, represents the sum of all sector 1 exports originating from region 1, -∑R1S1. The elements in this vector are used as binding constraints in the optimisation. This information is taken directly from the input-output tables generated by the GRIT system.
Vector z (1, (n x n - n) x m). This row vector denotes 'freight haulage times' between regions per dollar of trade flow. Freight haulage times were calculated using an origin/destination matrix. The major city/town in each region was used as the point of origin/destination. Haulage times between the North and South Islands were adjusted for the Cook Strait crossing.
Scalar v (1, 1). This scalar is the sum of row vector z. It represents the total freight travel time needed to move goods and services between all permutations of regions and sectors. Minimisation of this scalar is the objective function.
The optimisation is solved as follows:
Min: zw=v
subject to: Xw=y
w >= 0
where: w = column vector ((n x n - n) x m, 1) describing the flow ($) of m sector commodities between n regions, to be solved for.
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