Production systems and hierarchies of centres : The relationship between spatial and economic structures

1. Definition of the problem and results -- 1.1. Some comments on the structure of systems of centres -- 1.2. A brief survey of problems -- 1.3. Summary and main conclusions -- 2. Earlier works on systems of centres -- 2.1. Necessary conditions for hierarchy -- 2.2. Models for vertically integrated industries -- 2.3. Models with input-output coefficients -- 3. Hierarchy of centres in Sweden at the beginning of the 1970s — an empirical test on Swedish data -- 3.1. Background and presentation of the problem -- 3.2. An empirical investigation for Sweden -- Supplement. Component analysis — presentation and use -- 4. An attempt to extend the problem of Tinbergen and Bos — capacities of plants given -- 4.1. Presentation of the problem -- 4.2. Models for testing whether a centre should be split up -- 4.3. A programming model -- 4.4. Some numerical examples -- 4.5. Problems when decision-making is individual and based on price information -- Supplement. Balas’ method for combinatorial solution of integer problems -- 5. The system of centres when capacities of plants are not given — a further extension of the problem -- 5.1. Presentation of the problem -- 5.2. Control and co-ordination costs -- 5.3. A model for quadratic programming -- Appendix 1 -- Appendix 2 -- Appendix 3 -- Appendix 4 -- References.