water supply alternatives. This problem is a simplification of a water supply planning pro...

water supply alternatives. This problem is a simplification of a water supply planning problem faced by the adjacent cities of Beijing and Tianjin, China. These cities are experiencing a combination of growth in water demand and overdrafts of their groundwater supplies and so are reaching out for new supplies. Three fundamental types of alternatives are available:

(1) new reservoirs on rivers yet untapped,

(2) a diversion canal from the Yellow River, and

(3) water conservation/demand reduction.

Demand reduction has a number of dimensions, but in the case of these cities, the largest reduction can come about by changing local crop irrigation to a trickle irrigation system. This problem has all three of these elements, hut is obviously reduced in scope from the planning problem that the two cities face. Each alternative has a cost associated with it, and each has a volumetric yield in cubic kilometers per year. Even this system is made complicated, however, by the interaction of the components, as will be explained. The three components are shown below.

In the accompanying table. the water supply alternatives are numbered. and their cost and yield are listed. Interacting groups of the alternatives are also numbered, and their costs and yields are listed as well. Note that alternatives 6.7. R. and 9. the interacting groups, consist of combinations of reservoirs, The cosh of these combinations are nothing but the sum of the costs of the components. but the yields are not the sum of yicld-, By methods of water resources systems analysis. an engineer can calculate the firm yield of a pair or triplet of reservoirs, and this yield is typically larger than the sum of the individual yields. often by a good margin. Hence, the yield q7. for instance, is greater than , the sum of the yields of the component alternatives.

You are given a budget B that can be expended on all projects. What is the largest firm yield that can be obtained under this budget limitation? Structure ;:1 zero-one programming problem to show bow to solve this problem. There may be several different ways to formulate the problem.

Note that if reservoirs A and B are both built, it is equivalent to building alternative 6, and you must not count the individual yields, only the joint yield. Similarly, if A, Rand C are all built. the project/alternative 9 has been undertaken and the joint yield of I) has been achieved, not the sum of individual yields. That is, you cannot undertake alternative I) and any of alternatives 1,2. or 3 at the same time if you wish to count costs and yields correctly. You are given a zero-one variable. x,j., to represent whether alternative j is undertaken or not undertaken.