Costs of a Water Tank

GP > Posynomials > Example: Water Tank Model | Standard Forms | Applications

This is a simple example of a posynomial function, involving the construction and operating costs of a cylindrical liquid (say, oil, or water) storage tank.

A cylindrical water tank, with height and diameter . The tank includes a base, which is made of different material than the tank itself. In our model, the base's height does not depend on the tank's height. This is a reasonable approximation for heights not exceeding a certain value.

The parameters of the tank are its diameter and height . The costs to manufacture, and then operate during a given period of time (say, a year) the tank, include the following.

Filling costs represent the costs associated with supplying a volume of water in the given time period. These costs depends on the ratio $V_{rm supp}/V_{rm tank}$ , where $V_{rm supp}$ is the volume to be supplied, and $V_{rm tank} = (pi/4) hd^2$ is the volume of the tank (the smaller the volume of the tank is with respect to the volume to be supplied, the more often we have to refill the tank, and the larger the cost). Thus, the filling cost is inversely proportional to the tank's volume:

$C_{rm fill}(d,h) = alpha_1 frac{V_{rm supp}}{V_{rm tank}} = c_1 h^{-1}d^{-2},$

where alpha_1 is some positive constant, expressed in (say) dollars, and $c_1 := 4 alpha_1 V_{rm supp}/pi$ .

Construction costs include the costs associated with building a base for the tank, and costs associated with building the tank itself. In our model, the first type of costs depends only the base area , while the second depends on the surface of the tank, . (This assumes that we can use the same base height for a variety of tank heights.) Thus the construction cost function can be written

$C_{rm constr}(d,h) = C_{rm base}(d,h) + C_{rm tank}(d,h) = c_2 d^2 + c_3 dh,$

where the constants c_2 = alpha_2 pi/4 , c_3 = alpha_3 pi , with alpha_2>0, alpha_3>0 constants expressed in dollars per square meters.

The total manufacturing and operating cost function is thus the posynomial function

$C_{rm total}(d,h) = C_{rm fill}(d,h) + C_{rm constr}(d,h) = c_1 h^{-1}d^{-2} + c_2 d^2 + c_3 dh.$

Level sets of the cost function of the problem, corresponding to the values

$begin{array}{l} V_{rm supp} = 800,000 L, ;; alpha_1 = 10 : $, alpha_2 = 6 : $ / m^2, ;; alpha_3 = 2: $ / m^2. end{array}$

The function clearly appears not to be convex, since some level sets are not.

See also: Optimization of a water tank.