Selecting the Best Location for a Meteorological Tower: A Case Study of Multi-objective Constraint Optimization

Using the problem of selecting the best location for a meteorological tower as an example, we show that in multi-objective optimization under constraints, the traditional weighted average approach is often inadequate. We also show that natural invariance requirements lead to a more adequate approach – a generalization of Nash’s bargaining solution. Case study. We want to select the best location of a sophisticated multi-sensor meteorological tower. We have several criteria to satisfy. For example, the station should not be located too close to a road, so that the gas flux generated by the cars do not influence our measurements of atmospheric fluxes; in other words, the distance x1 to the road should be larger than a certain threshold t1: x1 > t1, or y1 def = x1 − t1 > 0. Also, the inclination x2 at the should be smaller than a corresponding threshold t2, because otherwise, the flux will be mostly determined by this inclination and will not be reflective of the atmospheric processes: x2 < t2, or y2 def = t2 − x2 > 0. General case. In general, we have several such differences y1, . . . , yn all of which have to be non-negative. For each of the differences yi, the larger its value, the better. Multi-criteria optimization. Our problem is a typical setting for multi-criteria optimization; see, e.g., [1, 4, 5]. Weighted average. A most widely used approach to multi-criteria optimization is weighted average, where we assign weights w1, . . . , wn > 0 to different criteria yi and select an alternative for which the weighted average w1 · y1 + . . .+wn · yn attains the largest possible value. Additional requirement. In our problem, we have an additional requirement – that all the values yi must be positive. Thus, we must only compare solutions with yi > 0 when selecting an alternative with the largest possible value of the weighted average. Limitations of the weighted average approach. In general, the weighted average approach often leads to reasonable solutions of the multi-criteria optimization problem. However, as we will show, in the presence of the additional positivity requirement, the weighted average approach is not fully satisfactory. A practical multi-criteria optimization must take into account that measurements are not absolutely accurate. Indeed, the values yi come from measurements, and measurements are never absolutely accurate. The results ỹi of the measurements are close to the actual (unknown) values yi of the measured quantities, but they are not exactly equal to these values. If – we measure the values yi with higher and higher accuracy and, – based on the resulting measurement results ỹi, we conclude that the alternative y = (y1, . . . , yn) is better than some other alternative y′ = (y′ 1, . . . , y ′ n), then we expect that the actual alternative y is indeed either better than y′ or at least of the same quality as y′. Otherwise, if we do not make this assumption, we will not be able to make any meaningful conclusions based on real-life (approximate) measurements. The above natural requirement is not always satisfied for weighted average. Let us show that for the weighted average, this “continuity” requirement is not satisfied even in the simplest case when we have only two criteria y1 and y2. Indeed, let w1 > 0 and w2 > 0 be the weights corresponding to these two criteria. Then, the resulting strict preference relation  has the following properties: – if y1 > 0, y2 > 0, y′ 1 > 0, and y ′ 2 > 0, and w1 · y1 + w2 · y2 > w1 · y′ 1 + w2 · y′ 2, then y = (y1, y2)  y′ = (y′ 1, y′ 2); – if y1 > 0, y2 > 0, and at least one of the values y′ 1 and y ′ 2 is non-positive, then y = (y1, y2)  y′ = (y′ 1, y′ 2). Let us consider, for every ε > 0, the tuple y(ε) def = ( ε, 1 + w1 w2 ) , with y1(ε) = ε and y2(ε) = 1 + w1 w2 , and also the comparison tuple y′ = (1, 1). In this case, for every ε > 0, we have w1 · y1(ε) + w2 · y2(ε) = w1 · ε + w2 + w2 · w1 w2 = w1 · (1 + ε) + w2 and w1 · y′ 1 + w2 · y′ 2 = w1 + w2, hence y(ε)  y′. However, in the limit ε → 0, we have y(0) = ( 0, 1 + w1 w2 ) , with y(0)1 = 0 and thus, y(0) ≺ y′. What we want: a precise description. We want to be able to compare different alternatives. Each alternative is characterized by a tuple of n values y = (y1, . . . , yn), and only alternatives for which all the values yi are positive are allowed. Thus, from the mathematical viewpoint, the set of all alternatives is the set (R) of all the tuples of positive numbers. For each two alternatives y and y′, we want to tell whether y is better than y′ (we will denote it by y  y′ or y′ ≺ y), or y′ is better than y (y′  y), or y and y′ are equally good (y′ ∼ y). These relations must satisfy natural properties. For example, if y is better than y′ and y′ is better than y′′, then y is better than y′′. In other words, the relation  must be transitive. Similarly, the relation ∼ must be transitive, symmetric, and reflexive (y ∼ y), i.e., in mathematical terms, an equivalence relation. So, we want to define a pair of relations  and ∼ such that  is transitive, ∼ is transitive, ∼ is an equivalence relation, and for every y and y′, one and only one of the following relations hold: y  y′, y′  y, or y ∼ y′. It is also reasonable to require that if each criterion is better, then the alternative is better as well, i.e., that if yi > y′ i for all i, then y  y′. Comment. Pairs of relations of the above type can be alternatively characterized by a pre-ordering relation a o b ⇔ (a  b ∨ a ∼ b). This relation must be transitive and – in our case – total (i.e., for every y and y′, we have y o y′ ∨ y′ o y. Once we know the pre-ordering relation o, we can reconstruct  and ∼ as follows: y  y′ ⇔ (y o y′& y′ 6o y); y ∼ y′ ⇔ (y o y′& y′ o y). Scale invariance: motivation. The quantities yi describe completely different physical notions, measured in completely different units. In our meteorological case, some of these values are wind velocities measured in meters per second, or in kilometers per hour, or miles per hour. Other values are elevations described in meters, kilometers, or feet, etc. Each of these quantities can be described in many different units. A priori, we do not know which units match each other, so it is reasonable to assume that the units used for measuring different quantities may not be exactly matched. It is therefore reasonable to require that the relations  and ∼ between the two alternatives y = (y1, . . . , yn) and y′ = (y′ 1, . . . , y ′ n) do not change if we simply change the units in which we measure each of the corresponding n quantities. Scale invariance: towards a precise description. When we replace a unit in which we measure a certain quantity q by a new measuring unit which is λ > 0 times smaller, then the numerical values of this quantity increase by a factor of λ: q → λ · q. For example, 1 cm is λ = 100 times smaller than 1 m, so the length q = 2 m, when measured in cm, becomes λ · q = 2 · 100 = 200 cm. Let λi denote the ratio of the old to the new units corresponding to the i-th quantity. Then, the quantity that had the value yi in the old units will be described by a numerical value λi ·yi in the new unit. Therefore, scale-invariance means that for all y, y ∈ (R) and for all λi > 0, we have y = (y1, . . . , yn)  y′ = (y′ 1, . . . , y′ n) ⇒ (λ1 ·y1, . . . , λn ·yn)  (λ1 ·y′ 1, . . . , λn ·y′ n)