6.5.6 Requirements Problems

So far we have addressed urban facility location problems of the type: "Where should I locate k facilities to maximize (or minimize) some (given) objective function ?" Very often, however, the question will be asked in quite different terms: "We would like to achieve certain standards of performance (either as specified by legislative fiat or as deemed necessary by service administrators). What is then the smallest (or least costly) number of facilities

that we need, and where should these facilities be located to achieve these standards ?"

In this section we shall discuss briefly procedures for dealing with this second type of question�which we shall refer to as a "requirements problem."

Clearly, our earlier work (and algorithms) can provide the building blocks for solving requirements problems. To take a concrete example, the Emergency Medical Service Systems Act passed by Congress in 1973 (EMSS Act PL93154) states in its guidelines that 95 percent of rural calls for emergency medical service should be reached within 30 minutes from the call and 95 percent of urban calls within 10 minutes. This is now a case where some standards of performance have been preset by legislation for an urban (and rural) service. At this point the analyst must take over. To determine appropriate locations for basing the emergency medical care facilities or an associated ambulance system, the foregoing specifications must be interpreted in more concrete operational terms. For instance, the following might be a reasonable interpretation of the standards of performance set by the EMSS Act: "It is required that 95 percent of all calls must be reached within 30 (10) minutes. We also know that, in most reasonable service systems, it might be expected that a certain percentage of calls for service will have to queue up for a period of time due to all the servers of the service system being busy. It might then be inferred that for the EMS system to have any hope of achieving the specified performance standards, it must be that all of the potential users of the service should be within 30 (or 10) minutes of travel time from their closest EMS facility."

We thus now require a set of locations such that no potential users are more than 30 (or 10) minutes away from at least one of them. This we recognize as a problem very similar to the k-centers problem. In this case, however, the number of required locations, k, is not given. Instead, we know the maximum acceptable distance that can be associated with our k-centers. In other words, in our notation, we are given the value of m(X*k) (= 30 or 10 minutes, depending on the case), and we are asked to find k and the locations X*k.

A possible approach to finding the least number and the locations of EMS facilities required to achieve m(X*k) = 30 (or 10) should now be obvious.18

EMS Coverage "Algorithm"
STEP 1: Set k = 1.
STEP 2: Solve the k-centers problem for the current value of k.
STEP 3: If m(X*k) 30 minutes (or 10 minutes), stop. The current value of k is the minimum number of facilities required and the appropriate locations of these facilities are the locations of the k-centers. If m(X*k) > 30 (or 10) minutes, go to Step 4.
STEP 4: Set k = k + 1 and return to Step 2.

In the procedure above we did not specify whether at Step 2 we are solving an absolute k-centers problem or a vertex k-centers problem. This will depend on whether the potential locations of the facilities are unrestricted or, as is so often the case in practice, the choice of locations is restricted to only a finite number of points (or general areas) in the region of interest. In the latter case the vertex k-centers problem is the appropriate one for Step 2.

We can now describe in quite general terms a more realistic "scenario" than hitherto, for facility location problems in the urban environment.
1. A certain primary objective is stated, usually in the form of a requirement for compliance with the performance standards set by an administrative, legislative, or other body (e.g., "achieve the performance standards set by the EMSS Act").
2. Some restrictions on the potentially acceptable facility locations are also specified. These restrictions are often due to local considerationsor conditions or to special requirements of the facilities to be constructed (e.g., "all ambulance stations must be adjacent to local hospitals").
3. One or more secondary objectives are also often specified. These secondary objectives are usually expressed in terms of costs, although other measures of effectiveness also appear sometimes (e.g., "once the performance standards in the primary objective are met, the least-cost system configuration should be selected").

Example 18: Ambulance Location/Allocation go a Rural Area

In a case reported recently [JARV 75a], a team of analysts were asked to assist the Bel-O-Mar Regional Council of Wheeling, West Virginia, in developing a regional emergency ambulance allocation plan. Bel-O-Mar is a regional planning agency responsible for a four-county region encompassing the Wheeling SMSA.19 This is a predominantly rural area of about 1,360 square miles straddling the Ohio River; its population of 200,000 people is dispersed among approximately 90 communities in the area.

The primary objective was to design a system that complies with the EMSS Act standards described earlier. The secondary objective--once the primary one was satisfied--was to minimize average response time, a fact that finally resulted in ambulances being allocated to the seven communities which were the largest demand-generators in the region. Finally, local considerations included the following: ambulances should not cross the Ohio River in responding to calls; ambulances must be located in both Wheeling and Bethlehem--the two largest communities in the area; and ambulances allocated to certain prespecified communities cannot provide service outside these communities.

Similar examples have been reported for fire departments, for sanitation departments, and for emergency facilities in general. It should also be emphasized that the secondary objective(s) often plays an important role in the determination of facility locations. The reason is that in many problems one finds numerous combinations of locations that achieve the primary objective of satisfying the preset performance standards. The ties must then be resolved with reference to the secondary objective(s). The secondary objective(s) usually calls for the solution of a k-medians type of problem, since, as we have stated, it is usually concerned with the minimization of some average-cost function.20

18 We are now assuming that link lengths are given in terms of travel time rather than distance.

19 Standard Metropolitan Statistical Area.

20 In many cases it is true that primary and secondary objectives are the reverse of what was stated above. That is, the primary objective is to minimize an average-cost function and the secondary one to achieve a given level of performance.