In many facility location problems, we are interested in simultaneously locating two types of facilities. For example, in a developing country, we might want to locate regional hospitals and local...

In many facility location problems, we are interested in simultaneously locating two types of facilities. For example, in a developing country, we might want to locate regional hospitals and local health care clinics.

Consider the following problem. We want to locate H hospitals and C clinics in a developing country. In general, we will have H


C (i.e., H is much less than C). For a variety of reasons, each clinic must be located within a distance Dhc (a critical hospital-to-clinic distance) of at least one of the hospitals. Some of the possible reasons for this requirement include the need to resupply clinics with supplies and medications from the hospitals and the need for physicians at a hospital to visit clinics on both a routine (inspection) basis and an emergency basis to deal with critically ill patients whose transport to the hospital may be impossible. We are given the populations, h_i, of each of a large number of rural villages whose people will be served by this health care system. Our objective is to minimize the total (over all villages) demand weighted average distance between a village and the nearest health care facility to the village. In other words, if one of the few hospitals is closer to a village than any of the clinics, then patients from that village will go to the hospital directly; otherwise, they will go to the nearest clinic.

Using the notation below, formulate this problem. Note that in each part of the problem you are given the critical constraints (or objective function) in words. You must formulate them using the notation defined below. Be sure all indices of summation are indicated clearly and that you indicate the indices for which each constraint applies (e.g., for all i 2 I).

Sets

I = set of villages or demand locations

J = set of candidate hospital sites

K = set of candidate clinic sites

Inputs

H = maximum number of hospitals to be located

C = maximum number of clinics to be located

d_ij
= distance between village i
 I and candidate hospital j
 J

d_ik
= distance between village i
 I and candidate clinic k
 K

d_jk
= distance between candidate hospital j
 J and candidate clinic k
 K

D_hc
= critical coverage distance between hospitals and clinics

hi = population of village i
 I

Decision Variables

The objective function and key constraints of the problem are listed below:

(a) Minimize the total demand-weighted distance between a village and the nearest health care facility.

(b) Locate exactly H hospitals.

(c) Locate exactly C clinics.

(d) Every village is assigned to exactly one facility (either a clinic or a hospital).

(e) Each clinic must be within D_hc
distance units of the nearest hospital.

(f) Demands at village i ∈ I can only be assigned to a hospital at candidate site j
 J if we locate a hospital at candidate site j
 J.

(g) Demands at village i
 I can only be assigned to a clinic at candidate site k
 K if we locate a clinic at candidate site k
 K.

(h) Integrality.