Facility location under service level constraints for heterogeneous customers

Abstract

We study the problem of locating service facilities to serve heterogeneous customers. Customers requiring service are classified as either high priority or low priority, where high priority customers are always served on a priority basis. The problem is to optimally locate service facilities and allocate their service zones to satisfy the following coverage and service level constraints: (1) each demand zone is served by a service facility within a given coverage radius; (2) at least $\alpha ^h$ proportion of the high priority customers at any service facility should be served without waiting; (3) at least $\alpha ^l$ proportion of the low priority cases at any service facility should not have to wait for more than $\tau ^l$ minutes. For this, we model the network of service facilities as spatially distributed priority queues, whose locations and user allocations need to be determined. The resulting integer programming problem is challenging to solve, especially in absence of any known analytical expression for the service level function of low priority customers. We develop a cutting plane based solution algorithm, exploiting the concavity of the service level function of low priority customers to outer-approximate its non-linearity using supporting planes, determined numerically using matrix geometric method. Using an illustrative example of locating emerging medical service facilities in Austin, Texas, we present computational results and managerial insights.

Appendix 1: Infinitesimal generator sub-matrices under non-preemptive priority

$$\begin{aligned} L_0 = \left( \begin{array}{c|ccccc} &{} (0,0) &{} (0,1) &{} (0,2) &{} (0,\ldots ) &{} (0,M) \\ \hline (0,0) &{}*&{}\varLambda ^h_j \\ (0,1) &{}\mu _j^h &{}*&{}\varLambda ^h_j \\ (0,2) &{} &{}\mu _j^h &{}*&{}\varLambda ^h_j \\ (0,\ldots ) &{} &{} &{}\ddots &{}\ddots &{}\ddots \\ (0,M) &{} &{} &{} &{}\mu _j^h &{}*\end{array} \right) \end{aligned}$$

$$\begin{aligned} F_0 = \left( \begin{array}{c|ccccccccc} &{} (1,0) &{} (1,h,1) &{} (1,h,2) &{} (1,h,\ldots ) &{} (1,h,M) &{} (1,l,1) &{} (1,l,2) &{} (1,l,\ldots ) &{} (1,l,M) \\ \hline (0,0) &{}\varLambda _j^l &{} \\ (0,1) &{} &{}\varLambda _j^l &{} \\ (0,2) &{} &{} &{}\varLambda _j^l &{}\\ (0,\ldots ) &{} &{} &{} &{}\ddots &{} \\ (0,M) &{} &{} &{} &{} &{}\varLambda _j^l \end{array} \right) \end{aligned}$$

$$\begin{aligned} B_0 = \left( \begin{array}{c|ccccc} &{} (0,0) &{} (0,1) &{} (0,2) &{} (0,\ldots ) &{} (0,M) \\ \hline (1,0) &{} \mu _j^l &{} &{} \\ (1,h,1) &{} &{} &{} &{} \\ (1,h,2) &{} &{} &{} &{} \\ (1,h,\ldots ) &{} &{} &{} &{} &{} \\ (1,h,M) &{} &{} &{} &{} &{}\\ (1,l,1) &{}0 &{}\mu _j^l &{} &{} &{}\\ 1,l,2) &{} &{} &{}\mu _j^l &{} &{}\\ (1,l,\ldots ) &{} &{} &{} &{}\ddots &{}\\ (1,l,M) &{} &{} &{} &{} &{}\mu _j^l\\ \end{array} \right) \end{aligned}$$

$$\begin{aligned} L = \left( \begin{array}{c|ccccccccc} &{} (k,0) &{} (k,h,1) &{} (k,h,2) &{} (k,h,\ldots ) &{} (k,h,M) &{} (k,l,1) &{} (k,l,2) &{} (k,l,\ldots ) &{} (k,l,M) \\ \hline (k,0) &{}*&{}\varLambda _j^h \\ (k,h,1) &{}\mu _j^h &{}*&{}\varLambda _j^h \\ (k,h,2) &{} &{}\mu _j^h &{}*&{}\varLambda _j^h\\ (k,h,\ldots ) &{} &{}&{}\ddots &{}\ddots &{}\ddots \\ (k,h,M) &{} &{} &{} &{}\mu _j^h &{}*&{}0 \\ (k,l,1) &{} &{} &{} &{} &{}0 &{}*&{}\varLambda _j^l \\ (k,l,2) &{} &{} &{} &{} &{} &{}0 &{}*&{}\varLambda _j^l \\ (k,l,\ldots ) &{} &{} &{} &{}&{} &{} &{}\ddots &{}\ddots &{}\ddots \\ (k,l,M) &{} &{} &{} &{} &{} &{} &{} &{} 0 &{} *\end{array} \right) \end{aligned}$$

$$\begin{aligned} F = \left( \begin{array}{c|ccccccccc} &{} (k+1,0) &{} (k+1,h,1) &{} (k+1,h,2) &{} (k+1,h,\ldots ) &{} (k+1,h,M) &{} (k+1,l,1) &{} (k+1,l,2) &{} (k+1,l,\ldots ) &{} (k+1,l,M) \\ \hline (k,0) &{}\varLambda _j^l &{} \\ (k,h,1) &{} &{}\varLambda _j^l &{} \\ (k,h,2) &{} &{} &{}\varLambda _j^l &{}\\ (k,h,\ldots ) &{} &{} &{} &{}\ddots &{} \\ (k,h,M) &{} &{} &{} &{} &{}\varLambda _j^l \\ (k,l,1) &{} &{} &{} &{}&{} &{}\varLambda _j^l &{} \\ (k,l,2) &{} &{} &{} &{} &{} &{} &{}\varLambda _j^l &{}\\ (0,l,\ldots ) &{} &{} &{} &{} &{} &{} &{} &{}\ddots &{} \\ (0,l,M) &{} &{} &{} &{} &{} &{} &{} &{} &{}\varLambda _j^l \end{array} \right) \end{aligned}$$

$$\begin{aligned} B = \left( \begin{array}{c|ccccccccc} &{} (k,0) &{} (k,h,1) &{} (k,h,2) &{} (k,h,\ldots ) &{} (k,h,M) &{} (k,l,1) &{} (k,l,2) &{} (k,l,\ldots ) &{} (k,l,M) \\ \hline (k-1,0) &{} \mu _j^l &{} &{} \\ (k-1,h,1) &{} &{} &{} &{} \\ (k-1,h,2) &{} &{} &{} &{} \\ (k-1,h,\ldots ) &{} &{} &{} &{} &{} \\ (k-1,h,M) &{} &{} &{} &{} &{}\\ (k-1,l,1) &{}0 &{}\mu _j^l &{} &{} &{}\\ (k-1,l,2) &{} &{} &{}\mu _j^l &{} &{}\\ (k-1,l,\ldots ) &{} &{} &{} &{}\ddots &{}\\ (k-1,l,M) &{} &{} &{} &{} &{}\mu _j^l\\ \end{array} \right) \end{aligned}$$

where $*$ is such that $A_0 \mathbf {e}$ + $B_0 \mathbf {e}$ = $\mathbf {0}$. $A_1 = B_0 - A_2$.

1.1 Appendix 2: Data

See Tables 4 and 5.

Table 4 5-Month period census tract level service call data for Austin, Texas (Daskin and Stern 1981)

Table 5 Census tract level travel time (in minutes) data for Austin, Texas (Daskin and Stern 1981)