Chapter 3, Problem 24RP

Explanation of Solution

 Formulation of a Linear Programming (LP) to determine the hospital optimal mix of DRGs:

 Let “i = 1” is diagnostic services, 2 is bed day service, 3 is nursing care use and 4 is drugs supplied.

 Let “x_ij” be the service “i” and “j” diagnostic-related groups (DRGs), for (i = 1, 2, 3, 4 and j = 1, 2, 3, 4).

 The objective is to find the optimal mix of DRGs.

 $z=\left[\begin{array}{c}\left(\text{profit/week from DRG1}\right)\left(\text{services from DRG1}\right)+\\ \left(\text{profit/week from DRG2}\right)\left(\text{services from DRG2}\right)+\\ \left(\text{profit/week from DRG3}\right)\left(\text{services from DRG3}\right)+\\ \left(\text{profit/week from DRG4}\right)\left(\text{services from DRG4}\right)\end{array}\right]$

 $=\left[\begin{array}{l}2000\left(x_{11}+x_{21}+x_{31}+x_{41}\right)+1500\left(x_{12}+x_{22}+x_{32}+x_{42}\right)+\\ 500\left(x_{13}+x_{23}+x_{33}+x_{43}\right)+500\left(x_{14}+x_{24}+x_{34}+x_{44}\right)\end{array}\right]$

 Therefore, the objective function to maximize the profit is,

 $\text{Maximize, }z=\left[\begin{array}{l}2000\left(x_{11}+x_{21}+x_{31}+x_{41}\right)+1500\left(x_{12}+x_{22}+x_{32}+x_{42}\right)+\\ 500\left(x_{13}+x_{23}+x_{33}+x_{43}\right)+500\left(x_{14}+x_{24}+x_{34}+x_{44}\right)\end{array}\right]$

 Constraint 1:

 At most, 570 hours of diagnostic service are available.

 $\begin{array}{c}\left[\begin{array}{c}\left(\text{diagostic service used in DRG1}\right)+\left(\text{diagostic service used in DRG2}\right)+\\ \left(\text{diagostic service used in DRG3}\right)+\left(\text{diagostic service used in DRG4}\right)\end{array}\right]\le 570\\ 7x_{11}+4x_{12}+2x_{13}+x_{14}\le 570\end{array}$

 Constraint 2:

 At most, 1000 bed-days are available.

 $\begin{array}{c}\left[\begin{array}{c}\left(\text{bed days used in DRG1}\right)+\left(\text{bed days used in DRG2}\right)+\\ \left(\text{bed days used in DRG3}\right)+\left(\text{bed days used in DRG4}\right)\end{array}\right]\le 1000\\ 5x_{21}+2x_{22}+x_{23}\le 1000\end{array}$

 Constraint 3:

 At most, 50000 hours of nursing are available.

 $\begin{array}{c}\left[\begin{array}{c}\left(\text{nursing hours used in DRG1}\right)+\left(\text{nursing hours used in DRG2}\right)+\\ \left(\text{nursing hours used in DRG3}\right)+\left(\text{nursing hours used in DRG4}\right)\end{array}\right]\le 50000\\ 30x_{31}+10x_{32}+5x_{33}+x_{34}\le 50000\end{array}$

 Constraint 4:

 At most, 50000 dollars of drugs are available