The Hospital Administrators at New Hope, County General, and City $\quad$

The hospital administrators at New Hope, County General, and City East recently received notice of an impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months.
$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Hospital}$
$\begin{array}{lccc}&\text { New Hope } &\text { County General }&\text {City East }\\\hline \text {Input Measures}\\ \text { Bed days unused (1000s) } & 83.0 & 105.0 & 104.1 \\\text { Supply expense }(\$ 1000 s) & 123.8 & 162.3 & 154.0 \\\text { Full-time staff } & 225.0 & 200.0 & 231.0 \\\text { Output Measures } & & & 82.7 \\\text { Patient-days (1000s) } & 105.0 & 71.0 & 175.0 \\\text { Nurses qualified } & 253.0 & 92.0 & 65.0 \\\text { Assistants on staff } & 125.0 & 45.0 & 830\end{array}$ Based on the following formulation, is City East efficient? If not, what input and output values should they aspire to in order to become efficient?
Let $\quad w _ { i } =$ weight assigned to output $j , j = 1 , \ldots , 4$
$\mathbf { v } _ { i } =$ weight assigned to input $i , i = 1 , \ldots , 3$
MAX: $\quad 82.7 w _ { 1 } + 175 .0 w _ { 2 } + 65.0 w _ { 3 } + 83 .0 w _ { 4 }$
Subject to:
$105.1 w _ { 1 } + 253.0 w _ { 2} + 125.0 w _ { 3 } + 98.0 w _ { 4 } - 83.0 v _ { 1 } - 123.8 v _ { 2 } - 225.0 v _ { 3 } \leq 0$
$71.0 w _ { 1 } + 92.0 w _ { 2 } + 45.0 w _ { 3 } + 88.0 w _ { 4 } - 105.0 v _ { 1 } - 162.3 v _ { 2 } - 200 v _ { 3 } \leq 0$
$82.7 w _ { 1 } + 175.0 w _ { 2 } + 65.0 w _ { 3 } + 83.0 w _ { 4 } - 104.1 v _ { 1 } - 154.0 v _ { 2 } - 231.0 v _ { 3 } \leq 0$
$104.1 \mathbf { v } _ { 1 } + 154.0 \mathbf { v } _ { \mathbf { 2 } } + 231.0 \mathbf { v } _ { \mathbf { 3 } } = 1$
$w _ { 1 , } w _ { 2 } w _ { 3 , } w _ { 4 } , \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \geq 0$