# Efficiency of Hospitals

## Contents

## Problem description

The administration of the hospitals in Paris decides to measure the efficiency of the surgery departments in four major hospitals with a desire to improve the service to the public. To keep this study anonymous, the hospitals are named H1 to H4. The method suggested to measure the efficiency is DEA (Data Envelopment Analysis). This method compares the performance of a fictitious hospital with the performances of the four hospitals.

Three initial indicators (resources) are taken into account: the number of non-medical personnel, the general expenses, and the available number of beds. In addition, four final indicators (services) are analyzed: the number of hospital admissions per day, the number of consultations in the outpatients’ clinic, the number of nurses on duty every day, and the number of interns and doctors on duty every day. The corresponding data have been analyzed over a period of two years and the numbers representing a day of average activity in every hospital are given in the following two tables.

Resource indicators

+---------------------+-----+------+-----+-----+ | | H1 | H2 | H3 | H4 | +---------------------+-----+------+-----+-----+ |Non-medical personnel| 90 | 87 | 51 | 66 | |General expenses (k$)|38.89|109.48|40.43|48.41| |Number of beds | 34 | 33 | 20 | 33 | +---------------------+-----+------+-----+-----+

Service indicators

+-------------------+-----+-----+-----+-----+ | | H1 | H2 | H3 | H4 | +-------------------+-----+-----+-----+-----+ |Admissions |30.12|18.54|20.88|10.42| |Consultations |13.54|14.45| 8.52|17.74| |Interns and doctors| 13 | 7 | 8 | 26 | |Nurses on duty | 79 | 55 | 47 | 50 | +-------------------+-----+-----+-----+-----+

Justify through the DEA method how hospital H2 is performing compared to the others.

## Variables

resources A matrix describing the resources services A matrix describing the services

## Reference

Applications of optimization... Gueret, Prins, Seveaux

% Marcus Edvall, Tomlab Optimization Inc, E-mail: tomlab@tomopt.com % Copyright (c) 2005-2009 by Tomlab Optimization Inc., $Release: 7.2.0$ % Written Oct 7, 2005. Last modified Apr 8, 2009.

## Problem setup

resources = [90 87 51 66;... 38.89 109.48 40.43 48.41;... 34 33 20 33]; services = [30.12 18.54 20.88 10.42;... 13.54 14.45 8.52 17.74;... 13 7 8 26;... 79 55 47 50]; h = size(resources,2); %coef s = size(services,1); %services r = size(resources,1); %resources coef = tom('coef',h,1); fserv = tom('fserv',s,1); fres = tom('fres',r,1); eff = tom('eff',1,1); % No variables are binary bnds = {coef >= 0, fserv >= 0, fres >= 0, eff >= 0}; % Coef constraints con1 = {sum(coef) == 1}; % Service constraint con2 = {services*coef == fserv}; % Resource constraint con3 = {resources*coef == fres}; % Objective objective = eff; indices = zeros(size(resources,2),1); for i=1:size(resources,2) % Service indicators, greater than ficticious ones con4 = {fserv >= services(:,i)}; % Efficiency relationship con5 = {fres <= resources(:,i)*eff}; constraints = {bnds, con1, con2, con3, con4, con5}; options = struct; options.solver = 'cplex'; options.name = 'Efficiency of Hospitals'; sol = ezsolve(objective,constraints,[],options); indices(i,1) = sol.eff; end PriLev = 1; if PriLev > 0 for i = 1:length(indices), disp(['H' num2str(i) ' is ' num2str(100*indices(i)) '% efficient']) end end % MODIFICATION LOG % % 051206 med Created % 060118 per Added documentation % 060125 per Moved disp to end % 090325 med Converted to tomSym

Problem type appears to be: lp ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Efficiency of Hospitals f_k 0.999999999999999890 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Dual Simplex LP solver Optimal solution found FuncEv 2 Iter 2 Problem type appears to be: lp ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Efficiency of Hospitals f_k 0.921049539123350750 sum(|constr|) 0.000000000000007105 f(x_k) + sum(|constr|) 0.921049539123357850 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Dual Simplex LP solver Optimal solution found FuncEv 5 Iter 5 Problem type appears to be: lp ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Efficiency of Hospitals f_k 1.000000000000000000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Dual Simplex LP solver Optimal solution found FuncEv 2 Iter 2 Problem type appears to be: lp ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2010-02-05 ===================================================================================== Problem: --- 1: Efficiency of Hospitals f_k 1.000000000000000000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Dual Simplex LP solver Optimal solution found FuncEv 3 Iter 3 H1 is 100% efficient H2 is 92.105% efficient H3 is 100% efficient H4 is 100% efficient