Qmdm Case Final

8/9/2019 Qmdm Case Final

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REG NO 10045

Case Study

Quantitative Methods for Decision Making

Saad Khawar1/3/2010

The document includes solution to the case study given in class MBA-1 in first semester course QMDM

taught by Dr. Ahmed Ali Shah

Submitted to: Dr. Ahmed Ali Shah


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2Saad Khawar Case Study - I - QMDM

1. Write summary of given and the result required?It is an application of linear programming used to measure the relative efficiency of operating units with

the same goals and objectives. This case is about the use Data Envelopment Analysis to measure the

relative efficiency of County hospital. For this purpose we have used a linear programming model to

construct a hypothetical composite hospital based on the outputs and inputs for the four hospitals inthe problem. For each operating unit that we want to measure the efficiency of, we must formulate and

solve a linear programming model similar to the linear program we solved to measure the relative

efficiency of County Hospital. The following step by step procedure should help you in formulating a

linear programming model for other types of DEA applications. Note that the operating unit we want to

measure the relative efficiency of is referred to as the jth operating unit.


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2. The theory used in setting the equationsA linear programming model is developed for each hospital whose efficiency has to be evaluated. Then a

composite hospital is created based on the outputs and inputs for all the four hospitals.

Output Measures

y Patient-days of service under Medicarey Patient-days of service not under Medicarey Number of nurses trainedy Number of interns trained

Input Measures

y The number of full-time equivalent (FTE) non physician personnely The amount spent on suppliesy The number of bed-days available

Variables to be determined

wg weight applied to inputs and outputs for General Hospital

wu weight applied to inputs and outputs for University Hospital

wc weight applied to inputs and outputs for County Hospital

ws weight applied to inputs and outputs for State Hospital

Constraints

The DEA approach requires that the sum of these weights equal 1. Thus, the first constraint is

wg +wu +wc +ws = 1

Output Constraints

Medicare Patient-Days for

composite hospital

= (Medicare Patient-Days for General Hospital)*wg +

(Medicare Patient-Days for County Hospital)*wc +

(Medicare Patient-Days for University hospital)*wu +

(Medicare Patient-Days for state hospital)*ws +


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Medicare patient-days

for Composite Hospital

The other output measures for the composite hospital are computed in a similar fashion.

General Format for the output

constraints

Output for the composite hospital >= Output for county hospital

Medicare Patient days 48.14 wg + 34.62 wu + 36.72 wc + 33.16 ws >= 36.72

Non medicare patient days 43.10 wg + 27.11 wu + 45.98 wc + 56.46 ws >= 45.98

Nurses 253 wg+148 wu+175 wc+l60 ws >= 175

Interns 41 wg+27 wu+23 wc+84 ws >= 23

Input Constraints

FTE nonphysicians

for Composite Hospital

The other input measures for the composite hospital are computed in a similar fashion.

General Format for the input

constraints

Output for the composite hospital >= Output for county hospital

FTE physicians 285.20wg + 162.30wu + 275.70wc + 210.40ws

Supple Expense 123.80wg + 128.70wu + 348.50wc + 154.10ws

Bed-days 106.70wg + 64.21wu + 104.10wc + 104.04ws

Objective Function

Minimize E = the fraction of County Hospitals input available to the composite hospital

E is also known as efficiency index

The input constraint corresponding to FTE non physicians, supplies and bed-days:

285.20wg + 162.30wu +275.70wc+210.40ws


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48.14wg+ 34.62wu +36.72wc+33.16ws >=36.72

43.10wg+ 27.11wu +45.98wc+33.16ws >=45.98

253wg + 148wu +175wc + 160ws >=175

41wg + 27wu +23wc + 84ws >=23

123.80wg+ 128.70wu +348.50wc+154.10ws


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Proof

wg 0.212

wu 0.260

ws 0.527

wc 0.000

E 0.905

Putting in the inequalities

253wg + 148wu + 175wc + 160ws 175

253(0.212) + 148(0.260) + 175(0) + 160(0.527) 175

176 175

Hence proved

Now checking for E

106.72wg+ 64.21wu+ 104.10wc+ 104.04ws-104.10E 0

106.72(0.212) + 64.21(0.260) + 104.10(0) + 104.04(0.527) - 104.10(0.905) 0

-0.06 0

Hence proved

s.no Constraints Limit Values

Calculated

Slack

Min E ( Objective) 0.905

1 wg + wu + wc+ ws = 1 1 1.000 0.000

2 48.14wg + 34.62wu + 36.72wc + 33.16ws 36.72 36.72 36.720 0.000

3 43.10wg + 27.11wu + 45.98wc + 56.46ws 45.98 45.98 45.980 0.000

4 253wg + 148wu + 175wc + 160ws 175 175 176.615 1.615

5 41wg + 27wu + 23wc + 84ws 23 23 60.027 37.027

6 285.20wg+ 162.30wu+ 275.70wc+ 210.40ws-275.70E 0 0.00 -35.824 35.824

7 123.80wg+ 128.70wu+ 348.50wc+ 154.10ws-348.50E 0 0.00 -174.422 174.422

8 106.72wg+ 64.21wu+ 104.10wc+ 104.04ws-104.10E 0 0.00 0.000 0.000

9 ws>0 0.00 0.527 0.000

10 wg>0 0.00 0.212 0.000

11 wc>0 0.00 0.000 0.000

12 wu>0 0.00 0.260 0.00013 E>0 0.00 0.905 0.000


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4. The conclusionGeneral hospital wg=0.212, University Hospital wu=0.260, State hospital ws=0.527 and County hospital

wc=0.000. Each input and output of the composite hospital is determined by the same weighted average

of the inputs and outputs of these three hospitals. The slack/surplus column provides some additional

information about the efficiency of County hospital compared to the Composite hospital. Specifically,

the Composite hospital has at least as much of each output as County hospital has (constraints 2-5) andprovides 1.6 more nurses trained (surplus for constraint 4) and 37 more interns trained (surplus for

constraint 5). The slack of 0 from constraint 8 shows that the Composite hospital uses approximately

90.5% of the bed-days used by County hospital. The slack values for constraints 6 & 7 shows that less

than 90.5% of the FTE non physician and the supplies expense resources used at County hospital are

used by Composite hospital.

Clearly, the Composite hospital is more efficient then County hospital, and we are justified in concluding

that County hospital is relatively inefficient compared to the other hospitals in the group. Given the

results of the DEA analysis hospital administrators should examine operations to determine how County

hospital resources can be more effectively utilized.

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