Question 68 Chapter 4 Introduction to Management Science 12t

Question# 68. Chapter 4. Introduction to Management Science 12th edition by Bernard W. Taylor III

In the event of a disaster situation at Tech from weather, an accident or terrorism, victims will be transported by emergency vehicles to three area hospitals: Montgomery Regional, Radford Memorial and Lewis Galt. Montgomery Regional is (on average) 10 minutes away from Tech, Radford Memorial is 20 minutes away and Lewis Galt is 35 minutes away. Tech wants to analyze a hypothetical disaster situation in which there are 15 victims with different types of injuries. The emergency facilities at Montgomery Regional can accommodate, at most, 8 victims; Radford memorial can handle 10 victims; and Lewis Galt can admit 7 victims. A priority has been assigned for each victim according to the hospital that would best treat the victim\'s type of injury, as shown in the table shown below(where 1 reflects the best treatment).

Priority

Patient

Montgomery Regional

Redford Memorial

Lewis Galt

1

1

2

3

2

1

2

3

3

2

3

1

4

2

3

1

5

2

1

3

6

1

3

2

7

3

3

1

8

3

1

2

9

3

1

2

10

1

1

2

11

3

3

1

12

3

3

1

13

2

2

1

14

1

2

2

15

3

3

1

For example, for victim 1\'s type of injury, the best hospital is Montgomery Regional, the next best is Radford Memorial and Lewis Galt is the 3rd best.

a) Formulate and solve a linear programming model that will send the victims to the hospital best suited to administer to their specific injuries while keeping the average transport time to 22 minutes or less.

b) Formulate and solve a linear programming model that will minimize the average transport time for victims while achieving an average hospital priority of at least 1.50 or better.

		Priority
Patient	Montgomery Regional	Redford Memorial	Lewis Galt
1	1	2	3
2	1	2	3
3	2	3	1
4	2	3	1
5	2	1	3
6	1	3	2
7	3	3	1
8	3	1	2
9	3	1	2
10	1	1	2
11	3	3	1
12	3	3	1
13	2	2	1
14	1	2	2
15	3	3	1

Solution

a) Formulation as linear programming modelnwith the objective to have best suited hospital while keeping the average transport time to 22 minutes or less.

Decision variables are in terms of which patient to be send to which hospital so these are double subscript like transpotation problems ( Xij from ith to jth place). Let me define Pim , Pir and Pil represents variables if ith patient is send to Montgomery, Radford and Lewis hospital respectively, like assignment problem it can take only two values zero and one depending upon wheather ith patient send to respective hospital or not. For example if patient 1 is send to Montgomery then P1m = 1 and P1r = P1l = 0

Objective Function: As stated above we are required to have best suited treatment and for that priority numbers are given for each patient and at each hospitals but least number is the best , therefore if Cim, Cir and Cil denote the priority score of ith patient with Montgomery, Radford and lewis hospitals respectively then sums of Pim*Cim, Pir*Cir and Pil*Cil for all vaues of i from 1 to 15, will be the total priority score which we want to minimize, hence

Objective function is Minimize Sigma Pim*Cim + Sigma Pir*Cir + Sigma Pil*Cil   Sigma over i =1,2,3.....15

Constraints Firstly one patient to one hospital( person can not be send to more than one place at a time ), therefore

Pim + Pir + Pil = 1 For all i = 1,2,3,....15

Further there are capacity constraints for each hospital, which are

Sigma Pim <= 8     Sigma Pir <= 10 and Sigma Pil <= 7 for all Sigmas (sums) i takes the values from 1 to 15

Then there is constraint about average time to be 22 minutes or less. Average time is

                     [ 10*(sigma Pim) + 20*(sigma Pir) + 35*(sigma Pil) ] / 15 and the constraint is

                   10*(sigma Pim) + 20*(sigma Pir) + 35*(sigma Pil) <= 22*15

Lastly as already said about decision variables, the allowable values are only 0 and 1

b) In this part the objective is to minimize the average transpotation time while achieving an average hospital priority of at least 1.50

Average transportation time is defined in part a), therefore

Objective function is Minimize (10/15) (sigma Pim) + (20/15) (sigma Pir) + (35/15) (sigma Pil)

Constraint about average hospital priority is at least 1.50 or better where better means lower values than 1.50 as best priority is assigned the value 1 and least priority as 3

Average priority is [ (sigma (Cim*Pim)/sigmaPim ) + ( sigma(Cir*Pir)/(sigmaPir) + (sigma (Cil*Pil)/(sigmaPil) ]/ 3

Sum of averages priorities of individual hospitals divided by number of hospitals, hence constraint is

sigma(Cim*Pim)/sigmaPim + sigma(Cir*Pir)/sigmaPir + sigma(Cil*Pil)/sigmaPil   <= 3*1.5

Solution is time consuming so only formulations are given