620-461 Modelling of Business, Management and Industrial Problems
th
The following question is an adaptation of questions 9.2.34 on page 508 and 14.16.4(a) on page 199 of
Winston (Edition 4, 2004) – Operations Research: Applications and Algorithms .
What to submit: Mathematical programming model with general expressions, Xpress-Mosel code with annotation/rationale for each part; discussions of any underlying assumptions; results generated by
Xpress-IVE; interpretation/discussions of results aimed at The Utopian Ambulance Service (UAS).
The Utopian Ambulance Service (UAS) is the main provider of ambulance services in the state of
Peaceful. The state of Peaceful has eight districts in total, called Atlantis-1, Atlantis-2, Atlantis-3, Atlantis-
4, Atlantis-5, Atlantis-6, Atlantis-7 and Atlantis-8. Currently, UAS has ten ambulance service centres
(ASC), providing services to the eight different districts.
UAS wishes to improve their services by purchasing new ambulances, and if needed, will upgrade their facilities to accommodate the increase in the number of ambulances. UAS has allocated $1,000,000 for this purpose. The penalty for not meeting the budget is $1 per dollar exceeding the budgeted amount.
Each district will be served by all ASC within a 10km radius from the centre of the district, as listed in
Table 1. Due to political reasons, at most one ASC in each district, i.e. within a 10km limit from the center
of the district, can be upgraded,. Whenever an ASC is upgraded, a fixed cost will be incurred. The number of ambulances currently stationed at the ASC, the capacity when the facility is upgraded and the
fixed cost of upgrade are shown for each ASC in Table 2. The cost of a new ambulance is $70,000 each.
All ASC’s are currently at their full capacity.
UAS will sign a quality-ofservice agreement with each district, stating that UAS’s average response time for service within that district should not exceed the Average Response Time Guarantee (ARTG)
specified in Table 1.
The response time of an ASC j can be estimated by
R
ASC j
 max 5 , 10
 j


5 j n j
(in minutes) for n j

0 , where n is the number of ambulances at ASC j and j
 j
is the number of districts ASC j is serving. The average response time for a district i is given by
Department of Mathematics and Statistics, The University of Melbourne

620-461 Modelling of Business, Management and Industrial Problems
R i
District 
 j

S i
R
ASC j
S i
(in minutes) where S i is the set of all ASC’s within a 10km radius from the centre of district i . For every minute that the average response time, R i
District
, exceeds the ARTG, UAS will be penalised at rates shown in Table
1. The total amount penalised will be paid to the district.
Table 1: ASC ’s within 10km radius of each district.
District
Atlantis-1
Atlantis-2
Atlantis-3
Atlantis-4
Atlantis-5
Atlantis-6
Atlantis-7
Atlantis-8
ASC within 10km radius
UC-1, UC-2
UC-2, UC-3
UC-4, UC-6
UC-1, UC-10
UC-4, UC-5, UC-9
UC-5, UC-8
UC-7, UC-9, UC-10
UC-3, UC-6
Average Response
Time Guarantee
(ARTG)
(minutes)
7
5
6
8
5
5
7
6
Table 2: Current number of ambulances and capacities at different ASC’s
Penalty per minute exceeding ARTG
($)
15000
12000
13000
18000
9000
13000
14000
12000
ASC
UC-1
UC-2
UC-3
UC-4
UC-5
UC-6
UC-7
UC-8
UC-9
UC-10
Number of ambulances
2
4
5
2
3
4
3
4
3
5
Capacity when facility is upgraded
(number of ambulances)
10
20
18
17
12
13
20
18
20
11
Fixed Cost when upgrading is performed
($)
100000
120000
160000
180000
130000
100000
180000
150000
180000
120000
(a) CALCULATE the current average response time in each district and the total ARTG penalty payable.
(b) FORMULATE , in a general sense, a linear integer program whose solution will tell UAS how to minimise the total cost. Define clearly all parameters and variables used.
(c) CODE and SOLVE the formulation in part (b) using Xpress MP using the sample code at http://www.more.ms.unimelb.edu.au/toolsandtechniques/educationsamples/educationsamples.htm
. COMMENT on the result.
(d) Should UAS proceed with the upgrade as suggested in part (c) and sign the agreement, if it was a 10year quality-of-service agreement and no upgrades are allowed for the duration of the agreement?
(e) UAS wish to evaluate the effect of the political decision to restrict at most one ASC in any district to be upgraded. REFORMULATE and SOLVE the linear integer program when at most two ASC’s in each district can be upgraded. ILLUSTRATE and COMPARE both results in parts (c) and (e). Do the same for the case for any number of ASC’s in a district could be upgraded. Does anything change?
Department of Mathematics and Statistics, The University of Melbourne