Spring
2014
A Case Study on
Strategic Location of
Best Buy Warehouses
using P-Median and PCenter approaches.
ISEN 601 Case Study 2
Team SphinX
Best Buy
Spring 2014
List of Contents
1. Introduction ........................................................................................................................................................ 3
2. Method of Approach ........................................................................................................................................... 3
3. Assumptions ........................................................................................................................................................ 4
3.1 General Assumptions .................................................................................................................................... 4
3.2 Assumptions specific for Random model ...................................................................................................... 5
4. Demand Calculation ............................................................................................................................................ 6
4.1 Assumptions specific for Demand Calculation .............................................................................................. 6
5. Observations and Discussion of Results .............................................................................................................. 7
5.1 Un-capacitated P-median Problem ............................................................................................................... 7
5.2 Un-Capacitated P-Center problems .............................................................................................................. 8
5.3 Capacitated P-Median Problem .................................................................................................................... 9
5.4 Capacitated P-Center problems .................................................................................................................. 10
6. Conclusions ....................................................................................................................................................... 12
Appendix ............................................................................................................................................................... 13
A.1 Appendix A (Network)..................................................................................................................................... 14
A.2 Table addressing each Best Buy store by a number along with population............................................... 14
A.3 Distance Matrix denoting the distance between each of the Best Buy store locations ............................. 15
B.1 Appendix B (Demand Calculation) .................................................................................................................. 17
C.1 Appendix C (Un-Capacitated P-Median) ......................................................................................................... 19
C.1 Ampl Code- Practical Case [2] ....................................................................................................................... 19
C.2 Ampl Code – Random Case ......................................................................................................................... 19
C.3 Tables comparing results for Practical and Random Cases for Un-capacitated P-Median......................... 20
Practical Case ................................................................................................................................................ 21
Random Case................................................................................................................................................. 21
C.4 Graphs comparing trend between Practical and Random Cases................................................................ 28
D. Appendix D (Un-Capacitated P-Center) ............................................................................................................ 30
D.1 Ampl Code – Practical Case [2] ..................................................................................................................... 30
D.2 Ampl Code – Random Case ......................................................................................................................... 30
D.3 Tables comparing results for Practical and Random Cases for Un-Capacitated P-Center ......................... 31
Practical Case ................................................................................................................................................ 32
Page |1
Random Case................................................................................................................................................. 32
D.4 Graphs comparing trend between Practical and Random Cases ............................................................... 39
E. Appendix E (Capacitated P-Median) ................................................................................................................. 41
E.1 Ampl Code – Practical Case [1] ..................................................................................................................... 41
E.2 Ampl Code – Random Case ......................................................................................................................... 41
E.3 Tables comparing results for Practical and Random Cases for Capacitated P-Median .............................. 42
Practical Case ................................................................................................................................................ 43
Random Case................................................................................................................................................. 43
E.4 Comparison between Practical and Random Cases .................................................................................... 50
F. Appendix F (Capacitated P- Center) .................................................................................................................. 52
F.1 Ampl Code- Practical Case [2] ....................................................................................................................... 52
F.3 Ampl Code - Random Case .......................................................................................................................... 52
F.3 Tables comparing results for Practical and Random Cases for Capacitated P-Center ................................ 53
Practical Case ................................................................................................................................................ 54
Random Case................................................................................................................................................. 54
F.4 Graph comparing Random and Practical Cases........................................................................................... 61
G. References ........................................................................................................................................................ 63
Figure 1 Initial Network diagram of Houston Best Buy stores connected through Free-ways............................... 3
Figure 2 Modified network used for Distance calculation [Appendix A.2] ............................................................. 7
Figure 3 A screenshot showing result for a capacitated P-center problem showing Objective function, Open
warehouses and percentage of demand satisfied ................................................................................................ 11
Table 1 Table denoting cost per PC per mile calculation ........................................................................................ 5
Page |2
1. Introduction
Best Buy Co., Inc. is an American multinational consumer electronics corporation headquartered
in Richfield, Minnesota, a Minneapolis suburb. It operates in the United States, Puerto Rico, Mexico,
Canada, and China. The company was founded by Richard M. Schulze and Gary Smoliak in 1966 as an
audio specialty store; in 1983, it was renamed and rebranded with more emphasis placed on
consumer electronics.
Best Buy sells consumer electronics and a variety of related merchandise, including software, video
games, music, DVDs, Blu-ray discs, mobile phones, digital cameras, car stereos and video cameras, in
addition to home appliances (washing machines, dryers, and refrigerators), in a noncommissioned sales environment.
Best Buy currently has nearly 1150 stores throughout the US of which 24 stores are in located in and
around Houston. The main focus of our case study is to locate warehouses to satisfy the demand at
the 24 stores located in and around Houston for the Personal Computer (PC) segment only.
Figure 1 Initial Network diagram of Houston Best Buy stores connected through Free-ways
2. Method of Approach
We are employing the p-median and p-center procedures to decide on the potential warehouse
locations according to the demand and distances. The main objective is to satisfy the demand at each
Page |3
of the stores with the lowest possible costs. Best Buy being the world’s largest multi-channel
consumer electronics retailer which handles the sales of different electronic goods.
The demands for PC’s have been calculated from the population data that has been approximated
using the census data which gives the population at each zip code.
In order to locate the warehouses the following approaches have been used.
P-Median
Uncapacitated
(Practical &
Random) cases
P-Center
Types of SubProblems solved
P-Median
Capacitated
(Practical &
Random) Cases
P-Center
3. Assumptions
3.1 General Assumptions
•
•
•
•
•
Trucks are assumed to be the only modes of transporting goods between different nodes.
Nodes are connected by freeways because trucks are used for transportation and their
movement is usually restricted to freeways.
Vertex Restricted – Considering only the vertex as the potential locations for the placement of
warehouses as without this assumption, the warehouses can be placed anywhere within the
complete network and the computational capabilities of our systems are limited.
This assumption seems valid as in the real time scenario, purchasing land in Houston is
expensive compared to upgrading an existing store with warehousing capabilities.
Fixed Costs and Labor costs are considered negligible for the same reason.
For the network, each node is connected to three of its closest nodes. (This is for minimum 3
degrees of connectivity). Distances to all other nodes is calculated using these points as ‘via’
points. This doesn’t change the three minimum distances but changes all other distances in
the final distance matrix.
Each truck is assumed to carry about 5000 PC’s at once based on the following calculations.
LTL(Less than Truckload) is assumed to cost the same as a FTL(Full Truck Load).
Page |4
Size of Each PC Package
Capacity of a Truck
Computers per Truck
Cost per PC per mile
Average Size of Laptop & Desktop
23.28 x 18.26 x 8.655 in
18396000 cubic inches
5000.0029648
$0.0016
Table 1 Table denoting cost per PC per mile calculation
•
•
The cost of transporting one PC across one mile is calculated using the website
www.freightquote.com, which provides the cost of transporting FTL (Full truck load) from
source to destination specified by the user. We used the following two locations since it was
the maximum distance traversed between any two nodes in our network (44.5 miles)
Best buy 7318 Cyprus Creek Parkway to Best Buy 19425 Gulf Freeway.
Quotes from many freight carriers were given and we chose to use FedEx since it is a major
carrier in US and has relatively stable costs.
Capacity Calculation for the capacitated problems is done first by assuming demand increases
linearly by the function given by Y = 5526.7*LN(2017)-41959 [Appendix B.1]
Where Y is the percentage of households with PC’s. This function is used to project demand
for 2017(3 years in the future) to account for increase in demand in the future.
The capacity for each warehouse is calculated by (Total projected demand)/(No of
warehouses)
Therefore the capacity is assumed to be equal for all warehouses.
3.2 Assumptions specific for Random model
•
•
•
•
•
Uniform distribution was used for generating random demand, warehouse capacity and
distances. The continuous uniform distribution represents a situation where all outcomes in a
range between a minimum and maximum value are equally likely.
Undirected network is formed by enforcing symmetry in the distance matrix i.e the transpose
of the matrix gives the original matrix. Each time a new random matrix is generated using
excel and loaded into the .dat file.
Demand in the random model had a Uniform (900, 2000) since the demand in the actual
model varied between these limits.
Capacity in random model was randomized by using a Uniform (35000, 48000) and then
dividing by the number of warehouses required as the number of warehouses to be opened
increased. The reason for these limits was to avoid repetitive occurrences of infeasible
solutions due to excess demand.
The network diagram remains unchanged. This is accomplished by manually adding the
distances between connecting nodes using the graph as reference in excel.
Page |5
4. Demand Calculation
Population covered
by a Best Buy retail
store [7]
Average number of
Households owning a
PC [4]
Average PC
replacement time
(to get estimated
demand per year)
Households buying
PC’s from Retail
stores [5]
Average number of
PC’s per household
[4]
Best Buy’s market
share in PC’s
The above flow chart represents the systematic approach that has been adapted for predicting
demand for PC’s at each of the Best Buy retail stores for Houston area. Appendix B contains full
details of the approach.
4.1 Assumptions specific for Demand Calculation
•
•
•
Population of Houston in a particular Zip-code is being served by Best-Buy store in that
particular area assuming that when a customer tries to locate the nearest best buy store on a
Global Positioning System (GPS), he types in the Zip Code of his residence.
Zip codes in Houston area that are not served by a Best Buy are randomly distributed across
the city. This population is added up and divided equally among all the other best buy
locations.
Assumptions that lead us to arrive at the forecasted demand for 2014 are gathered from
various references and resources mentioned in Appendix B.1
Page |6
Figure 2 Modified network used for Distance calculation [Appendix A.2]
5. Observations and Discussion of Results
5.1 Un-capacitated P-median Problem
•
•
•
•
Objective function value decreases as the number of warehouses increase as the total distance to be
travelled decreases with more Open Warehouses. As the number of Open Warehouses increases the
Transportation cost decreases as the demand of a particular store is satisfied by an open warehouse
located either at that store or close to that store.
Objective function for the Practical and Random cases are not compared because the units are
different due to use of different algorithms. [Appendix C.1, Appendix C.2]
Elapsed time to solve the problems remains negligible (both for practical and random case) because
of the low complexity of Un-capacitated P-Median problem, the computer is able to solve them
almost instantaneously. [Appendix C.4]
Number of branches in the solution process was always below 10 and 0 for majority of the cases for
both practical and random data. This again points to the simplicity of the Un-capacitated P-median
algorithm. The reason for number of branches being 0 is because the program was able to tighten the
bounds to such an extent that the variables became fixed. [Appendix C.4]
Page |7
•
Number of MIP Iterations was observed to be higher for lower values of P that is when the number
of open Warehouses were lower. This points to an increased difficulty in locating fewer warehouses.
The time to solve these is also marginally higher as compared to opening higher number of
warehouses. [Appendix C.4]
Transportation cost for total demand
(in dollars)
Transportation Cost vs Number of open warehouses (Practical Case)
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0
P-median
1
2
3
4
5
15
16
17
18
20
21
22
23
5
10
15
20
25
Number of Open Facilities
Objective Function
Location of Open Warehouses
Value
2026.59
12
1684.99
12,21
1356.36
1,12,21
1225.83
1,11,19,21
1098.51
1,11,15,19,21
256.2
2,3,4,6,7,8,10,12,13,15,19,21,22,23,24
202.35
2,3,4,6,7,8,10,12,13,14,15,19,21,22,23,24
153.3
2,3,4,6,7,8,10,12,13,14,15,19,20,21,22,23,24
114.14
2,3,4,6,7,8,9,10,12,13,14,15,19,20,21,22,23,24
54.9
1,2,4,5,6,7,8,9,10,12,13,14,15,17,19,20,21,22,23,24
33.78
1,2,4,5,6,7,8,9,10,12,13,14,15,17,18,19,20,21,22,23,24
13.38
2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24
4.97
1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24
5.2 Un-Capacitated P-Center problems
•
The center problems are more complex than the median problems since here the aim is to reduce the
maximum distance between a store and a warehouse. Each time a distance is reduced a new
maximum distance is found between another store and warehouse which again needs to be reduced.
Page |8
•
•
This is an iterative process. Hence the number of MIP iterations are very high compared to the pmedian problems.
Elapsed time is higher owing to the increased complexity of the center problems and in some
Random cases runs close to a minute (2,3,4,5 center). [Appendix D.4]
Number of branches and MIP Iterations are higher for the same centers which validates the fact that
as the number of branches and MIP Iterations increase, the Elapsed time increases. Also as the
number of warehouses to be opened approached (n-3), (n-2), Elapsed time decreased as the number
of branches were less. [Appendix D.4]
Objective Function vs Number of Open Warehouses
Objective Function (in miles)
30
25
20
15
10
5
0
0
5
10
15
20
25
Number of Open Warehouses
5.3 Capacitated P-Median Problem
•
•
Objective function value decreases as the number of warehouses increases.
Elapsed time is mostly minimal and equal for the practical case with a marginal increase for lower
values of P in the Random case. This was observed in the un-capacitated problem as well. [Appendix
E.4]
•
Number of branches are considerably higher when compared to the un-capacitated case as an
additional capacity constraint has been added. [Appendix E.4]
•
Number of MIP Iterations for lower values of P are observed to be higher which is similar to the Uncapacitated case. But the iterations on an average are still higher since the bounds on the variables
have become tighter due to addition of extra capacity constraint. [Appendix E.4]
Page |9
Number of Branch and Bounds
Number of Branch and Bounds for Capacitated P-Median
200
150
100
50
0
1
2
3
4
5
15
16
17
18
20
21
22
23
21
22
23
Number of Open Warehouses
Practical Case
Random Case
Number of MIP Iterations
Number of MIP Iterations for Capacitated P-Median
1000
800
600
400
200
0
1
2
3
4
5
15
16
17
18
20
Number of Open Warehouses
Practical Case
Random Case
5.4 Capacitated P-Center problems
•
•
•
The capacitated P-Center problems are again more complex than the Capacitated P-median problems
since here the aim is to reduce the maximum distance between a store and a warehouse. Each time a
weighted distance is reduced a new maximum weighted distance is found between another store and
warehouse which again needs to be reduced. This is an iterative process. Hence the number of MIP
iterations are very high compared to the p-median problems.
The Practical and Random cases for this type of problem took the most Elapsed time, higher number
of Branches and MIP Iterations to solve owing to the addition of an extra capacitated constraint over
the Un-capacitated P-Center problem (the next most complex problem)
Elapsed time in Practical case for the first time reached close to 1 second in one case and finished on
an average of 0.5 seconds. However in the Random cases the highest solving time was 2895.99 for an
iteration in 15-center problem. This is due to the higher number of Branches and MIP iteration
required for a 15-center problem. As we approach the extreme ends, computation becomes tougher
P a g e | 10
•
•
owing to the increased complexity of the center problems and in some Random cases runs close to a
minute (2,3,4,5 center). [Appendix F.4]
Number of branches and MIP Iterations are higher for higher centers which validates the fact that as
the number of branches and MIP Iterations increase, the Elapsed time increases. Also as the number
of warehouses to be opened approached (n-3), (n-2) Elapsed time decreased as the number of
branches were less. [Appendix F.4]
The below screen shot shows how results look in Ampl. The Objective function indicates 12.3. The
variable y[j] indicates the open warehouse. For example in the below simulation, warehouses at
location 3, 13, 19, 21 and 23 are open. The variable x[i,j] indicates the amount of demand satisfied by
open warehouse j to existing store at i. For example in the below case, 57.1% of demand at store 3 is
satisfied by open warehouse at 3 and 42.8% of demand is satisfied by open warehouse at 19.
Figure 3 A screenshot showing result for a capacitated P-center problem showing Objective function, Open
warehouses and percentage of demand satisfied
P a g e | 11
6. Conclusions
•
•
•
•
•
•
The choice between Median problem or a Center problem is dictated by the goal of the
management which can be to either reduce the total cost of transportation or to reduce the
lead time. (Time for transporting goods from warehouse to retail store). If the main aim of
management is to reduce Transportation cost, a P-Median approach is suggested and If the
objective is to reduce the Lead Time, a P-Center approach is recommended
The formation of the initial network diagram i.e the connection of the nodes by arcs plays an
important role in the final solution. A change in the arc connections will change the arc
lengths which could result in different answers. Hence formation of network diagram is critical
for successful implementation of warehouse location problems.
The use of Random data to validate the models should be done with caution as the usage of
improper upper and lower bounds may lead to infeasible solutions. A good model might be
made to look impractical.
The variation of elapsed time when under one second might not be due to the complexity of
the problem but rather due to the processing capability of the computer. For example, a
difference of 0.250s and 0.255s in the Elapsed times can be considered negligible. Higher
Elapsed times over 15 - 20 seconds is due to the higher complexities of the problem which is
further verified by the fact that number of Branches and MIP iterations are more.
As the number of warehouses that are to be opened approaches (n-3), (n-2) and so on, the
problem becomes easier to solve and takes lesser time as the complexity decreases.
Vertex restriction for our case helps to reduce the complexity of the problem as this is a small
model (maximum distance 44.5 miles) and it is economical for Best Buy to expand the existing
store location to a small adjacent warehouse rather than locating a large number of
warehouses across the city of Houston. However, when considering locating New facilities
across wider areas (Ex. State or Country), Vertex restriction is not ideal as potential warehouse
locations might increase manifold and can be computationally challenging.
P a g e | 12
Appendix
P a g e | 13
A.1 Appendix A (Network)
A.2 Table addressing each Best Buy store by a number along with population
•
Each retail store has been assigned a number for identification purpose. This makes it easier to identify
the warehouse in the general solution
Number as
depicted on the
Network
Store Location
Location
Population
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
5133 Richmond Avenue
510 GulfGate Center Mall
13238 Northwest Fwy
10777 North Fwy
7034 Highway 6N
2480 Highway 6N
9670 Katy Fwy
6006 E Sam Houston Pkwy N
7318 Cypress Creek Pkwy
8210 S Gessner Dr
100 Meyerland Plaza Mall
10780 Kempwood Dr
904 Lathrop St
10047 Westpark Dr
12089 Beechnut St
2632 Smith Ranch Road
5692 Fairmont Pkwy
16980 Southwest Fwy
19425 Gulf Fwy
5340 W Grand Pkwy S
25525 Highway 290
2000 Willowbrook Mall #
1550
5135 W Alabama St#7210
171 N Pasadena Blvd
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Houston,TX
Pearland, TX
Pasadena,TX
Sugarland,TX
Webster,TX
Richmond,TX
Cypress,TX
Houston,TX
59336
86399
95081
69819
116502
102151
91989
78521
73008
121969
82628
73358
75464
86385
106526
120228
73223
124514
73059
83682
122264
73008
Houston,TX
Pasadena,TX
59336
85977
23
24
P a g e | 14
A.3 Distance Matrix denoting the distance between each of the Best Buy store
locations
•
•
•
The below table contains the distance between the nodes.
The distance from Node 1 to Node 24 is assumed to be equal to distance between nodes 24 to
Node 1 (undirected nodes). Hence the distance between same nodes are 0. This explains the
reason for diagonal elements to be 0.
All the distances are given in miles.
X
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
P a g e | 15
1
0
9.2
1.1
10.3
8
11.5
11.7
31.8
16.4
26.1
21.2
20.9
22.5
19.5
31.6
22.8
22.4
24.1
29.6
35.6
43.3
29.9
37.8
44.5
2
9.2
0
9.1
18.2
9.1
9.4
8.7
36.1
13.6
25
19.1
18.8
10.3
16.7
19.3
16.6
19.6
22
28.5
35.3
41
21
35
43.4
3
1.1
9.1
0
9.8
11.7
10.3
10.8
31
14.7
23.3
20.3
20
21.6
18.6
30.7
21.9
21.5
23.2
28.7
34.7
41.2
29
37
43.6
4
10.3
18.2
9.8
0
22.1
10.4
17.2
22
16.4
15.5
15.8
15.5
29.7
24.7
37.4
28.3
20.9
18.6
19
23.9
31.5
32.8
27.4
33.9
5
8
9.1
12
22
0
3.5
6.8
28
7.5
19
13
13
19
16
29
19
18
16
23
29
35
26
30
38
6
12
9.4
10
10
3.5
0
5.7
26
6.4
16
9.7
10
19
16
29
19
16
13
20
26
32
25
27
35
7
11.7
8.7
10.8
17.2
6.8
5.7
0
31
5.1
19.3
11.9
13
10.9
8.2
21.1
11.5
10.9
16.2
22.8
28.8
35.3
18.6
26.5
37.7
8
32
36
31
22
28
26
31
0
25
11
22
22
36
29
42
32
27
25
14
10
12
38
30
23
9
16.4
13.6
14.7
16.4
7.5
6.4
5.1
25.4
0
14.8
7.4
8
10.5
5.9
19.5
10.6
7.9
11.6
18.3
24.3
30.8
17.7
25.4
33.2
10
26.1
25
23.3
15.5
19.1
16.3
19.3
11.1
14.8
0
12.3
11.5
24.1
18.1
30.8
21.7
16.8
14.4
6.8
10.1
18.2
26.2
18.8
21.7
11
21.2
19.1
20.3
15.8
13.1
9.7
11.9
21.6
7.4
12.3
0
0.8
11.4
6.3
20.5
10.8
6.4
4
12.5
18.5
25
15.8
17.9
27.4
12
20.9
18.8
20
15.5
12.9
10.2
13
22.2
8
11.5
0.8
0
12.6
7
19.7
10.1
5.6
3.4
12.1
18.1
24.6
15.1
17.2
27
13
22.5
10.3
21.6
29.7
18.6
18.6
10.9
35.7
10.5
24.1
11.4
12.6
0
7.2
9.1
6.7
10.3
14.8
24.2
30.7
37.2
11.1
25.7
39.6
P a g e | 16
14
19.5
16.7
18.6
24.7
15.5
15.5
8.2
28.9
5.9
18.1
6.3
7
7.2
0
15.2
5.6
2.7
6.8
16.9
22.8
29.3
12.4
20.2
31.7
15
31.6
19.3
30.7
37.4
28.5
28.5
21.1
42.4
19.5
30.8
20.5
19.7
9.1
15.2
0
13.3
17.3
21.4
31.3
37.3
43.8
13.9
31.7
46.1
16
22.8
16.6
21.9
28.3
18.9
18.9
11.5
32.4
10.6
21.7
10.8
10.1
6.7
5.6
13.3
0
3.5
8
20.4
26.4
31
8.7
18.9
33.3
17
22.4
19.6
21.5
20.9
18.3
15.6
10.9
27.4
7.9
16.8
6.4
5.6
10.3
2.7
17.3
3.5
0
4.7
17.1
23.1
29.6
9.6
17.5
31.9
18
19
20
21
22
23
24
24.1 29.6 35.6 43.3 29.9 37.8 44.5
22 28.5 35.3 41
21
35 43.4
23.2 28.7 34.7 41.2 29
37 43.6
18.6 19 23.9 31.5 32.8 27.4 33.9
16 22.6 28.6 35.1 26 29.8 37.5
13.3 19.9 25.8 32.3 25 27.1 34.7
16.2 22.8 28.8 35.3 18.6 26.5 37.7
25.1 13.9 10.4 12.3 37.8 29.5 22.5
11.6 18.3 24.3 30.8 17.7 25.4 33.2
14.4 6.8 10.1 18.2 26.2 18.8 21.7
4
12.5 18.5 25 15.8 17.9 27.4
3.4 12.1 18.1 24.6 15.1 17.2 27
14.8 24.2 30.7 37.2 11.1 25.7 39.6
6.8 16.9 22.8 29.3 12.4 20.2 31.7
21.4 31.3 37.3 43.8 13.9 31.7 46.1
8
20..4 26.4 31
8.7 18.9 33.3
4.7 17.1 23.1 29.6 9.6 17.5 31.9
0
11.2 16.9 23.5 14.2 14.4 25.9
11.2
0
6.7 12.7 24.2 14.7 15.1
16.9 6.7
0
6.4
30 20.6 14.2
23.5 12.7 6.4
0
34.5 18.3 11.2
14.2 24.2 30 34.5
0
18.3 35.7
14.4 14.7 20.6 18.3 18.3
0
18.6
25.9 15.1 14.2 11.2 35.7 18.6
0
B.1 Appendix B (Demand Calculation)
The following steps have been taken to find the demand at each selected store.
1. Proper location of the stores with addresses.
2. Population that is served by each store.
3. Number of households that are present in each area served by the stores. Houston
demographics state that on an average about 2.67 people stay in a house. [7]
No of Households = Population/2.67
4. The number of households owning a personal computer has always been on the rise since
1984. Projecting this trend using the data obtained from the recent studies has helped us
predict the percentage of households that own a PC for the year 2014 (87.40%) and also the
next five years. [3]
No of Households that own a PC in 2014 = No of households * 87.40%
Projected
Demand
P a g e | 17
Year
1984
1989
1993
1997
2003
2007
2010
2012
Percentage
8.2
15
22.9
36.6
61.8
69.7
76.7
78.9
2014
2017
87.4
95.6
5. It is also known that on an average each household has nearly 2 PC’s and hence [4]
Total number of PC’s = No of households that own a PC * 2
6. The demand for a product is always distributed among the Market Share of the retailer. The
Best Buy Market share is about 19.30% till the year 2013. [5] Hence
Best Buy Share in PC Market = Total number of PC’s * 19.3%
7. With the increase in e-commerce, the sales that happen through online shopping is almost on
par with the in-store sales. For our scenario we are only considering the in-store sales alone.
Statistics state that about 51.74% of the people buy products in the store and rest purchase
online. [5] Hence
Households buying from Best Buy Store = Best Buy Share in PC Market * 51.74%
8. The final values obtained from the above equation gives us the demand values for PC’s at
each of the existing locations.
Size of Each PC
Package
Laptops Size
19.29 x 13.15 x 3.31
inches
Desktop Size
27.28 x 23.37 x 14
inches
Cost for Full Truck Load (FTL)- 358 $
Capacity of one truck = 18,396,000 cubic inches
Average volume of 1 PC = 3679.178 cubic inches
No. of computers per truck = (capacity of truck) / (average volume of PC)
= (18396000)/(3679.178)
= 5000.003 ~ 5000
The cost of transporting one pc by one mile= 358/(5000*44.5) = 0.0016 $
Therefore number of PC’s in one FTL is 5000
P a g e | 18
Average Size of PC
23.28 x 18.26 x 8.655
C.1 Appendix C (Un-Capacitated P-Median)
C.1 Ampl Code- Practical Case [2]
ampl: reset;
ampl: model uncapmed.mod;
ampl: data uncapmed.dat;
ampl: solve;
param dwd{i in 1..24, j in 1..24};
param p = 1;
var x {i in 1..24, j in 1..24} binary;
var y {j in 1..24} binary;
minimize Total_cost: sum {i in 1..24, j in 1..24} dwd[i,j] * x[i,j];
subject to demand {i in 1..24}: sum {j in 1..24} x[i,j] = 1;
subject to prevent {i in 1..24, j in 1..24} : x[i,j] <= y[j];
subject to facilities {i in 1..24}: sum {j in 1..24} y[j] <= p;
ampl: display y;
ampl: display x;
C.2 Ampl Code – Random Case
ampl: reset;
ampl: model uncapmed.mod;
ampl: data uncapmed.dat;
ampl: solve;
param dwd {i in 1..24, j in 1..24} := Uniform (0, 2000);
param p = 1;
var x {i in 1..24, j in 1..24} binary;
P a g e | 19
var y {j in 1..24} binary;
minimize Total_cost: sum {i in 1..24, j in 1..24} dwd[i,j] * x[i,j];
subject to demand {i in 1..24}: sum {j in 1..24} x[i,j] = 1;
subject to prevent {i in 1..24, j in 1..24} : x[i,j] <= y[j];
subject to facilities {i in 1..24}: sum {j in 1..24} y[j] <= 1;
ampl: display y;
ampl: display x;
Refer: Attachment Un-Capacitated P-Median.zip for complete solutions (screen-shots)
C.3 Tables comparing results for Practical and Random Cases for Un-capacitated PMedian
pmedian
1
2
3
4
5
15
16
17
18
20
21
22
23
Objective
P a g e | 20
2026.59
1684.99
1356
1225.83
1098.51
256.2
202.35
153.3
144.14
54.9
33.78
13.3
4.97
Practical
Random
Elapsed Time
0.203
0.2
0.2
0.2
0.2
0.21
0.222
0.16
0.2
0.221
0.2
0.21
0.2
0.1557
0.5211
0.2791
0.2872
0.1466
0.1265
0.1377
0.1234
0.126
0.125
0.124
0.132
0.127
Practical
Random
Branch and Bounds
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6.9
3.4
9.2
0.2
0
0
0
0
0
0
0
0
Practical
Random
MIP Iterations
131
133
98
106
93
43
36
35
32
27
28
26
25
396.6
601
323
308.6
125.5
41.8
39.4
38.9
40.2
39.1
38.7
39.2
39.4
Practical Case
S.No
1
2
3
4
5
15
16
17
18
20
21
22
23
Objective
2026.59
1684.99
1356
1225.83
1098.51
256.2
202.35
153.3
144.14
54.9
33.78
13.3
4.97
Time( seconds)
0.203
0.2
0.2
0.2
0.2
0.21
0.222
0.16
0.2
0.221
0.2
0.21
0.2
Branch and Bounds
0
0
0
0
0
0
0
0
0
0
0
0
0
MIP (Iterations)
131
133
98
106
93
43
36
35
32
27
28
26
25
Random Case
1-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 21
Iteration No
Elapsed Time
Objective
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard Deviation
0.187
0.14
0.14
0.14
0.14
0.249
0.156
0.14
0.125
0.14
0.1557
0.00121061
0.034793821
18773.47
16081.09
17552.27
17419.12
16889.1
19722.63
19092.28
18361.37
17586.24
17302.88
17878.045
1085201.537
1041.730069
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of MIP
Iterations
420
354
400
417
355
411
439
367
385
418
396.6
799.44
28.27437002
2-Median
S.No
1
2
3
4
5
6
7
8
9
10
3-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 22
Iteration No
Elapsed Time
Objective
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard Deviation
0.624
0.796
0.53
0.546
0.734
0.437
0.359
0.14
0.421
0.624
0.5211
0.03313789
0.182038155
10840.84
10189.51
10189.53
10539.81
12098.49
10442.4
9942.5
9124.49
11002.4
11642.1
10601.207
652624.4054
807.8517224
Iteration No
Elapsed Time
Objective
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard Deviation
0.546
0.312
0.312
0.218
0.296
0.203
0.452
0.14
0.187
0.125
0.2791
0.01646829
0.128328835
6194.61
7720.42
7043.98
6782.77
6806
7689.79
6714.06
5601.06
6434.22
6940
6792.691
366295.8987
605.2238418
No of
Branches
1
13
11
4
22
4
2
0
2
10
6.9
43.89
6.62495283
No of MIP
Iterations
447
784
596
537
1068
510
410
345
582
731
601
40697.4
201.7359661
No of
Branches
12
2
2
2
2
2
10
0
2
0
3.4
15.24
3.903844259
No of MIP
Iterations
445
384
334
245
301
357
456
161
306
241
323
7739.6
87.97499645
4-Median
S.No
1
2
3
4
5
6
7
8
9
10
5-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 23
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Elapsed
Time
0.53
0.218
0.219
0.406
0.25
0.312
0.125
0.125
0.406
0.281
0.2872
0.01512736
Standard Deviation
0.122993333
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Elapsed Time
Objective
5431.98
4178.28
3929.16
4721.09
4702.92
5085.17
5757.89
4998.79
5507.19
5970.13
5028.26
394663.638
1
628.222602
3
Objective
0.296
3799.29
0.109
4073.08
0.125
3091.22
0.156
5293.73
0.109
3771.62
0.218
5313.21
0.109
3855.76
0.109
4115.21
0.11
3692.7
0.125
4338.94
0.1466
4134.476
0.00354344 438447.571
0.059526801 662.1537367
No of
Branches
18
0
7
29
10
6
0
0
20
2
9.2
90.76
No of MIP
Iterations
463
169
232
575
247
338
170
155
483
254
308.6
20102.24
9.526804291
141.7823684
No of
Branches
0
0
0
0
0
2
0
0
0
0
0.2
0.36
0.6
No of MIP
Iterations
151
116
122
167
101
188
95
95
104
116
125.5
939.45
30.65044861
15-Median
S.No
1
2
3
4
5
6
7
8
9
10
16-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 24
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Elapsed
Objective
Time
0.203
2165.96
0.124
2172.27
0.109
1716.53
0.11
2453.49
0.11
1117.46
0.125
1647.69
0.109
2967.81
0.141
1981.9
0.125
2120.26
0.109
1712.89
0.1265
2005.626
0.00075365 227978.919
0.027452687 477.4713803
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of MIP
Iterations
38
42
42
39
41
41
46
38
47
44
41.8
8.76
2.959729717
Elapsed
Objective
Time
0.202
1803.51
0.125
1322.00
0.125
1858.84
0.141
2117.27
0.124
2242.92
0.125
2040.87
0.109
2114.39
0.109
2386.81
0.125
1806.62
0.172
2035.51
0.1357
1972.87357
0.00077621 78664.91572
0.027860546 280.4726648
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of MIP
Iterations
40
37
42
39
39
40
40
42
37
38
39.4
2.84
1.685229955
17-Median
S.No
1
2
3
4
5
6
7
8
9
10
18-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 25
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Elapsed
Objective
Time
0.234
1741.19
0.125
1671.29
0.109
1603.62
0.109
2346.74
0.11
1348.82
0.11
1703.89
0.125
1469.30
0.109
2171.14
0.109
2777.16
0.094
2313.58
0.1234
1914.672041
0.00142904 190453.153
0.037802645 436.4093869
Elapsed
Time
0.22
0.11
0.11
0.12
0.12
0.13
0.12
0.11
0.11
0.11
0.126
0.001024
0.032
Objective
2165.14
2327.49
1808.18
2325.07
2035.16
1744.30
2020.69
974.56
1876.32
1932.97
1920.988671
135292.3838
367.8211301
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of MIP
Iterations
37
38
39
39
41
39
39
39
39
39
38.9
0.89
0.943398113
No of MIP
Iterations
42
38
41
41
39
40
40
41
42
38
40.2
1.96
1.4
20-Median
S.No
1
2
3
4
5
6
7
8
9
10
21-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 26
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Elapsed
Objective
Time
0.22
1958.25
0.11
1402.34
0.11
2614.35
0.11
2571.50
0.11
2301.51
0.17
1809.96
0.11
1898.57
0.1
1494.75
0.1
2529.58
0.11
2193.27
0.125
2077.407116
0.001365
171345.3724
0.036945906 413.938851
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of MIP
Iterations
40
38
41
41
41
37
39
40
38
36
39.1
2.89
1.7
Elapsed
Objective
Time
0.22
1646.71
0.11
1720.78
0.11
1807.94
0.11
2068.06
0.11
1827.66
0.13
1701.84
0.11
1690.02
0.11
2082.48
0.12
1968.03
0.11
2289.98
0.124
1880.351381
0.001064
40773.95535
0.032619013 201.9256184
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of MIP
Iterations
39
38
37
39
39
39
38
40
39
39
38.7
0.61
0.781024968
22-Median
S.No
1
2
3
4
5
6
7
8
9
10
23-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 27
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
Standard
Deviation
Elapsed
Objective
Time
0.22
1962.35
0.16
2107.82
0.16
1956.87
0.11
2093.12
0.11
2038.41
0.12
1474.39
0.11
2524.77
0.11
2073.25
0.11
2093.63
0.11
2145.88
0.132
2047.04843
0.001236
58998.03324
0.035156792 242.8951075
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of MIP
Iterations
38
39
39
41
41
40
37
38
38
41
39.2
1.96
1.4
Elapsed
Objective
Time
0.22
1662.84
0.11
2398.12
0.11
1848.12
0.11
2187.57
0.11
1954.60
0.14
2497.38
0.12
1896.25
0.12
2499.71
0.12
1625.81
0.11
2145.88
0.127
2071.627082
0.001041
95034.03651
0.032264532 308.2759097
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of MIP
Iterations
37
40
42
41
39
41
37
38
38
41
39.4
3.04
1.743559577
C.4 Graphs comparing trend between Practical and Random Cases
Elapsed Time for Uncapacitated P-Median
0.6
Elapsed TIme (s)
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
15
16
17
18
20
21
22
23
Number of Open Warehouses
Number of Branch and Bounds
Practical Case
Random Case
Number of Branch and Bounds for Uncapacitated
P-Median
10
9
8
7
6
5
4
3
2
1
0
1
2
3
4
5
15
16
17
18
Number of Open Warehouses
Practical Case
P a g e | 28
Random Case
20
21
22
23
Number of MIP Iterations
Number of MIP Iterations for Uncapacitated PMedian
700
600
500
400
300
200
100
0
1
2
3
4
5
15
16
17
18
Number of Open Warehouses
Practical Case
P a g e | 29
Random Case
20
21
22
23
D. Appendix D (Un-Capacitated P-Center)
D.1 Ampl Code – Practical Case [2]
ampl: reset;
ampl: model uncapcen.mod;
ampl: data uncapcen.dat;
ampl: solve;
param dis{i in 1..24, j in 1..24};
param p = 1;
var x {i in 1..24, j in 1..24} binary;
var y {j in 1..24} binary;
var z >= 0;
minimize Total_distance: z;
subject to demand {i in 1..24}: sum {j in 1..24} x[i,j] = 1;
subject to prevent {i in 1..24, j in 1..24}: x[i,j] <= y[j];
subject to facilities {i in 1..24}: sum {j in 1..24} y[j] <= p;
subject to disvar {i in 1..24}: sum {j in 1..24} dis[i,j] * x[i,j] <= z;
ampl: display y;
ampl: display x;
D.2 Ampl Code – Random Case
ampl: reset;
ampl: model uncapcen.mod;
ampl: data uncapcen.dat;
ampl: solve;
param dis{i in 1..24, j in 1..24} := Uniform (0, 46);
param p = 1;
var x {i in 1..24, j in 1..24} binary;
var y {j in 1..24} binary;
var z >= 0;
minimize Total_distance: z;
subject to demand {i in 1..24}: sum {j in 1..24} x[i,j] = 1;
P a g e | 30
subject to prevent {i in 1..24, j in 1..24}: x[i,j] <= y[j];
subject to facilities: sum {j in 1..24} y[j] <= p;
subject to disvar {j in 1..24}: sum {i in 1..24} dis[i,j] * x[i,j] <= z
ampl: display x;
ampl: display y;
Refer: Attachment Un-Capacitated P-Center.zip for complete solutions (screen-shots)
D.3 Tables comparing results for Practical and Random Cases for Un-Capacitated PCenter
Practical
p-center
Objective
value
1
2
3
4
5
15
16
17
18
20
21
22
23
25.9
19.5
15.1
14.2
12.3
6.4
5.9
5.1
3.5
3.4
2.7
1.1
0.8
P a g e | 31
Random
Elapsed time
0.344
0.468
0.609
0.562
0.5
0.343
0.344
0.328
0.28
0.296
0.312
0.281
0.187
0.942
37.32
46.91
71.31
36.11
0.4008
0.368
0.319
0.319
0.272
0.279
0.293
0.265
Practical
Random
Branch & Bound
nodes
0
3
16
9
11
3
15
4
0
0
0
4
0
46
33273.5
39978.8
63113.8
18992.9
47.7
16.8
5.1
6.2
0.1
2.8
4.2
1.6
Practical
Random
MIP Iterations
637
1294
1271
1127
861
272
291
260
257
245
250
243
255
1056.7
510877.8
648261.9
1060446
685059
1252.1
1085.6
932.2
1008.2
883.2
989.6
1055.4
1004.6
Practical Case
P-Center
1
2
3
4
5
15
16
17
18
20
21
22
23
Objective
value
25.9
19.5
15.1
14.2
12.3
6.4
5.9
5.1
3.5
3.4
2.7
1.1
0.8
Elapsed
time
0.344
0.468
0.609
0.562
0.5
0.343
0.344
0.328
0.28
0.296
0.312
0.281
0.187
Number of Branch &
Bound nodes
0
3
16
9
11
3
15
4
0
0
0
4
0
MIP Iterations
637
1294
1271
1127
861
272
291
260
257
245
250
243
255
Random Case
1-Center
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 32
Iterations
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
Elapsed Time
1.04
0.88
0.91
0.94
0.92
1.04
0.95
0.94
0.91
0.89
0.942
0.002836
0.053254108
Objective
406.24
422.13
450.23
445.28
388.09
424.6
434.35
438.57
411.32
399.04
421.985
381.224305
19.5249662
No of Branches
46
46
46
46
46
46
46
46
46
46
46
0
0
MIP Iterations
1102
1031
1060
1083
1012
1114
1040
995
1070
1060
1056.7
1297.01
36.01402505
2-Center
S.No
1
2
3
4
5
6
7
8
9
10
3-Center
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 33
Iteration
No
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
Elapsed Time
Objective
No of Branches
8.09
9.5
10.661
52.63
30.363
35.385
33.989
28.698
39.088
124.816
37.322
1036.9698
32.20201546
121.99
125.77
124.98
57.71
55.81
47.49
48.96
52.8
60.1
56.52
75.213
1044.164201
32.31352969
7134
7449
7729
54575
19427
25269
28566
21817
33847
126922
33273.5
1163603363
34111.6309
Iteration No
Elapsed Time
Objective
No of Branches
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
37.503
68.463
35.185
94.096
70.149
29.168
20.167
41.298
45.298
27.835
46.9162
488.4983742
22.10199932
68.66
54.38
59.52
47.63
59.78
67.25
51.14
63.03
52.02
58.98
58.239
43.382829
6.586564279
33990
69443
36137
65384
76866
16881
12357
33750
41628
13352
39978.8
497183043.4
22297.60174
MIP
Iterations
81445
96916
113514
728048
418820
479138
441928
380992
510915
1857062
510877.8
2.40416E+11
490321.9491
MIP
Iterations
505459
952003
460894
1373581
926901
431667
250460
582727
596125
402802
648261.9
1.02391E+11
319985.7721
4-Center
S.No
1
2
3
4
5
6
7
8
9
10
5-Center
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 34
Iteration
No
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
Iteration
No
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
Elapsed Time
Objective
No of Branches
13.75
74.71
49.641
39.115
136.272
100.286
75.901
190.279
8.723
24.515
71.3192
3017.20512
54.92909174
26.39
39.28
33.14
29.43
36.05
36.17
37.55
29.48
23.06
28.88
31.943
25.294481
5.029361888
5591
65128
36345
17074
146360
92958
72117
179485
3234
12846
63113.8
3370552341
58056.45822
Objective
No of
Branches
10140
3824
7155
51926
14221
5602
31574
48012
10893
6582
18992.9
295531261.1
17191.02269
Elapsed Time
21.328
21
8.955
20.34
18.346
20.89
84.348
21.74
27.258
19.84
13.832
19.63
48.575
24.52
102.836
26.36
20.654
19.78
15.054
18.54
36.1186
21.264
946.301071
5.218084
30.76200694 2.284312588
No of MIP
Iterations
219217
1074354
769825
631517
1960346
1528607
1116982
2809058
108165
386391
1060446.2
6.42732E+11
801705.9648
No of MIP
Iterations
380972
133868
328535
1637411
530142
221535
838771
2096544
381816
300996
685059
3.9246E+11
626465.9513
15-Center
S.No
1
2
3
4
5
6
7
8
9
10
16-Center
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 35
Iteration No
Elapsed Time
Objective
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
0.421
0.343
0.328
0.405
0.406
0.484
0.577
0.405
0.28
0.359
0.4008
0.00634196
0.079636424
5.06
5..43
8.07
5.95
4.85
7.31
6.34
7.67
7.82
8.34
6.823333333
1.533466667
1.238332212
Elapsed Time
Objective
No of Branches
0.437
0.39
0.297
0.515
0.358
0.374
0.343
0.343
0.296
0.328
0.3681
0.00400649
0.06329684
6.71
6.27
2.99
8.01
4.73
7.32
6.78
7.48
4.95
5.76
6.1
2.08914
1.445385762
12
7
3
79
0
43
0
7
7
10
16.8
566.76
23.80672174
Iteration
No
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
No of
Branches
1
25
0
29
19
10
372
21
0
0
47.7
11800.01
108.6278509
No of MIP Iterations
932
1181
832
1025
1189
880
3514
1320
530
1118
1252.1
613823.09
783.4686273
No of MIP
Iterations
896
846
986
1635
738
1304
1154
1234
943
1120
1085.6
62186.04
249.3712894
17-Center
S.No
1
2
3
4
5
6
7
8
9
10
18-Center
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 36
Iteration
No
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
Elapsed Time
Objective
0.483
0.375
0.265
0.296
0.234
0.265
0.312
0.327
0.343
0.296
0.3196
0.00448324
0.066957001
6.66
5.53
7.4
8.37
5.36
6.69
6.11
3.7
4.35
8.37
6.254
2.211544
1.487126087
Iteration
No
1
2
3
4
5
6
7
8
9
10
Average
Variance
SD
Elapsed Time
Objective
0.39
0.265
0.312
0.39
0.312
0.281
0.297
0.312
0.312
0.328
0.3199
0.00151949
0.038980636
4.67
5.85
8.71
8.02
8.32
4.98
7.93
7.32
6.47
4.86
6.713
2.154281
1.467746913
No of
Branches
6
3
0
0
0
1
13
1
20
7
5.1
40.49
6.363175308
No of MIP
Iterations
1209
983
758
899
771
827
1239
855
792
989
932.2
27066.76
164.519786
No of
Branches
0
2
0
0
1
0
1
55
0
3
6.2
265.56
16.29601178
No of MIP
Iterations
1244
795
890
1093
661
754
1117
1323
1143
1062
1008.2
43938.56
209.6152666
20-Center
S.No
1
2
3
4
5
6
7
8
9
10
Iteration
Elapsed
Objective
No
Time
1
0.374
11.57
2
0.234
7.2
3
0.234
5.14
4
0.249
4.41
5
0.265
6.63
6
0.312
7.31
7
0.234
5.5
8
0.265
5.49
9
0.296
5.06
10
0.265
5.93
Average
0.2728
6.424
Variance 0.00175816
3.744644
SD
0.041930419 1.935108266
No of
Branches
0
0
0
0
0
0
0
0
1
0
0.1
0.09
0.3
No of MIP
Iterations
931
606
1024
1073
575
758
701
1107
1148
909
883.2
39970.36
199.9258863
21-Center
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 37
Iteration
Elapsed
Objective
No of
No of MIP
No
Time
Branches
Iterations
1
0.359
4.72
0
849
2
0.265
6.3
3
1089
3
0.281
4.6
0
1140
4
0.234
6.42
0
813
5
0.297
6.73
0
968
6
0.297
6.56
23
957
7
0.265
6.03
2
1082
8
0.265
5.31
0
883
9
0.28
6.36
0
1015
10
0.25
8.64
0
1100
Average
0.2793
6.167
2.8
989.6
Variance 0.00105061
1.200861
46.36
11772.04
SD
0.032413115 1.095838035 6.808817812 108.4990323
22-Center
S.No
1
2
3
4
5
6
7
8
9
10
Iteration
Elapsed
Objective
No of
No of MIP
No
Time
Branches
Iterations
1
0.312
4.61
0
931
2
0.28
8.36
0
1132
3
0.312
6.84
28
1138
4
0.281
13.415
0
1144
5
0.297
7.5
0
1055
6
0.28
6.5
0
1016
7
0.297
8.38
8
1220
8
0.297
4.5
0
1212
9
0.296
7.28
6
1094
10
0.28
8.83
0
612
Average
0.2932
7.6215
4.2
1055.4
Variance 0.00014296 5.69586
70.76
28727.84
SD
0.011956588 2.3866 8.411896338 169.4928907
23-Center
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 38
Iteration
Elapsed
Objective
No of
No of MIP
No
Time
Branches
Iterations
1
0.359
9.62
0
1059
2
0.234
7.54
0
924
3
0.218
7.64
0
1122
4
0.234
8.52
0
934
5
0.265
5.42
0
1037
6
0.265
4.99
12
1145
7
0.281
7.9
1
862
8
0.234
6.37
3
865
9
0.296
5.72
0
1027
10
0.265
5.64
0
1071
Average
0.2651
6.936
1.6
1004.6
Variance 0.00151449
2.106644
12.84
9371.84
SD
0.038916449 1.451428262 3.583294573 96.80826411
P-center
1
2
3
4
5
15
16
17
18
20
21
22
23
Objective
Function Value
25.9
19.5
15.1
14.2
12.3
6.4
5.9
5.1
3.5
3.4
2.7
1.1
0.8
Location of Open Warehouses
18
14,19
6,16,19
6,16,20,23
6,12,13,21,23
1,2,4,5,7,8,10,13,15,17,19,20,21,22,23,24
1,2,4,6,8,10,11,13,14,15,19,20,21,22,23,24
2,3,4,5,8,9,10,12,13,15,17,19,20,21,22,23,24
1,2,4,5,7,8,9,10,12,13,15,17,19,20,21,22,23,24
1,2,4,5,6,7,8,9,10,12,13,15,16,17,19,20,21,22,23,24
1,2,4,5,6,7,8,9,10,11,13,15,16,17,18,19,20,21,22,23,24
1,2,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,24
1,2,3,4,5,6,7,8,9,10,12,13,14,15,16,17,18,19,20,21,22,23,24
D.4 Graphs comparing trend between Practical and Random Cases
P a g e | 39
P a g e | 40
E. Appendix E (Capacitated P-Median)
E.1 Ampl Code – Practical Case [1]
ampl: reset;
ampl: model capmed.mod;
ampl: data capmed.dat;
ampl: solve;
param dis{i in 1..24, j in 1..24};
param demand{i in 1..24};
param p = 1;
param c = 40000;
var x {i in 1..24, j in 1..24} >=0, <=1;
var y {j in 1..24} binary;
minimize Total_cost: sum {i in 1..24, j in 1..24} dis[i,j] * x[i,j];
subject to demandconstraint {i in 1..24}: sum {j in 1..24} x[i,j] = 1;
subject to prevent {j in 1..24}: sum {i in 1..24} demand[i] * x[i,j] <= c * y[j];
subject to facilities: sum {j in 1..24} y[j] <= p;
ampl: display y;
ampl: display x;
E.2 Ampl Code – Random Case
ampl: reset;
ampl: model capmed.mod;
ampl: data capmed.dat;
ampl: solve;
param dis{i in 1..24, j in 1..24} := Uniform (0, 46);
param demand{i in 1..24} := Uniform (900, 2000);
param p = 1;
var x {i in 1..24, j in 1..24} >=0, <=1;
P a g e | 41
var y {j in 1..24} binary;
minimize Total_cost: sum {i in 1..24, j in 1..24} dis[i,j] * x[i,j];
subject to demandconstraint {i in 1..24}: sum {j in 1..24} x[i,j] = 1;
subject to prevent {j in 1..24}: sum {i in 1..24} demand[i] * x[i,j] <= 2000 * y[j];
subject to facilities: sum {j in 1..24} y[j] <= p;
ampl: display y;
ampl: display x;
Refer: Attachment Capacitated P-Median.zip for complete solutions (screen-shots)
E.3 Tables comparing results for Practical and Random Cases for Capacitated PMedian
Practical
pObjective
median
value
1
2
3
4
5
15
16
17
18
20
21
22
23
P a g e | 42
326.3
252.7
184.08
165.3
143.84
48.5
40.14
33.52
26.84
17.33
13.09
9.65
6.61
Random
Elapsed time
0.266
0.265
0.266
0.281
0.234
0.265
0.25
0.265
0.25
0.25
0.25
0.25
0.234
0.372
0.403
0.351
0.306
0.309
0.31
0.304
0.32
0.275
0.264
0.28
0.288
0.292
Practical
Random
Branch & Bound
0
3
0
7
0
135
160
147
78
190
151
55
19
31.3
28.8
18.5
3.2
8.4
2.3
3.9
8.5
3.3
5.2
2.4
0.7
0.9
Practical
Random
MIP Iterations
89
344
228
249
155
520
546
376
263
625
491
219
72
796.4
786.9
602.1
212.6
293.8
82.5
74.5
100.9
72.1
73.5
62.1
63.1
50.5
Practical Case
pmedian
Objective value
Elapsed
time
Branch & Bound
nodes
MIP
Iterations
1
2
3
4
5
15
16
17
18
20
21
22
23
326.3
252.7
184.08
165.3
143.84
48.5
40.14
33.52
26.84
17.33
13.09
9.65
6.61
0.266
0.265
0.266
0.281
0.234
0.265
0.25
0.265
0.25
0.25
0.25
0.25
0.234
0
3
0
7
0
135
160
147
78
190
151
55
19
89
344
228
249
155
520
546
376
263
625
491
219
72
Random Case
1-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 43
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
Elapsed
Objective
No of
Time
Branches
0.453
248.21
28
0.437
237.91
66
0.566
241.44
51
0.176
Infeasible
0
0.212
Infeasible
0
0.345
244.51
24
0.38
0.38
0
0.403
249.01
96
0.433
248.89
45
0.321
220.82
3
0.3726
211.39625
31.3
0.01205104 6436.827348
977.01
0.109777229 80.22984076 31.25715918
No of MIP
Iterations
853
1682
890
48
52
705
52
2321
936
425
796.4
495480.24
703.9035729
2-Median
S.No
Iteration No
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
Average
Variance
S.D
3-Median
S.No
Iteration
No
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
Average
Variance
S.D
P a g e | 44
Elapsed Time
0.555
0.58
0.357
0.301
0.48
0.162
0.53
0.164
0.441
0.461
0.4031
0.02090209
0.144575551
Objective
233.26
221.93
223.18
238.28
221.86
Infeasible
232.06
Infeasible
230.02
241.4
230.24875
48.84898594
6.989204957
Elapsed
Objective
Time
0.548
175.94
0.269
181.44
0.433
142.06
0.35
157.22
0.273
Infeasible
0.336
137.68
0.381
126.127
0.185
Infeasible
0.356
165.74
0.387
197.62
0.3518
160.478375
0.00883376 516.9091235
0.093988084 22.73563554
No of Branches
4
7
50
20
60
0
15
0
12
120
28.8
1303.96
36.11038632
No of MIP Iterations
489
562
1124
690
1394
50
637
50
662
2211
786.9
378309.49
615.0686872
No of Branches
No of MIP Iterations
42
15
0
14
0
19
1
0
24
70
18.5
460.05
21.44877619
1117
668
331
707
50
629
417
49
750
1303
602.1
151281.89
388.9497268
4-Median
S.No
1
2
3
4
5
6
7
8
9
10
Iteration
No
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
Elapsed Time
Objective
No of Branches
No of MIP Iterations
0.588
0.262
0.32
0.246
0.314
0.29
0.195
0.235
0.383
0.233
0.3066
0.01141924
0.106860844
116.27
Infeasible
120.15
90.72
138.73
98
Infeasible
Infeasible
108.91
132.77
115.0785714
260.3692122
16.13596022
7
0
21
0
1
1
0
0
1
1
3.2
39.16
6.257795139
356
51
21
220
275
312
52
49
374
416
212.6
21719.64
147.3758461
5-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 45
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
Elapsed Time
0.511
0.216
0.416
0.257
0.272
0.23
0.269
0.37
0.286
0.271
0.3098
0.00785036
0.088602257
Objective
103.78
105.47
90.53
74.76
Infeasible
Infeasible
82.53
89.34
Infeasible
95.41
91.68857143
103.8990694
10.1930893
No of Branches
11
3
25
26
0
0
0
1
0
18
8.4
105.04
10.24890238
No of MIP Iterations
368
247
426
646
53
51
208
430
50
459
293.8
37879.56
194.6267196
15-Median
S.No
Iteration No
Elapsed Time
Objective
No of Branches
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
0.512
0.213
0.365
0.273
0.319
0.227
0.361
0.173
0.337
0.328
0.3108
0.00839936
0.091648022
46.62
48.72
39.79
65.9
44.86
43.73
54.72
40.26
46.12
79.07
50.979
140.353029
11.8470683
7
9
1
3
2
1
0
4
1
2
0
2.3
6.41
2.53179778
S.No
Iteration No
Elapsed Time
Objective
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
0.481
0.267
0.369
0.234
0.303
0.305
0.338
0.227
0.281
0.241
0.3046
0.00534044
0.073078314
34.51
42.1
49.95
54.6
60.36
45.32
Infeasible
68.1
Infeasible
38.78
49.215
112.66415
10.614337
No of MIP
Iterations
106
84
76
102
76
88
96
58
67
72
82.5
218.25
14.7732867
16-Median
P a g e | 46
No of
Branches
10
11
3
8
0
1
0
2
0
4
3.9
16.29
4.036087214
No of MIP Iterations
85
90
75
103
60
53
56
70
60
93
74.5
275.05
16.58463144
17-Median
S.No
Iteration No
Elapsed Time
Objective
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
0.594
0.296
0.253
0.287
0.177
0.367
0.178
0.451
0.303
0.296
0.3202
0.01420776
0.119196309
64.49
52.65
51.27
44.65
52.08
49.21
49.98
53.73
56.42
Infeasible
52.72
26.74584444
5.171638468
No of Branches
2
0
1
5
12
1
9
45
10
0
8.5
165.85
12.87827628
No of MIP
Iterations
73
67
77
82
104
76
116
252
105
57
100.9
2848.89
53.37499415
18-Median
S.No
Iteration No
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
P a g e | 47
Elapsed
Objective
No of Branches No of MIP Iterations
Time
0.371
59.93
1
66
0.22
57.28
1
70
0.168
54.41
5
65
0.315
35.06
7
78
0.227
48.74
0
60
0.502
55.78
4
68
0.351
49.58
7
93
0.204
46.93
5
84
0.216
57.67
2
72
0.176
57.94
1
65
0.275
52.332
3.3
72.1
0.0103222
50.634256
6.21
91.89
0.101598228 7.115775151 2.491987159
9.585927185
20-Median
S.No
Iteration No
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
Elapsed
Time
0.359
0.403
0.238
0.263
0.186
0.31
0.225
0.181
0.296
0.179
0.264
0.0053882
0.07340436
Objective
59.19
63.4
55.26
63.03
49.16
61.65
63.59
58.9
51.48
46.3
57.196
35.984704
5.998725198
No of
Branches
2
3
1
0
3
0
5
36
2
0
5.2
107.76
10.38075142
No of MIP Iterations
64
63
64
68
53
63
82
158
68
52
73.5
855.65
29.25149569
21-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 48
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
Elapsed Time
0.592
0.341
0.172
0.18
0.343
0.159
0.283
0.191
0.287
0.254
0.2802
0.01499936
0.122471874
Objective
57.18
46.62
57.84
55.73
52.32
64.66
57.51
40.54
56.76
60.47
54.963
43.246981
6.576243685
No of Branches
0
0
0
1
9
0
11
3
0
0
2.4
15.44
3.929376541
No of MIP Iterations
52
61
54
62
63
62
94
69
61
43
62.1
160.09
12.6526677
22-Median
S.No
Iteration No
Elapsed Time
Objective
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
0.479
0.205
0.19
0.326
0.326
0.216
0.183
0.3
0.33
0.331
0.2886
0.00758244
0.087077207
77.1
64.29
64.13
57.08
48.67
52.28
54.76
44.65
40.49
65.08
56.853
109.485521
10.46353291
No of
Branches
1
0
2
0
0
2
0
0
0
2
0.7
0.81
0.9
No of MIP Iterations
66
83
82
51
50
67
68
61
42
61
63.1
157.29
12.54153101
23-Median
S.No
1
2
3
4
5
6
7
8
9
10
P a g e | 49
Iteration No
1
2
3
4
5
6
7
8
9
10
Average
Variance
S.D
Elapsed Time
0.482
0.205
0.316
0.259
0.318
0.219
0.18
0.37
0.281
0.292
0.2922
0.00700876
0.083718337
Objective
60.57
65.4
49.94
61.99
65.36
48.44
53.18
50.95
84.33
67.12
60.728
106.277016
10.30907445
No of Branches
0
1
1
0
0
0
0
0
7
0
0.9
4.29
2.071231518
No of MIP Iterations
61
73
61
52
51
35
61
43
7
61
50.5
313.85
17.71581215
E.4 Comparison between Practical and Random Cases
Elapsed Time for Capacitated P-Median
0.45
0.4
Elapsed TIme (s)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1
2
3
4
5
15
16
17
18
20
21
22
23
Number of Open Warehouses
Practical Case
Random Case
Number of Branch and Bounds
Number of Branch and Bounds for Capacitated P-Median
200
180
160
140
120
100
80
60
40
20
0
1
2
3
4
5
15
16
17
18
Number of Open Warehouses
Practical Case
P a g e | 50
Random Case
20
21
22
23
Number of MIP Iterations
Number of MIP Iterations for Capacitated P-Median
900
800
700
600
500
400
300
200
100
0
1
2
3
4
5
15
16
17
18
Number of Open Warehouses
Practical Case
P a g e | 51
Random Case
20
21
22
23
F. Appendix F (Capacitated P- Center)
F.1 Ampl Code- Practical Case [2]
ampl: reset;
ampl: model capcen.mod;
ampl: data capcen.dat;
ampl: solve;
param dis{i in 1..24, j in 1..24};
param demand{i in 1..24};
param p = 1;
param c = 40000;
var x {i in 1..24, j in 1..24}>=0,<=1;
var y {j in 1..24} binary;
var z >= 0;
minimize Total_distance: z;
subject to demconstaint {i in 1..24}: sum {j in 1..24} x[i,j] = 1;
subject to prevent {i in 1..24, j in 1..24}: x[i,j] <= y[j];
subject to facilities: sum {j in 1..24} y[j] <= p;
subject to disvar {i in 1..24}: sum {j in 1..24} dis[i,j] * x[i,j] <= z;
subject to cap {j in 1..24}: sum {i in 1..24} demand[i] * x[i,j] <= c;
ampl: display x;
ampl: display y;
F.3 Ampl Code - Random Case
ampl: reset;
ampl: model capcen.mod;
ampl: data capcen.dat;
ampl: solve;
param dis{i in 1..24, j in 1..24} := Uniform (0, 46);
param demand{i in 1..24} := Uniform (900, 2000);
param q = Uniform (35000, 48000);
param p = 1;
P a g e | 52
var x {i in 1..24, j in 1..24}>=0,<=1;
var y {j in 1..24} binary; var z >= 0;
minimize Total_distance: z;
subject to demconstaint {i in 1..24}: sum {j in 1..24} x[i,j] = 1;
subject to prevent {i in 1..24, j in 1..24}: x[i,j] <= y[j];
subject to facilities: sum {j in 1..24} y[j] <= p;
subject to disvar {i in 1..24}: sum {j in 1..24} dis[i,j] * x[i,j] <= z;
subject to cap {j in 1..24}: sum {i in 1..24} demand[i] * x[i,j] <= q;
ampl: display x;
ampl: display y;
F.3 Tables comparing results for Practical and Random Cases for Capacitated PCenter
Refer: Attachment Capacitated P-Center.zip for complete solutions (screen-shots)
Practical
pObjective
center
value
1
2
3
4
5
15
16
17
18
20
21
22
23
P a g e | 53
25.9
19.5
15.1
14.2
12.3
6.4
6.311
5.268
5.089
4.01
2.7
2.6
1.63
Random
Elapsed time
0.265
0.546
0.826
0.983
1.186
0.624
0.515
0.656
0.359
0.375
0.312
0.265
0.25
0.2272
1.58
0.75
0.82
43.79
289.79
97.87
0.17
51.83
0.21
0.45
0.21
0.19
Practical
Random
Branch &
Bound
nodes
8
104
112
138
150
223
162
224
72
49
0
3
13
Practical
Random
MIP Iterations
23.5
292.5
241.6
253.8
8844.8
113525.4
27360.3
0
26675.5
0
231.6
0
3
892
4717
8416
7746
10888
6200
3567
8217
1163
1503
573
506
700
941.9
23846.9
9766.5
9916.3
930392.1
6320178
1625163
503.4
1190168.8
588.6
59127.1
674
845
Practical Case
p-center
Objective
value
Elapsed
time
Branch & Bound
nodes
MIP
Iterations
1
2
3
4
5
15
16
17
18
20
21
22
23
25.9
19.5
15.1
14.2
12.3
6.4
6.311
5.268
5.089
4.01
2.7
2.6
1.63
0.265
0.546
0.826
0.983
1.186
0.624
0.515
0.656
0.359
0.375
0.312
0.265
0.25
8
104
112
138
150
223
162
224
72
49
0
3
13
892
4717
8416
7746
10888
6200
3567
8217
1163
1503
573
506
700
Random Case
1-Center
Sno
Iterations
Elapsed Time
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
0.305
0.162
0.238
0.255
0.143
0.225
0.24
0.213
0.218
0.273
0.2272
0.00208356
0.045646029
P a g e | 54
Objective
No of
Branches
39.2
33
Infeasible
0
37.38
29
35.73
23
Infeasible
0
41.8
35
38.5
35
33.26
28
35.91
26
35.68
26
37.1825
23.5
6.06736875
152.25
2.463202945 12.33896268
No of
Iterations
1023
994
1000
820
836
983
936
892
941
994
941.9
4567.09
67.58024859
2-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
Elapsed
Time
4.147
0.74
0.688
0.602
0.742
0.742
3.96
0.545
0.77
2.901
1.5837
1.95883541
1.399584013
Objective
Infeasible
24.54
23.87
23.75
24.39
27.92
Infeasible
Infeasible
22.19
Infeasible
24.44333333
2.997722222
1.731393145
No of
No of
Branches
Iterations
543
72808
182
8479
238
7972
227
6257
221
7040
322
8231
498
65582
0
3566
161
7115
533
51419
292.5
23846.9
29112.25
691441756.9
170.6231227 26295.28013
3-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
P a g e | 55
Elapsed
Objective
No of
Time
Branches
0.962
13.62
129
0.978
18.74
396
0.852
18.38
421
0.387
Infeasible
0
0.72
16.24
215
0.782
16.05
295
0.613
16.18
157
0.79
16.43
197
0.337
Infeasible
0
1.11
19.17
606
0.7531
16.85125
241.6
0.05614669 2.920710937
33093.64
0.236952928 1.709008759 181.9165743
No of
Iterations
10550
15162
12257
2018
8749
9422
7984
11490
1388
18645
9766.5
25229590.45
5022.906574
4-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
Elapsed
Objective
No of
Time
Branches
1.113
12.97
377
1.085
13.01
579
0.325
7.31
34
0.583
10.78
199
1.232
11.47
550
0.32
Infeasible
0
2.195
14.13
586
0.27
8.26
12
0.822
12.17
201
0.295
Infeasible
0
0.824
11.2625
253.8
0.3345606
4.99106875
56042.36
0.578412137 2.234069997 236.7326762
No of
Iterations
15720
19742
2330
8938
2531
1947
34059
932
11430
1534
9916.3
104293058.2
10212.39728
Elapsed
Objective
Time
0.303
18
0.468
9.19
1.888
10.69
1.462
10.21
0.487
9.03
0.88
8.91
428.749
Infeasible
0.753
11.69
2.074
10.86
0.873
11.4
43.7937
11.10888889
16465.94784 6.866032099
128.3197095 2.620311451
No of
Iterations
366
3299
35191
23767
6165
11073
9168965
10704
32200
12191
930392.1
7.54169E+12
2746213.748
5-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
P a g e | 56
No of
Branches
0
88
894
483
112
158
85584
385
488
256
8844.8
654384586.8
25580.94187
15-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
Elapsed
Objective
Time
2895.99
Infeasible
0.212
4.87
0.192
7
0.31
4.63
0.228
4.77
0.175
5.14
0.225
6.24
0.17
6.39
0.192
5.78
0.245
3.67
289.7939
5.387777778
754695.3472 0.968883951
868.7320342 0.984319029
No of
Branches
1135232
0
0
19
0
0
0
0
0
3
113525.4
1.15987E+11
340568.8667
No of
Iterations
63195942
574
511
1474
660
468
459
454
539
701
6320178.2
3.59428E+14
18958587.94
Elapsed
Objective
Time
0.218
4.54
0.156
4.94
0.14
4.64
0.156
5.92
0.125
7.66
0.156
8.99
0.14
5.89
0.141
6.17
0.266
6.25
0.219
5.75
0.1717
6.075
0.00191461
1.691665
0.043756257 1.300640227
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of
Iterations
509
555
369
568
420
451
372
481
748
561
503.4
11554.64
107.4925114
17-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
P a g e | 57
18-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
Elapsed
Objective
Time
516.696
Infeasible
0.218
6.70
0.172
9.78
0.172
7.84
0.171
9.21
0.219
5.51
0.187
5.73
0.156
4.50
0.171
6.39
0.156
6.46
51.8318
6.902222222
24010.96983 2.674306173
154.9547348 1.6353306
No of
Branches
266755
0
0
0
0
0
0
0
0
0
26675.5
6404240702
80026.5
No of
Iterations
11896508
695
609
627
617
747
570
480
460
375
1190168.8
1.27362E+13
3568779.735
Elapsed
Objective
Time
0.203
7.42
0.156
7.34
0.141
5.03
0.171
4.75
0.156
5.46
0.14
4.55
0.39
4.79
0.171
6.97
0.141
5.03
0.484
4.91
0.2153
5.625
0.01305401
1.183725
0.114254147 1.087991268
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of
Iterations
360
759
557
763
499
687
621
649
426
565
588.6
16211.24
127.3233678
20-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
P a g e | 58
21-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
Elapsed
Objective
Time
0.234
6.18
0.156
4.84
3.089
Infeasible
0.125
4.46
0.14
5.39
0.14
7.99
0.156
4.48
0.163
5.47
0.22
7.85
0.15
10.01
0.4573
6.296666667
0.77064101 3.264844444
0.877861612 1.806888055
No of
Branches
0
0
2316
0
0
0
0
0
0
0
231.6
482747.04
694.8
No of
Iterations
649
591
585262
575
515
331
601
573
1549
625
59127.1
30757642455
175378.569
Elapsed
Objective
Time
0.234
6.79
0.203
6.01
0.188
4.63
0.188
3.19
0.188
5.30
0.156
6.23
0.156
8.07
0.516
5.73
0.141
6.52
0.172
3.49
0.2142
5.596
0.01075736
2.023624
0.103717694 1.422541388
No of
Branches
0
0
0
0
0
0
0
0
0
0
0
0
0
No of
Iterations
659
1036
483
809
862
645
496
679
447
624
674
30789.8
175.4702254
22- Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
P a g e | 59
23-Center
Sno
Iterations
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Avg
Var
SD
P a g e | 60
Elapsed
Objective
Time
0.265
4.87
0.328
8.36
0.14
5.82
0.156
7.27
0.156
5.63
0.141
7.17
0.156
5.66
0.265
Infeasible
0.156
6.86
0.156
5.43
0.1919
6.341111111
0.00409389 1.125609877
0.063983514 1.060947631
No of
Branches
0
0
0
0
0
0
0
30
0
0
3
81
9
No of
Iterations
623
2188
553
729
626
494
581
1454
593
609
845
268259.2
517.937448
F.4 Graph comparing Random and Practical Cases
Elapsed Time for Capacitated P-Center
350
Elapsed TIme (s)
300
250
200
150
100
50
0
1
2
3
4
5
15
16
17
18
20
21
22
23
22
23
Number of Open Warehouses
Practical Case
Random Case
Number of Branch and Bounds
Number of Branch and Bounds for Capacitated P-Center
120000
100000
80000
60000
40000
20000
0
1
2
3
4
5
15
16
17
18
20
Number of Open Warehouses
Practical Case
P a g e | 61
Random Case
21
Number of MIP Iterations
Number of MIP Iterations for Capacitated P-Center
7000000
6000000
5000000
4000000
3000000
2000000
1000000
0
1
2
3
4
5
15
16
17
Number of Open Warehouses
Practical Case
P a g e | 62
Random Case
18
20
21
22
23
G. References
[1] Alberto Ceselli, Giovanni Righini (2005) A Branch-and-Price Algorithm for the Capacitated pMedian Problem, Networks-An International Journal. Volume 45, Issue 3, pp 125–142
[2] Aykut Ozsoy F. (2003) Algorithms for some discrete location problems, M.Sc. Thesis Report,
Bilkent University, Turkey
[3] Maria Albareda-Sambola, Juan A. Díaz, Elena Fernández., (2010) Lagrangean duals and exact
solution to the capacitated p-center problem (2010), European Journal of Operational Research.
Volume 201, Issue 1, pp 71-81
[3] United States Census Bureau., Computer and Internet Trends in America March 02, 2012 Web. 02
April 2014.
[4] Computer and Internet Use in the United States May 2013 Web 02 April 2014.
[5] U.S Census Bureau Electronic Shopping and Mail-Order Houses—Total and E-Commerce Sales by
Merchandise Line: 2008 and 2009, 2012, Web 03 April 2014.
[6] Robert Fourer, David M. Gay, Brain W. Kernighan., (2003) AMPL Reference Manual
[7] U.S. Census Bureau: State and County QuickFacts March 27 2014 Web 04 April 2014.
P a g e | 63