Minimizing Product Cost Crashing using Graph Network System
NESHAR BASUMATARY, ABDUR ROHMAN, SURAJIT KR. NATH*
1Department of Mathematical Sciences,
Bodoland University, Kokrajhar, 783370,
INDIA
*Corresponding Author
Abstract: - Graph is mathematical representation which can use in any real life situation where a relationship is
present between the objects/elements. In this article, we try to find minimizing product cost crashing using
graph network system. The minimizing product cost crashing is a system that used by businesses to reduce the
expenses related with the manufacturing unit and product processes. Here we, Analyzed the various cost
reduction strategies and their impact on product development cycles and identified efficient method for
lowering production expenses while maintaining quality standards. The outcome of this article contributes to
enhancing cost management practices and improving overall profitability in the industry.
Key-Words: - Graph theory, Graph Network, Optimization, Crashing, Critical Path, Slope.
Received: August 24, 2023. Revised: November 17, 2023. Accepted: December 13, 2023. Published: December 31, 2023.
1 Introduction
Minimizing product cost is the system that identifies
and reduces expenses related to a running business.
The main focus of minimizing product cost is to
lower of the overall costs of a business without
compromising quality and negative impact in the
area of the company. A graph Network system, [1],
is involved in optimizing the transformation of
goods from one location to another while taking into
account factors such as the cost of transformation,
the distance traveled, and the availability of
transformation routes.
Crashing, [2], on the other hand, is a technique
when it appears, an additional costs related to
crashing are viewed against a minimum possible
benefit to complete a project within a short period. It
helps to speed up the timeline of a project through
additional resources. Crashing is a one way process
to compress the rest of the path and to make up for
delays in the beginning.
In some situations, [3], minimizing product cost
and crashing may be related. For example, if a
company is scheduled on a project it may need to
allocate its resources to certain activities to speed up
its completion and meet the deadline however this
may result in additional costs that need to be
factored into the overall product cost.
In mathematics, graph theory, [4], is a branch
of mathematics that deals with the study of graphs,
which are mathematical structures used to model
pairwise relationships between any two objects. A
graph is determined as a mathematical structure that
represents a particular function by connecting a set
of points. It involves analyzing properties and
characteristics of graphs such as connectivity, paths,
cycles, and graph coloring, to solve in various fields
including computer science, operations research,
social networks, etc. In mathematics, all these
networks are called graphs. By using the graph
theory we can find a critical path. The concept of
minimizing product cost crashing is various
strategies that can be employed to achieve this
objective.
2 Preliminary
Definitions, [4], [5], [6], [7]
Graph: Graph theory is the study of points and
lines. It is a pictorial representation that represents
the Mathematical truth. Graph theory is the study of
the relationship between the vertices(nodes) and
edges(lines). Formally, a graph is denoted as a pair
G = (V, E) where V represents the finite set of
vertices and E represents the finite set of edges.
Subgraph: A graph of G is a subgraph having all of
its points and lines in G.
Clearly from Figure 1 and Figure 2, we see that
Figure 1 is a example of Graph and Figure 2 is a
example of subgraph.
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Neshar Basumatary, Abdur Rohman, Surajit Kr. Nath
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Fig. 1: Graph
Fig. 2: Subgraph
Degree of vertex: The degree of a vertex in an
undirected graph is the number of links incident
with it, with the exception that a loop at a vertex
contributes twice to the degree of that vertex. It is
denoted by deg(v), where v is the vertex of the
graph.
A vertex whose degree is zero is called an isolated
vertex.
Clearly, we see that Figure 3 is a example of degree
of vertices.
Fig. 3: Example of degree of vertices
In this figure the degree of the vertices are
deg(v1) = 2, deg(v2) = 3, deg(v3) = 1, deg(v4) = 2.
Directed and undirected graph: If in a graph G(V,
E) each edge of a graph G has a direction, then G is
called a directed graph.
If each edge of graph G has no direction, then graph
G is called an undirected graph.
Example: Clearly, Figure 4 is a example of
undirected graph and Figure 5 is a example of
directed graph.
Fig. 4: Undirected graph
Fig. 5: Directed graph
Weighted graph: A weighted graph is a graph in
which each line is given a numerical weight. A
weighted graph is, therefore, a special type of
labeled graph in which the labels are numbers
usually taken to be positive.
Example: Here Figure 6 is a example of weighted
graph.
Fig. 6: Example of Weighted Graph
Walk: A walk of graph G is an alternating sequence
of points and lines beginning and ending with
points, in which each line is independent with the
two points immediately preceding and following it.
Trail: A trail is a walk with no repeated edge.
Path: A path in a graph is an infinite or finite
sequence of edges that connect a sequence of
vertices which by most definitions are all distinct
from one another.
Cycle: A trail of a non-zero length from a vertex V
to itself in a graph is called a cycle except for the
beginning and the ending vertices that are both
equal to V there are no repeated vertices in the walk.
In Figure 7, we clearly mention walk, trail, path,
cycle.
Fig. 7: Example of Cycle
v1 v2 v4 v3 v2 v1 v4 (walk)
v1 v2 v4 v3 v6 v4 v1 (walk) (trail)
v4 v3 v2 v4 v6 v7 (walk) (trail)
v1 v2 v3 v6 v5 v4 (walk) (trail) (path)
v1 v2 v3 v6 v5 v4 v1 (walk) (trail) (cycle)
Critical path: The critical path is the sequence of
tasks that determine the project’s shortest duration
1
v
2
v
3
v
4
v
5
v
6
v
7
v
12
8
13
1
v
2
v
3
v
4
v
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and must be completed on time for the project to be
completed successfully.
Transportation: Transportation is crucial for
connecting people and goods or services from one
place to another. It can include various modes like
cars, trains, planes, buses, bicycles, and walking.
Slope: In mathematics, the slope refers to the
measure of the steepness of a line it represents the
ratio of vertical change (rise) to horizontal change
(run)between two points on the line.
3 Minimizing Product Cost Crashing
using Graph Network
Minimizing product cost crashing means reducing
the project completion time by adding extra
resources to it. The project may crash by reducing
the normal completion time of critical activities is
called the crashing of activities. This can be
obtained by increasing resources to perform.
Many companies suffer from challenges and
difficulties due to complex projects. Because of
dependency on popular ways to plan schedule and
control the project development, control of time,
cost and good performance also need to be
completed on time with good quality within the
allocated budget.
For this research paper, we have given an
example below in Table 1 about finding the product
cost and crashing of a network with two critical path
in Figure 8, Figure 9, Figure 10 and Figure 11 and
we find Crash limit and Slope in Table 2, Table 3
and Table 4.
Table 1. Product Cost
Let the indirect cost per week is Rs.200
Fig. 8: Graph of Critical Path
Critical Path:
Normal project completion time = 18 weeks
Critical path:
and
Crash limit and Slope:
Table 2. Crash limit and Slope
(i). Iteration:
Fig. 9: Graph of Critical Path
Activity
Normal
time(week)
Crash
time(weeks)
Normal
cost(Rs)
Crash
cost(Rs)
Cost
Slope
1-2
7
4
700
850
50
1-3
5
3
500
700
100
1-4
8
5
600
1200
200
2-5
9
7
800
1250
225
3-5
5
3
700
1000
150
3-6
6
5
1100
1300
200
4-6
7
5
1200
1400
125
5-7
2
1
400
500
100
6-7
3
2
500
850
350
Total
= 6500
Critical path
Critical
Activity
Crash
Limit
Cost
Slope
3
50
2
225
1
100
3
200
2
125
1
350
1
2
3
4
5
6
7
6
9
5
2
8
6
5
6
3
1
2
3
4
5
6
7
7
9
5
2
8
7
5
6
3
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Critical Path:
Project Completion time = 17 weeks
Critical path:
and
Total Cost = pre. Total cost + direct cost(cost slope)
– indirect cost
=10,100 +[50+125]- 200
=10,075
Crash limit and Slope:
Table 3. Crash limit and slope
(ii). Iteration:
Fig. 10: Graph of Critical path
Critical Path:
Project Completion time = 16 weeks
Critical path:
and
Total Cost = pre. Total cost + direct cost(cost slope)
– indirect cost
=10,075 +[50+125]- 200
=10,050
Crash limit and Slope:
Table 4. Crash limit and slope
Critical path
Critical
Activity
Crash
Limit
Cost Slope
1
50
2
225
1
100
3
200
0
125
1
350
(iii). Iteration:
Fig. 11: Graph of critical path
Critical Path:
Project Completion time = 16 weeks
Critical path:
and
Total Cost = pre. Total cost + direct cost(cost slope)
– indirect cost
=10,050+[50+125]- 200
=10,100
Final Result:
Since the total cost of this iteration (iii) is more than
that of the previous iteration, stop the procedure and
treat the solution of the previous iteration (ii) as the
best solution for implementation. The final crashed
project completion time is 16 weeks. Corresponding
critical paths are-
and
4 Conclusion
Minimizing product cost involves optimizing the
cost of producing and delivering goods or services
to customers while crashing is a technique used to
reduce the duration of critical activities in a project
Critical path
Critical
Activity
Crash
Limit
Cost Slope
2
50
2
225
1
100
3
200
1
125
1
350
1
2
3
4
5
6
7
4
9
5
2
7
5
5
6
3
1
2
3
4
5
6
7
5
9
5
2
8
5
5
6
3
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Volume 22, 2023
schedule to meet a deadline minimizing product cost
could involve optimizing the transportation of goods
from one location to another. While taking into
account factors such as the cost of transportation
distance traveled and available transportation routes.
If a company is behind schedule on a project it
may need to allocate more resources to certain
activities to speed up their completion and meet the
deadline which could result in additional costs that
need to be factored into the overall product costs.
Companies need to find a balance between
minimizing product costs and crashing to meet
project deadlines. Careful planning analysis and
decision-making are essential to achieve the desired
result and minimize cost without compromising
project timelines or quality
References:
[1] Petrina C. Feng, “Impact of Graphs and
Network in Minimizing Project and Product
cost”, The International Journal of Business
Management and Technology, vol.2, no. 1, pp.
01-6, 2018.
[2] M.N. Islam, B. Rana, S. Rafique, T. Aziza,
“Crashing Project Time with Least Cost: A
Linear Programming Approach”, Journal of
Business Research, vol. 6, 2004.
[3] C.L. Karmaker and P. Halder, “Scheduling
Project Crashing Time Using Linear
Programming Approach: Case Study”,
International Journal of Research in
Industrial Engineering, 2017.
[4] D.B.West, “Introduction to Graph Theory”,
Pearson, 2000.
[5] R. Likaj, A. Shala, M. Mehmetaj, P. Hyseni,
X. Bajrami, “Application of Graph theory to
find the optimal paths for the transportation
problem,” IFAC Proceedings Vol., vol.46,
Issue 8, pp. 235-240, 2013.
[6] R. Rehab and R.I. Carr, “time-cost trade-off
among related activities”,.J.construct.Eng.
manage.,1977.
[7] G. Ambrus, P.Hajnal, “The Slope parameter
of Graphs”, Acta Scientiarum
Mathematicarum, 2006.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors contributed to the present research, at
all stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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WSEAS TRANSACTIONS on COMPUTERS
DOI: 10.37394/23205.2023.22.40
Neshar Basumatary, Abdur Rohman, Surajit Kr. Nath
E-ISSN: 2224-2872
356
Volume 22, 2023
