# Mathematics and Business Relationship Research Paper

## Introduction

Mathematics is a field of examine which has been very useful in various sorts of study leading to enterprise decision making. In this paper, the primary focus is on making use of mathematical tools and methods to enterprise decision-making. The paper begins with a discussion of PERT and CPM Techniques. Then the usage of this system in project planning is elaborated. Finally, each thee techniques are mixed as an utility and a enterprise determination is made. Relevant examples are discussed for instance the hyperlink of PERT/CPM to business decision-making. Graphical modeling of a sample project is completed.

## Critical Path Method (CPM)

This methodology is used to establish the shortest path out there. In enterprise, various processes are recognized for doing a selected job and the shortest way to finish the job is identified utilizing CPM.

Critical Path Method (CPM) charts are just like PERT charts and are sometimes known as PERT/CPM. In a CPM chart, the important path is indicated. A crucial path has a set of dependent tasks (each depending on the previous one) which together take the longest time to finish. Although it is not usually done, a CPM chart can outline multiple, equally critical paths. Tasks that fall on the crucial path must be noted indirectly, so that they might be given special consideration. One way is to draw important path duties with a double line as a substitute of a single line.

Tasks that fall on the important path should obtain particular attention from both the project manager and the personnel assigned to them. The important path for any given methodology could shift because the project progresses; this will occur when tasks are accomplished either behind or forward of schedule, causing other duties which can still be on schedule to fall on the new critical path.

## Project Evaluation and Review Technique (PERT)

Program analysis and evaluation approach (PERT) charts depict task, period, and dependency data. Each chart begins with an initiation node from which the first task, or tasks, originates. If a quantity of duties begin on the same time, they’re all started from the node or department, or fork out from the begin line. Each task is represented by a line which states its name or another identifier, its length, the variety of folks assigned to it, and in some circumstances the initials of the personnel assigned. The other finish of the task line is terminated by another node which identifies the start of another task, or the start of any slack time, that’s, ready time between duties.

Each task is connected to its successor duties on this manner forming a community of nodes and connecting traces. The chart is full when all ultimate duties come collectively on the completion node. When slack time exists between the top of 1 task and the start of another, the usual methodology is to draw a damaged or dotted line between the top of the first task and the beginning of the next dependent task.

A PERT chart could have a number of parallel or interconnecting networks of duties. If the scheduled project has milestones, checkpoints, or review factors (all of that are extremely really helpful in any project schedule), the PERT chart will observe that each one tasks up to that time terminate at the evaluate node. It ought to be noted at this point that the project evaluate, approvals, user critiques, and so forth all take time. This time should never be underestimated when drawing up the project plan. It isn’t unusual for a evaluate to take 1 or 2 weeks. Obtaining administration and consumer approvals may take even longer

## Project Planning and Management: CPM/PERT

Consider the starting stage of a large-scale project (for example, the construction of Disneyland in HK). Successful completion of the project requires completion of lots of of different actions by several different groups or corporations (e.g. MTR builds the rail hyperlinks, HK Government builds roads, KMB builds bus routes, SHK builds motels, a roller-coaster company from Japan builds one journey within the park, McDonald’s builds some eating places, and so on.) Ideally, we want to full the project as quickly as potential.

This could probably be accomplished if each exercise was accomplished in parallel. However, it’s usually inconceivable to begin some activities earlier than another exercise has been completed. For example, construction of buildings cannot begin until the forest cover over the Penny Bay space (site of Disneyland) has been cleared. Thus the entire project is damaged right into a set of well-defined activities. Further, the time taken to complete every activity is to be estimated.

The questions that the CPM/PERT analysis can answer embrace the following.

- What is the earliest time to complete the entire project?
- Given an activity, what is the earliest time to do it?
- Which actions are the bottlenecks? These are activities that, if their completion is delayed, the whole project might be delayed.
- What is the most recent time that to begin an activity, and nonetheless not delay the project?

For instance, it’ll take six months to rent and train a set of individuals to put on stuffed animal costumes and dance on the street; Disneyland will open in December 2008. So hiring and training needn’t begin until the summer season of 2005.

## Graphical Modeling of Project Activities

In classical modeling of CPM/PERT as graphs, it is convenient to represent each activity as an edge. The weight of the edge indicates the time required to finish it. Each node represents a milestone, which is the cut-off date when a selected exercise is accomplished and/or some exercise is begun. The edges are directed, and the top of the edge is incident on the node representing the milestone similar to the completion of the activity.

The requirement that some activity can not start before one other exercise has been completed imposes a *precedence relationship*. Therefore, the arrows in our graph will depict the sequence during which the activities may be completed (equally: the paths point out the sequence during which milestones may be achieved).

The graph additionally models the priority relations of the milestones. If activity u must be accomplished before beginning exercise v, we point out this by an arrow from the milestone (complete u) to the milestone (begin v). However, there may be no real exercise between these two occasions – during which case this edge may have a weight = 0; such edges are referred to as *dummy edges* since they solely characterize priority, however not a real activity.

To successfully mannequin a project when it comes to a PERT community, we should first listing out all priority relationships. It is sufficient, for every activity, to establish its immediate predecessors.

### Example

Consider a project to construct a small Hydroelectric Power Plant. The details of the completely different activities, their priority constraints, and the estimated times are shown within the desk beneath.

[newline]

Activity | Description | Immediate Predecessor |
Duration |

a | Ecological survey | – | 6.2 |

b | File environmental impact report; get approval | a | 9.1 |

c | Economic feasibility study | a | 7.3 |

d | Preliminary design and value estimation | c | 4.2 |

e | Project approval and funding commitments | b, d | 10.2 |

f | Call quotation for tools (turbines, generators) | e | 4.3 |

g | Select provider for equipment | f | 3.1 |

h | The ultimate design of the project | e | 6.5 |

i | Select construction contractor | e | 2.7 |

j | Arrange building supplies supply | h, i | 5.2 |

k | Dam building | j | 24.8 |

l | Power station building | j | 18.4 |

m | Power traces erection | g, h | 20.3 |

n | Turbines, mills installation | g, l | 6.8 |

o | Build-up reservoir water level | k | 2.1 |

p | Commission the generators | n, o | 1.2 |

q | Start supplying water | m, p | 1.1 |

Once the project plan information is in a type corresponding to above, it can be expressed in the form of a directed graph utilizing the next technique.

- Construct a node,
*s*, representing the start of the project. - For every exercise that has no quick predecessor, make an edge incident from
*s*. - Other actions are added to extend the graph in accordance with the listing of instant predecessors; when there are two activities such that a few of their instant predecessors are the same, however some are not, we need to introduce a dummy exercise to symbolize the priority constraint.
- In the end, all actions that don’t have any successor are connected to a common node that represents the end of the project.

Construct the graph for our hydroelectric plant project. In the graph, denote the sides by the activity names, in decrease case letters. Each node represents the end of the exercise or activities which are depicted by the perimeters incident to it; likewise, it depicts the point in time to begin any activity whose corresponding edge is incident from this node. The weight of the edge is the length of the activity it represents. The dashed lines are the dummy edges: these edges usually are not given any name and have weight=0. They are used to depict the precedence constraints.

Figure 33a reveals the primary few steps in constructing the PERT graph. We start with node s, and add the edge *a,* for exercise *a,* which can take 6.2 weeks. Since activity *a* is an instantaneous predecessor for each *b* and *c*, so the sides for these two actions are incident from the node depicting the end of exercise *a*, particularly node A.

Figure 33b reveals the state of affairs where a dummy node is required. Activity *c* is an instantaneous predecessor of exercise *c*. However, both activities *b* and *d* are quick predecessors of task *e*. Therefore we want a dummy edge connecting the events depicted by nodes B and D. Since task *e* can only begin when all actions depicted by the perimeters incident to its starting node (namely node D within the figure), this dummy edge ensures that task *b* can additionally be completed earlier than task *e* begins.

*Figure 34. The complete graph is for the example.*Once the PERT graph is totally constructed, we course of it in two phases. In the first stage, referred to as the *Forward pass*, the *earliest attainable begin time* for each exercise is estimated. In the second stage, referred to as the *Backward pass*, the *latest allowed begin time* for every exercise is estimated.

### Forward pass

Begin from the beginning node, and compute the earliest time that each successive exercise can start. Earliest start time at node s = 0. Any edge (activity) that’s incident from this node can be traversed in time equal to its weight. This provides the completion time of the exercise. The most completion time as a end result of each edge incident on a node is the earliest begin time of any exercise that begins from this node. This is the required and sufficient situation for all priority constraints to be glad.

Thus in the graph, the earliest finish time of activity *a* = 6.2; this is written in square brackets subsequent to the node for “end of exercise *a*”, namely node A.

Notice that exercise *e* has two immediate predecessors: b (which cannot be completed earlier than 15.3), and *d* (which can’t be accomplished before 17.7). Thus, task e, which begins from node D, can’t begin earlier than the *maximum of these two*, specifically 17.7; due to this fact *e* can’t end before 17.7 + 10.2 = 27.9. Following this logic, the forward move is completed as shown in Figure 36. From the graph, it is clear that the earliest time after we can full the project is sixty eight.8.

### Backward pass

For the backward cross, start on the end node (in our case, node Q). The objective is to reply the following query: given the *earliest completion time for the general project*, what is the latest time to start a given exercise, without causing any delay to the project?

Certainly, the project cannot be accomplished earlier than sixty eight.8 weeks. The final exercise that we carry out is q, which takes 1.1 weeks and can’t start till all other activities are accomplished. Thus if *q* is acknowledged at any time after sixty eight.eight – 1.1 = 67.7 weeks, then the project will get delayed. Since the start of activity *q* is at node M, we write this limit, sixty seven.7, in the sq. bracket next to node M (see Figure 37a).

The latest allowed start time for each node is labored out utilizing this logic. The only case where we need to pay some attention to this process is depicted in Figure 37b. Node L signifies that activity *l*¸ whose period is eighteen.four weeks, should be accomplished no later than 59.7. Therefore it must start at a time no later than 59.7 – 18.4 = forty one.3 weeks. Using the identical logic for exercise k, the most recent begin time for exercise k is sixty four.four – 24.8 = 39.6. So which of these two figures, forty one.3 and 39.6, is the correct figure for occasion J ? Clearly, we must decide the smaller of the two numbers – since if we denote the latest time at node J by forty one.3, then we might only complete the exercise k at forty one.3 + 24.4 = 65.7, which will delay the project completion time (if we reach node K at any time later than 64.4, it’ll delay the project).

Slack time: The slack time for an occasion is the distinction between its newest and earliest time.

## Limitations of CPM/Pert Methods

While good in theory, CPM/PERT are generally less useful in follow due to two reasons.

- It is usually very tough to estimate the exact time for any activity. Hence most fashionable PERT strategies associate three-time values with every exercise: the best case time, the anticipated time, and the worst-case time. Then a probabilistic model is built, utilizing the worst case, finest case, and anticipated period of every task. However, such estimates are based mostly on a collection of assumptions, lots of which will not be practically correct.
- The second downside is that in developing PERT graphs, we only contemplate the priority relations between duties. However, in real-life initiatives, a quantity of tasks might share some physical resources (e.g. cranes could additionally be utilized in completely different tasks during the project). If two or extra jobs require the same resource, it may not be possible to do them simultaneously. This sort of constraint isn’t captured by PERT. To contemplate such instances, we must use a unique kind of problem model. Such fashions are studied in a special area of IELM, referred to as resource-constrained scheduling. It is typically possible to construct LP formulations of such fashions.

## Reference

R.K. Sharma, Operation Research – Theory and Application, First Edition, Macmillan Publishers.

V.Sunderasan, Resource Management Techniques, K.R. Publications.

Paneerselvam, Operation Research, second version, Prentice Hall of India.