Software Process and Different Models Involved In This Process for MI0024 MBA Assignment

“Explain the software process and different models involved in this process.” This is MI0024 SMU MBA question assignment. The question has been taken from Software Engineering for Information System (IS) of SMU MBA MI0024. I have decided to write the assignment question after Branch and Bound Technique to solve an I.P.P. problem.

In the recent years, there has been a significant emphasis on “process maturity.” The Software Engineering Institute (SEI) has developed a comprehensive model predicated on a set of software engineering capabilities that should be present as organizations reach different levels of process maturity. To determine an organization’s current state of process maturity, the SEI uses an assessment that results in a five point grading scheme.

To solve actual problems is an industry setting, a software engineer or a team of engineers must incorporate a development strategy that encompasses the process, methods, and tools layers. All software development can be characterized as a problem solving loop in which four distinct stages are encountered: status quo, problem definition, technical development and solution integration.

This problem solving loop applies to software engineering work at many different levels of resolution. It can be used at the macro level when the entire application is considered, at a mid-level when program components are being engineered and even at the line of code level.

Software Process Models:

The Linear Sequential Model: Sometimes called the classic life cycle or the waterfall model, the linear sequential model suggest a systematic, sequential approach to software development that begins at the system level and progress through analysis, design, coding, testing and support.

The Prototyping Model: Often, a customer defines a set of general objectives for software but does not identify detailed input, processing, or output requirements.

The Incremental Model: The incremental model combines elements of the linear sequential model with the iterative philosophy of prototyping.

The Spiral Model: The spiral model, originally proposed by Boehm [BOE88], is an evolutionary software process model that couples the iterative nature of prototyping with the controlled and systematic aspects of the linear sequential model.

The RAD Model: Rapid application development (RAD) is an incremental software development process model that emphasizes an extremely short development cycle.

The intent of software engineering is to provide a framework for building software with higher quality. Software engineering is a discipline that integrates process, methods and tools for the development of computer software.

Branch and Bound Technique to solve an I.P.P. Problem for MB0032 SMU MBA Assignment

“Describe the Branch and Bound Technique to solve an I.P.P. problem.” This is MB0032 SMU MBA question assignment. It is solved assignment of Operation Research chapter of Sikkim Manipal University. I already have shared North-West Corner rule, Matrix Minimum Method and Integer Programming Problem assignments also of SMU MBA MB0032.

Sometimes a few or all the variables of an IPP are constrained by their upper or lower bounds or by both. The most general technique for the solution of such constrained optimization problems is the branch and bound technique.

To explain how this partitioning helps, let us assume that there were no integer restrictions, and suppose that this then yields an optimal solution to LPP.

Then the solution having the larger value for z is clearly optimum for the given IPP. However, it usually happens that one of these problems has no optimal solution satisfying and thus some more computations are necessary. We now discuss step wise the algorithm that specifies how to apply the partitioning and in a systematic manner to finally arrive at an optimum solution.

We start an initial lower bound for z, say z at the first iteration which I less than or equal to the optimal value z, this lower bound may be taken as the starting Lj for some Xj.

In addition to the lower bound z, we also have a list of LPP’s differing only in the bounds. To start with the master list contains a single LPP consisting of 1, 2, 3, 4 and 5. We now discuss below, the step by step procedure that specifies how the partitioning 6 and 7 can be applied systematically to eventually get an optimum integer – valued solution.

Now, on terming we find that only two feasible integer solution namely 5 and 6 have been recorded. The best of these gives the optimum solution to the given IPP. Hence the optimum integer solution to the given IPP is:

Z0 = 55, X1 = 4, X2 = 3.

North West Corner Rule for Finding the Initial Basic Feasible Solution in the Transportation Problem from MB0032 Assignment

“Describe the North-West Corner rule for finding the initial basic feasible solution in the transportation problem.” The question has been taken from MB0032 MBA assignment of SMU. This question has been taken from Operation Research chapter of Sikkim Manipal University. We already have discussed about MODI method, Matrix Minimum Method and Integer Programming Problem in this series.

North West Corner Rule:

Step 1: The first assignment is made in the cell occupying the upper left hand (North West) corner of the transportation table. The maximum feasible amount is allocated there, that is X11 = min (a1, b1).

So that either the capacity of origin O1 is used up or the requirement at destination D1 is satisfied or both. This value of X11 is entered in the upper left hand corner (Small Square) of cell (1, 1) in the transportation table.

Step 2: If b1 > a1 the capacity of origin O, is exhausted but the requirement at destination D1 is still not satisfied, so that at least one more other variable in the first column will have to take on a positive value. Move down vertically to the second row and make the second allocation of magnitude X21 = min (a2, b1 – x21) in the cell (2, 1). This either exhausts the capacity of origin O2 or satisfies the remaining demand at destination D1.

If a1 > b1 the requirement at destination D1 is satisfied but the capacity of origin O1 is not completely exhausted. Move to the right horizontally to the second column and make the second allocation of magnitude X12 = min (a1 – x11, b2) in the cell (1, 2). This either exhausts the remaining capacity of origin O1 or satisfies the demand at destination D2.

If b1 = a1, the origin capacity of O1 is completely exhausted as well as the requirement at destination is completely satisfied. There is a tie for second allocation; an arbitrary tie breaking choice is made. Make the second allocation of magnitude X12 = min (a1 – a1, b2) = 0 in the cell (1, 2) or X21 = min (a2, b1 – b2) = 0 in the cell (2, 1).

Step 3: Start from the new north west corner of the transportation table satisfying destination requirements and exhausting the origin capacities one at a time, move down towards the lower right corner of the transportation table until all the rim requirements are satisfied.

Transportation Problem and MODI Method of Solving Transportation Problem From MB0032 Assignment

The question is – “What do you understand by the transportation problem? What is the basic assumption behind the transportation problem? Describe the MODI method of solving transportation problem.” It has been taken from MB0032 assignment of SMU MBA. It is the question of Operation Research book of Sikkim Manipal University. I am back with this question after the - Matrix Minimum Method, Integer Programming Problem and Penalty Cost Method or Big-M Method for Solving LPP.

Transportation Problem:

Here we study an important class of linear programs called the transportation model. This model studies the minimization of the cost of transporting a commodity from a number of sources to several destinations. The supply at each source and the demand at each destination are known.

The objective is to develop an integral transportation schedule that meets all demands from the inventory at a minimum total transportation cost.

Basic Assumption behind Transportation Problem:

Let us consider a T.P involving m-origins and n-destinations. Since the sum of origin capacities equals the sum of destination requirements, a feasible solution always exists. Any feasible solution satisfying m+n-1 of the m + n constraints is a redundant one and hence can be deleted. This also means that a feasible solution to a T.P can have at the most only m + n – 1 strictly positive components, otherwise the solution will degenerate.

It is always possible to assign an initial feasible solution to a T. P. in such a manner that the rim requirements are satisfied. This can be achieved either by inspection or by following some simple rules. We begin by imagining that the transportation table is blank i.e. initially all Xij = 0. The simplest procedures for initial allocation discussed in the following section.

MODI Method of Solving Transportation Problem:

The first approximation to (2) is always integral and therefore always a feasible solution. Rather than determining a first approximation by a direct application of the simplex method it is more efficient to work with the table given below called the transportation table. The transportation algorithm is the simplex specialized to the format of table it involves:

a) Finding an integral basic feasible solution
b) Testing the solution for optimality
c) Improving the solution, when it is not optimal
d) Repeating steps (1) and (2) until the optimal solution is obtained

The solution of T.P. is obtained in two stages. In the first stage we find basic feasible solution by any one of the following methods a) North-west corner rale b) Matrix Minima method or least cost method c) Vogel’s approximation method. In the second stage we test the B.Fs for its optimality either by MODI metod or by stepping stone method.

Integer Programming Problem and Gomory’s All-I.P.P. Method for Solving the I.P.P. Problem from MB0032 MBA Assignment

This is the most probable assignment question of Operation Research for MB0032 SMU MBA. The question is “What do you understand by the Integer Programming Problem? Describe the Gomory’s All-I.P.P. method for solving the I.P.P. problem.” You already have taken many solved assignments from SMU MBA MB002 such as - classification of Operations Research models, Matrix Minimum Method and Penalty Cost Method or Big-M Method for Solving LPP.

An integer programming problem can be described as follows:

Determine the value of unknowns X1, X2, …, Xn

So as to optimize z = C1X1 + C2X2 + … CnXn

Subject to the constraints

ai1 X1 + ai2 X2 + … + ain Xn = bi, i = 1,2, …, m

and xj ≥ 0 j = 1, 2, …, n

where xj being an integral value for j = 1, 2, …, k ≤ n.

If all the variables are constrained to take only integral value i.e. k = n, it is called an all (or pure) integer programming problem. In case only some of the variables are restricted to take integral value of rest (n – k) variables are free to take any one negative values, then the problem is known as mixed integer programming problem.

Gomory’s All – IPP Method:

An optimum solution to an I. P. P. is first obtained by using simplex method ignoring the restriction of integral values. In the optimum solution if all the variables have integer values, the current solution will be the desired optimum integer solution. Otherwise the given IPP is modified by inserting a new constraint called Gomory’s or secondary constraint which represents necessary condition for integrability and eliminates some non integer solution without losing any integral solution.

After adding the secondary constraint, the problem is then solved by dual simplex method to get an optimum integral solution. If all the values of the variables in this solution are integers, an optimum inter-solution is obtained, otherwise another new constrained is added to the modified L P P and the procedure is repeated.

An optimum integer solution will be reached eventually after introducing enough new constraints to eliminate all the superior non integer solutions. The construction of additional constraints, called secondary or Gomory’s constraints is so very important that it needs special attention.

Matrix Minimum Method and The Initial Basic Feasible Solution In The Transportation Problem for MB0032 Assignment

This question comes from Operation Research assignment of MB0032 for SMU MBA. The question is “Describe the Matrix Minimum method of finding the initial basic feasible solution in the transportation problem.” From the SMU MBA MB002 assignment I already have shared about classification of Operations Research models and Penalty Cost Method or Big-M Method for Solving LPP.

Matrix Minimum Method:

Step 1: Determine the smallest cost in the cost matrix of the transportation table. Let it be Cij, Allocate Xij = min (aj, bj) in the cell (i, j).

Step 2: If Xij = aj cross off the ith row of the transportation table and decrease bj by ai go to step 3.

If xij = bj cross off the ith column of the transportation table and decrease ai by bj go to step 3.

If Xij = ai = bj crosss off either the ith row or the ith column but not both.

Step 3: Repeat steps 1 and 2 for the resulting reduced transportation table until all the rim requirements are satisfied whenever the minimum cost is not unique make an arbitrary choice among the minima.

The Initial Basic Feasible Solution:

Let us consider a T.P involving m-origins and n-destinations. Since the sum of origin capacities equals the sum of destination requirements, a feasible solution always exists. Any feasible solution satisfying m+n -1 of the m+n constraints is a redundant one and hence can be deleted. This also means that a feasible solution to a T.P can have at the most only m + n -1 strictly positive component, otherwise the solution will degenerate.

It is always possible to assign an initial feasible solution to a T.P. in such a manner that the rim requirements are satisfied. This can be achieved either by inspection or by following some simple rules. We begin by imagining that the transportation table is blank i.e. initially all Xij = o. The simplest procedures for initial allocation discussed in the following section.

Penalty Cost Method or Big-M Method for Solving LPP for MBA MB0032 Assignment

Write the question from Operation Research assignment of MB0032 for SMU MBA. The question is “Describe the Penalty Cost method (Big M Method) for solving L.P.P.” From the SMU MBA MB002 assignment you already have gotten classification of Operations Research models.

Consider a L.P.P. when at least one of the constraints is of the type ≥ or =. While expressing in the standard form, add a non negative variable to each of such constraints. These variables are called artificial variables. Their addition causes violation of the corresponding constraints, since they are added to only one side of an equation, the new system is equivalent to the old system of constraints if and only if the artificial variables are zero. To guarantee such assignments in the optimal solution, artificial variables are incorporated into the objective function with large positive coefficients in a minimization program or very large negative coefficients in a maximization program. These coefficients are denoted by ± M.

Whenever artificial variables are part of the initial solution Xo, the last row of simplex table will contain the penalty cost M. The following modifications are made in the simplex method to minimize the error of incorporating the penalty cost in the object function. This method is called Big M-method to penalty cost method.

1. The last row of the simplex table is decomposed into two rows, the first of which involves those terms not containing M, while the second involves those containing M.

2. The step 1 of the simplex method is applied to the last row created in the above modification and followed by steps 2,3 and 4 until this row contains no negative elements. Then step 1 of simplex algorithm is applied to those elements next to the last row that are positioned over zero in the last row.

3. Whenever an artificial variable ceases to be basic, it is removed from the first column of the table as a result of step 4, it is also deleted from the top row of the table as is the entire column under it.

4. The last row is removed from the table whenever it contains all zones.

5. If non-zero artificial variables are present in the final basic set, then the program has no solution. In contrast, zero valued artificial variables in the final solution may exist when one or more of the original constraints equations are redundant.

Classification of OR Models and Operation Research Problem Solving for MB0032 Assignment

The question is related to Operation Research assignment of MB0032 for SMU MBA. The question is “Describe the broad classification of Operations Research models in details. Name the different steps needed in OR approach of problem solving?” However, we already have solved significance of reviewing ROI.

A model is a representation of the reality. It is an idealized representation or abstraction of a real life system. A model is helpful in decision making as it provides a simplified description of complexities and uncertainties of a problem in logical structure.

A) Physical Model: It includes all form of diagrams, graphs and charts. They are designed to deal with specific problems. They bring out significant factors and inter-relationship in pictorial firm so as to facilitate analysis.

B) Mathematical Model: It is known as symbolic models also. It employs a set of mathematical symbols to represent the decision variable of the system.

C) By Nature of Environment: We have 1) Deterministic model in which every thing is defined and the results are certain. Eg: EOQ model 2) Probabilistic Models in which the input and output variables follow a probability distribution. Eg: Games Theory.

D) By the extent of Generality: The two models belonging to this class are 1) General models can be applied in general and does not pertain to one problem only. Eg: Linear programming 2) Specific model is applicable under specific condition only. Eg: Sales response curve or equation as a function of advertising is applicable in the marketing function alone.

Methodology of Operations Research:

The basic dominate characteristic feature of operations research is that it employs mathematical representations or model to analyze problems. This distinctive approach represents an adaptation of the scientific methodology used by the physical sciences. The scientific method translates a real given problem into a mathematical representation which is solved and transformed into the original context. The OR approach problem solving consists of the following steps:

Definition of the Problem

Construction of the Model

Solution of the Model

Validation of the Model

Implementation of the Final Result