February 2011Master of Computer Application (MCA) – Semester 4

MC0079 – Computer Based Optimization Methods – 4 Credits (Book ID: B0902)

Assignment Set – 1 (60 Marks)

1. Write down the algorithm of the graphical method to solve a L.P.P.Ans:

2. Explain the algorithm for solving a linear programming problem by graphical method.Ans: Linear programming (LP) is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear equations.

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Given a polytope and a real-valued affine function defined on this polytope, a linear programming method will find a point on the polytope where this function has the smallest (or largest) value if such point exists, by searching through the polytope vertices.An Algorithm for solving a linear programming problem by Graphical Method:(This algorithm can be applied only for problems with two variables).Step – I: Formulate the linear programming problem with two variables (if the given problem has more than two variables, then we cannot solve it by graphical method).Step – II: Consider a given inequality. Suppose it is in the forma1x1 + a2x2 <= b (or a1x1 + a2x2 >= b). Then consider the relation a1x1+ a2x2= b. Find two distinct points (k, l), (c, d) that lie on the straight line a1x1+ a2x2= b. This can be found easily: If x1= 0, then x2 = b / a2. If x2=0, then x1 = b / a1. Therefore (k, l) = (0, b / a2) and (c, d) = (b / a1, 0) are two points on the straight line a1x1+a2x2= b.Step – III: Represent these two points (k, l), (c, d) on the graph which denotes X–Y-axis plane. Join these two points and extend this line to get the straight line which represents a1x1+ a2x2= b.Step – IV: a1x1 + a2x2= b divides the whole plane into two half planes, which are a1x1+ a2x2 <= b (one side) and a1x1+ a2x2 >= b (another side). Find the half plane that is related to the given inequality.Step – V: Do step-II to step-IV for all the inequalities given in the problem. The intersection of the half-planes related to all the inequalities and x1 >= 0,x2 >= 0 , is called the feasible region (or feasible solution space). Now find this feasible region.Step – VI: The feasible region is a multisided figure with corner points A, B,C, … (say). Find the co-ordinates for all these corner points. These corner points are called as extreme points.Step – VII: Find the values of the objective function at all these corner/extreme points.Step – VIII: If the problem is a maximization (minimization) problem, then the maximum (minimum) value of z among the values of z at the corner/extreme points of the feasible region is the optimal value of z. If the optimal value exists at the corner/extreme point, say A (u, v), then we say that the solution x1= u and x2= v is an optimal feasible solution.Step – IX: Write the conclusion (that include the optimum value of z, and the co-ordinates of the corner point at which the optimum value of z exists).

3. A manufacturing firm has discontinued production of a certain unprofitable product line. This created considerable excess production capacity. Management is considering to devote this excess capacity to one or more of three products: call them product 1, 2 and 3. The available capacity on the machines which might limit output are given below :

Machine Type Available Time(in machine hours per week)

Milling Machine 250

Lathe 150

Grinder 50

The number of machine-hours required for each unit of the respective product is given below :

Productivity (in Machine hours/Unit)

Machine Type Product 1 Product 2 Product 3

Milling Machine 8 2 3

Lathe 4 3 0

Grinder 2 – 1

The unit profit would be Rs. 20, Rs. 6 and Rs. 8 for products 1, 2 and 3. Find how much of each product the firm should produce in order to maximize profit ?

Ans:

4. Determine optimal solution to the problem given below. Obtain the initial solution by VAM.Ans: 5. Explain Project Management (PERT). Write down the differences between PERT and CPM.

Ans:

CPM was developed by Du Pont and the emphasis was on the trade-off between the cost of the project and its overall completion time (e.g. for certain activities it may be possible to decrease their completion times by spending more money - how does this affect the overall completion time of the project?)

Definition: In CPM activities are shown as a network of precedence relationships using activity-on-node network construction– Single estimate of activity time– Deterministic activity times

USED IN : Production management - for the jobs of repetitive in nature where the activity time estimates can be predicted with considerable certainty due to the existence of past experience.

PERT was developed by the US Navy for the planning and control of the Polaris missile program and the emphasis was on completing the program in the shortest possible time. In addition PERT had the ability to cope with uncertain activity completion times (e.g. for a particular activity the most likely completion time is 4 weeks but it could be anywhere between 3 weeks and 8 weeks).

Basic difference between PERT and CPM: Though there are no essential differences between PERT and CPM as both of them share in common the determination of a critical path and are based on the network representation of activities and their scheduling that determines the most critical activities to be controlled so as to meet the completion date of the project.

PERT:1. Since PERT was developed in connection with an R and D work, therefore it had to cope with the uncertainties which are associated with R and D activities. In PERT, total project duration is regarded as a random variable and therefore associated probabilities are calculated so as to characterize it.2. It is an event-oriented network because in the analysis of network emphasis is given an important stages of completion of task rather than the activities required to be performed to reach to a particular event or task.3. PERT is normally used for projects involving activities of non-repetitive nature in which time estimates are uncertain.4. It helps in pinpointing critical areas in a project so that necessary adjustment can be made to meet the scheduled completion date of the project.

CPM:1. Since CPM was developed in connection with a construction project which consisted of routine tasks whose resources requirement and duration was known with certainty, therefore it is basically deterministic.2. CPM is suitable for establishing a trade-off for optimum balancing between schedule time and cost of the project.

3. CPM is used for projects involving activities of repetitive nature.

Project scheduling by PERT-CPM:It consists of three basic phases: planning, scheduling and controlling.1. Project Planning.2. Scheduling.3. Project Control.