Subject Operations Research

Assignment MB0048 – Set 1

Q.1. (a) what is linear programming problem?

(b) A toy company manufactures two types of dolls, a basic version doll- A and a deluxe version doll-B. Each doll of type B takes twice as long to produce as one of type A, and the company would have time to make maximum of 1000 per day. The supply of plastic is sufficient to produce 1000 dolls per day (both A & B combined). The deluxe version requires a fancy dress of which there are only 500 per day available. If the company makes a profit of Rs 3.00 and Rs 5.per doll, respectively on doll A and B, then how many of each doll should be produced per day in order to maximise the total profit. Formulate this problem.

(a): A linear program (LP) is a minimization problem where we are asked to minimize a given linear Function subject to one or more linear inequality constraints. The linear function is also called the objective function.

(b):

Formulation maximize z =3x1 +5x2

Constraints x1+2x2<=2000

X1+x2<=1500

X2<=600

X1,x2>=0

[x1=1000,x2=500,max z= 5500]

What are the advantages of Linear programming techniques?

1. The linear programming technique helps to make the best possible use of available productive resources (such as time, labor, machines etc.) 2. It improves the quality of decisions. The individual who makes use of linear programming methods becomes more objective than subjective. 3. It also helps in providing better tools for adjustment to meet changing conditions. 4. In a production process, bottle necks may occur. For example, in a factory some machines may be in great demand while others may lie idle for some time. A significant advantage of linear programming is highlighting of such bottle necks. 5. Most business problems involve constraints like raw materials availability, market demand etc. which must be taken into consideration. Just we can produce so many units of product does not mean that they can be sold. Linear programming can handle such situation also.

3. Write a note on Monte-Carlo simulation.

The Monte-Carlo method is a simulation technique in which statistical distribution functions are created by using a series of random numbers. This approach has the ability to develop many months or years of data in a matter of few minutes on a digital computer. The method is generally used to solve the problems which cannot be adequately represented by mathematical models, or, where solution of the mode, is not possible by analytical method .The Monte-Carlo simulation procedure can be summarized in the following steps:

Step 1: Define the problem:

a) Identify the objectives of the problem, and

b) Identify the main factors which have the greatest effects on the objectives of the problem Step 2: Construct an appropriate model:

a) Specify the variables and parameters of the model

.b) Formulate the appropriate decision rules, i.e., state the conditions under which the experiment is to be performed. c) Identify the type of distribution that will be used –Models use either theoretical distributions or empirical distributions to state the patterns the occurrence associated with the variables. d) Specify the manner in which time will change.

e) Define the relationship between the variables and parameters.

Step 3: Prepare the model for experimentation:

a) Define the starting conditions for the simulation, and

b) Specify the number of runs of simulation to be made.

Step 4: Using step 1 to 3, experiment with the model:

a) Define a coding system that will correlate the factors defined in step 1 with the random numbers to be generated for the simulation. b) Select a random number generator and create the random numbers to be used in the simulation. c) Associate the generated random numbers with the factors identified in step 1 and coded instep 4 (a).

Step 5: Summarize and examine the results obtained in...