Linear programming Is a specific case of mathematical programming (mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. (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 for which there are only 500 per day available. If the company makes a profit of RSI 3. 0 and RSI 5 per doll, respectively on doll A and B, then how many of each doll should be produced per day in order to maximize the total profit. Formulate this problem. Mans. Let XSL and XX be the number of dolls produced per day of type A and B, respectively. Let the A require t hrs. So that the doll B require at hrs. So the total time to manufacture XSL and XX dolls should not exceed 20th hrs. Therefore, + text s 20th Other constraints are simple.

Then the linear programming problem becomes: Maximize p = ex. ; 5 XA Subject to restrictions, XSL + XX 1500 (Plastic constraint) XX 600 (Dress constraint) And non-negatively restrictions 2. What are the advantages of Linear programming techniques? Mans. Advantages-? 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 ay 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. Mans. Simulation is also called experimentation in the management laboratory. While dealing with business problems, simulation is often referred to as 'Monte Carlo Analysis'. Two American mathematicians, Von Neumann and Ulna, in the late sass found a problem in the field of nuclear physics too complex for analytical solution and too dangerous for actual experimentation. They arrived at an approximate solution by sampling. The method they used had resemblance to the gambler's betting systems on the roulette table, hence the name 'Monte Carlo' has stuck.

Imagine a betting game where the stakes are based on correct prediction of the number of heads, which occur when five coins are tossed. If it were only a question of one coin; most people know that there is an equal likelihood of a head or a tail occurring, that is the probability of a head is h. However, without the application of probability theory, it would be difficult to predict the chances of getting various numbers of heads, when five coins are tossed. Why don't you take five coins and toss them repeatedly.

Note down the outcomes of each toss after every ten tosses, approximate the probabilities of various outcomes. As you know, the values of these probabilities will initially fluctuate, but they would tend to stabilize as the number of tosses are increased. This approach in effect is a method of sampling, but is not very invention. Instead of actually tossing the coins, you can conduct the experiment using random numbers. Random numbers have the property that any number is equally likely to occur, irrespective of the digit that has already occurred.

The deluxe version requires a fancy dress for which there are only 500 per day available. If the company makes a profit of RSI 3. 0 and RSI 5 per doll, respectively on doll A and B, then how many of each doll should be produced per day in order to maximize the total profit. Formulate this problem. Mans. Let XSL and XX be the number of dolls produced per day of type A and B, respectively. Let the A require t hrs. So that the doll B require at hrs. So the total time to manufacture XSL and XX dolls should not exceed 20th hrs. Therefore, + text s 20th Other constraints are simple.

Then the linear programming problem becomes: Maximize p = ex. ; 5 XA Subject to restrictions, XSL + XX 1500 (Plastic constraint) XX 600 (Dress constraint) And non-negatively restrictions 2. What are the advantages of Linear programming techniques? Mans. Advantages-? 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 ay 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. Mans. Simulation is also called experimentation in the management laboratory. While dealing with business problems, simulation is often referred to as 'Monte Carlo Analysis'. Two American mathematicians, Von Neumann and Ulna, in the late sass found a problem in the field of nuclear physics too complex for analytical solution and too dangerous for actual experimentation. They arrived at an approximate solution by sampling. The method they used had resemblance to the gambler's betting systems on the roulette table, hence the name 'Monte Carlo' has stuck.

Imagine a betting game where the stakes are based on correct prediction of the number of heads, which occur when five coins are tossed. If it were only a question of one coin; most people know that there is an equal likelihood of a head or a tail occurring, that is the probability of a head is h. However, without the application of probability theory, it would be difficult to predict the chances of getting various numbers of heads, when five coins are tossed. Why don't you take five coins and toss them repeatedly.

Note down the outcomes of each toss after every ten tosses, approximate the probabilities of various outcomes. As you know, the values of these probabilities will initially fluctuate, but they would tend to stabilize as the number of tosses are increased. This approach in effect is a method of sampling, but is not very invention. Instead of actually tossing the coins, you can conduct the experiment using random numbers. Random numbers have the property that any number is equally likely to occur, irrespective of the digit that has already occurred.