Answer:

Solve the equations and construct the coordinates for the following as shown below:

…… (1)

…… (2)

Construct the coordinates for the equation (1) and (2) as shown below:

Calculate the coordinates for the equation (1) by assuming the value of p=0 shown as follows:

Calculate the coordinates for the equation (1) by assuming the value of a=0 shown as follows:

Calculate the coordinates for the equation (2) by assuming the value of p=0 shown as follows:

Calculate the coordinates for the equation (2) by assuming the value of a=0 shown as follows:

Therefore, the coordinates for the equation (1) are (15, 0) and (0, 12)

Therefore, the coordinates for the equation (1) are (12, 0) and (0, 15)

The first equation represents the picking constraint. There are a total of 60 minutes of picking available from the single picking in one hour.

The variables a , p represent the number of apples and pears harvested, respectively.

The second equation represents the packing constraint. The graph of these two equations is as follows.

The graph indicates that 6.7 apples and 6.7 pears should be harvested to maximize the use of labor. However, the profit for harvesting these many fruits is $33.5. If only apples are harvested, then 12 apples can be harvested in one hour, which gives a profit of $36.

Thus, optimal profit decision is to harvest only apples.

Answer:

Given

In the given problem a company has two options of either to buy new machinery or to modify the existing one in order to produce the product X. There are two probabilities that are associated with the two outcomes. There is a possibility of prosperity (0.6) and that of recession (0.4) that has an impact on the two options.

The table given below gives the two possible options and the payoffs associated with the respective options.

A decision tree based on the data as provided in the table given above can be drawn as given below:

Thus, the expected payoff that is obtained from the option of buying new machinery in order to produce product X is $450,000.

On the other hand the expected payoff obtained from the option of modifying the existing machinery is $540,000.

Thus, the decision that should be taken is to modify the existing machinery in order to produce product X as the expected payoff in that case is more than the expected payoff obtained in the case of buying a new machinery.

Answer:

Linear programming model is used to obtain maximum profits, by utilizing minimum resources available. It helps the decision makers to take effective decisions by using the resources productively. It helps in minimizing cost of operations. It consists of objective function, model constraints and decision variables.

Formulate the LPP to solve this problem graphically, first use the following constraint equations and plot a graph.

The first equation represents the painting constraint. There is a total of 80 hours of painting available from the two painters in one week.

The two variable b and t represent the number of blocks and trucks produced, respectively.

The second equation represents the wood working constraint.

Construct a graph using the following methodology as explained below:

Consider B as x and T as Y for the coordinate calculation.

Calculate coordinates for equation (1) as shown below:

Step: 1 Calculate the coordinates for the equation (1) by assuming the value of y=0 shown as follows:

Calculate the coordinates for the equation (1) by assuming the value of x=0 shown as follows:

Therefore, the coordinates for the equation (1) are (40, 0) and (0,80).

Calculate coordinates for equation (2) as shown below:

Step: 1 Calculate the coordinates for the equation (2) by assuming the value of y=0 shown as follows:

Calculate the coordinates for the equation (2) by assuming the value of x=0 shown as follows:

Therefore, the coordinates for the equation (1) are (120, 0) and (0,40).

Construct a graph by using the above coordinates as shown below:

From the above graph, it can be identified that points O, D and B are in the feasible region. Notice the green region having coordinates of x 1 =24 and x 2 = 32, which satisfies the objective function as follows:

Substitute the values of x 1 and x 2 in the objective function as shown below:

Therefore, from the above graphical analysis it indicates that the optimal number of blocks is 24 , and the optimal number of trucks is 32. Solving the system of equations will give the same result.