One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes that can be made from 5kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

Let there be x cakes of first kind and y cakes of second kind. Therefore, x≥0 and y≥0 The given information can be compiled in a table as follows. Flour(g) Flour(g) Cakes of first kind,x Cakes of second kind,y Availability 200 25 100 50 5000 1000 ∴200x+100y≤5000 ⇒2x+y≤50 25x+50y≤1000 ⇒x+2y≤40 Total numbers of cakes,Z,that can be made are, Z=x+y The mathematical formulation of the given problem is Maximize Z=x+y...(1) subject to the constraints, 2x+y≤50...(2) x+2y≤40...(3) x,y≥0...(4) The feasible region determined by the system of constraints is as follows. Thus the maximum numbers of cakes that can be made is 30(20 of one kind and 10 of the other kind).

One kind of cake requires 200g flour and 25g of fat,and another kind of cake requires 100g of flour and 50g of fat.Find the maximum number of cakes which can be made from 5kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes?

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Notes on Linear Programming Problem

Concepts Used:

Linear Programming Problems

The Linear Programming Problems (LPP) is a problem that is concerned with finding the optimal value of the given linear function. The optimal value can be either maximum value or minimum value. Here, the given linear function is considered an objective function. The objective function can contain several variables, which are subjected to the conditions and it has to satisfy the set of linear inequalities called linear constraints.

Linear Programming Simplex Method

Step 1: Establish a given problem. (i.e.,) write the inequality constraints and objective function.

Step 2: Convert the given inequalities to equations by adding the slack variable to each inequality expression.

Step 3: Create the initial simplex tableau. Write the objective function at the bottom row. Here, each inequality constraint appears in its own row. Now, we can represent the problem in the form of an augmented matrix, which is called the initial simplex tableau.

Step 4: Identify the greatest negative entry in the bottom row, which helps to identify the pivot column. The greatest negative entry in the bottom row defines the largest coefficient in the objective function, which will help us to increase the value of the objective function as fastest as possible.

Step 5: Compute the quotients. To calculate the quotient, we need to divide the entries in the far right column by the entries in the first column, excluding the bottom row. The smallest quotient identifies the row. The row identified in this step and the element identified in the step will be taken as the pivot element.

Step 6: Carry out pivoting to make all other entries in column is zero.

Step 7: If there are no negative entries in the bottom row, end the process. Otherwise, start from step 4.

Step 8: Finally, determine the solution associated with the final simplex tableau.