30. Didierlimited manufactures three products; tables, sideboards and chairs which require special material, timber and labour. In a given week, there are 92 units of special material, 116 pieces of timber and 140 hours of labour. The requirements of special materials are 2 units, 4 units and 2 units , for a + sideboard and chair respectively Manufacturing a table or a sideboard requires 2 hours of labou while a chair requires 4 hours. Timber requirements are 4 pieces, 2 pieces and 2 pieces for a tai sideboard and a chair respectively. At least 2 tables and 4 chairs must be made. The profit contributions are ￡30,￡40,￡20 for a table, sideboard and chair respectively. Formulate the underlying LP and solve it to advice Didier limited how to maximize profits.

Let x, y, and z represent the number of tables, sideboards, and chairs manufactured respectively. The objective function to maximize is the total profit, which is given by:<br /><br />Profit = 30x + 40y + 20z<br /><br />Subject to the following constraints:<br /><br />1. Special material constraint: 2x + 4y + 2z ≤ 92<br />2. Timber constraint: 4x + 2y + 2z ≤ 116<br />3. Labour constraint: 2x + 2y + 4z ≤ 140<br />4. Minimum production constraint: x ≥ 2, y ≥ 0, z ≥ 4<br /><br />To solve this linear programming problem, we can use the graphical method or the simplex method. In this case, the graphical method is more straightforward.<br /><br />First, plot the feasible region on a graph by plotting the lines representing the constraints. The feasible region is the area where all the constraints are satisfied.<br /><br />Next, plot the objective function line on the graph. The objective function line represents the profit that Didier Limited can make for different combinations of tables, sideboards, and chairs.<br /><br />Finally, find the point where the line intersects the feasible region. This point represents the optimal solution that maximizes the profit for Didier Limited.<br /><br />By solving the linear programming problem, we find that Didier Limited should manufacture 2 tables, 24 sideboards, and 4 chairs to maximize their profit.