30. Didier limited 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.<br /><br />Objective function: Maximize profit = 30x + 40y + 20z<br /><br />Constraints:<br /><br />1. Special material: 2x + 4y + 2z ≤ 92<br />2. Timber: 4x + 2y + 2z ≤ 116<br />3. Labour: 2x + 2y + 4z ≤ 140<br />4. At least 2 tables: x ≥ 2<br />5. At least 4 chairs: 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 suitable due to the small number of variables and constraints.<br /><br />By plotting the constraints on a graph, we can find the feasible region. The feasible region is the area where all the constraints are satisfied. In this case, the feasible region is a polygon with vertices at (2, 0, 4), (0, 28, 4), and (0, 0, 35).<br /><br />Next, we evaluate the objective function at each vertex of the feasible region:<br /><br />1. At (2, 0, 4): Profit = 30(2) + 40(0) + 20(4) = 120<br />2. At (0, 28, 4): Profit = 30(0) + 40(28) + 20(4) = 1120<br />3. At (0, 0, 35): Profit = 30(0) + 40(0) + 20(35) = 700<br /><br />Therefore, Didier Limited should manufacture 0 tables, 28 sideboards, and 4 chairs to maximize profits, which is $1120.