(a) A oompany purchasing scrap material has two types of scarp materials available. The first type has 30% of material X,20% of material Y and 50% of material Z by weight. The second type has 40% of material X,10% of material Y and 30% of material Z. The costs of the two scraps are K120 and K160 per kg respectively. The company requires at least 240kg of material X,100kg of material Y and 290kg of material Z. Find the optimum quantities of the two scraps to be purchased so that the company requirements of the three materials are satisfied at a minimum cost. [6 marks] (b) A mining company extracts gravel, the basic product it sells, from three mines, Mopani Copper Mines (M1), Konkola Copper Mines (M2) and Kariba Minerals Ltd (M3). The monthly production of each mine is 45,150 and 75 tones of gravel respectively. The gravel has to be transported to five main consumers, Lusaka (C1), Choma (C2], Kabwe( C3), Kitwe (C4) and Livingstone (C3) requiring for their needs 100,60,40,75 and 25 tones of gravel per week respectively. The problem that concerns the company's management is the minimization of the required cost for the transportation of the product to the consumers. For this purpose a detailed cost analysis was carried out which gave the results of the following table (the numbers denote the transportation cost in Kwacha per ton of gravel). Flnd the basic feasible solution using; (i) North-West Corner Method (NWCM) [5 marks] (ii) Least Cost Method (LCM) [5 marks] (iii) Vogel Approximation Method (VAM)