12. A company may manufacture n different products, each of which uses various amounts of m limited resources. Each unit of product i yields a profit of c; dollars and uses dji units of the th resource. The available amount of the jth resource is bj. To maximize profit the company selects the quantities x; to be manufactured of each product by solving maximize cx subject to Ax = b, x > 0. The unit profits calready take into account the variable cost associated with man- ufacturing each unit. In addition to that cost, the company incurs a fixed overhead H, and for accounting purposes it wants to allocate this overhead to each of its products. In other words, it wants to adjust the unit profits so as to account for the overhead. Such an overhead allocation scheme must satisfy two conditions: (1) Since H is fixed regardless of the product mix, the overhead allocation scheme must not alter the optimal solution, (2) All the overhead must be allocated; that is, the optimal value of the objective with the modified cost coefficients must be H dollars lower than 7the original optimal value of the objective. a) Consider the allocation scheme in which the unit profits are modified accord- ing to = c ry, A, where yo is the optimal solution to the original dual and r = H/20 (assume H 0 be the amount of overhead associated with the ith resource, where H; 0. The unit profits calready take into account the variable cost associated with man- ufacturing each unit. In addition to that cost, the company incurs a fixed overhead H, and for accounting purposes it wants to allocate this overhead to each of its products. In other words, it wants to adjust the unit profits so as to account for the overhead. Such an overhead allocation scheme must satisfy two conditions: (1) Since H is fixed regardless of the product mix, the overhead allocation scheme must not alter the optimal solution, (2) All the overhead must be allocated; that is, the optimal value of the objective with the modified cost coefficients must be H dollars lower than 7the original optimal value of the objective. a) Consider the allocation scheme in which the unit profits are modified accord- ing to = c ry, A, where yo is the optimal solution to the original dual and r = H/20 (assume H 0 be the amount of overhead associated with the ith resource, where H;