| # A Drug Production Problem ## Problem description A company produces two kinds of drugs, DrugI and DrugII, containing a specific active agent A, which is extracted from raw materials purchased on the market. There are two kinds of raw materials, RawI and RawII, which can be used as sources of the active agent. The related production, cost and resource data are given in the tables below. The goal is to find the production plan which maximizes the profit of the company. ## Problem data Drug production data: Contents of raw materials: Resources: ## Linear programming formulation ### Variables Let us denote by , the amounts (in 1000 of packs) of Drug I and II produced, while , denote the amounts (in kg) of raw materials to be purchased. ### Objective function According to the problem data, the objective to be minimized in this problem has the form , where represents the purchasing and operational costs, and represents the income from selling the drugs. ### Constraints We have total of five inequality constraints, and additional sign constraints on the variables. Balance of active agent: This constraint says that the amount of raw material must be enough to produce the drugs. Storage constraint: This constraint says that the capacity of storage for the raw materials is limited. Manpower constraint: which expresses the fact that the resources in manpower are limited, we cannot allocate more than hours to the project. Equipment constraint: This says that the resources in equipment are limited. Budget constraint: This limits the total budget. Sign constraints: A CVX implementation of the problem is as follows. CVX implementation
    % min c'*x : A*x <= b, x >= 0
    % x = (RawI, RawII, DrugI, DrugII)
    % data:
    % objective function: we minimize cost minus profit
    c = [100 199.9  -5500 -6100]';
    % constraints
    A = [-0.01 -0.02 0.500 0.600; % balance
        1 1 0 0; % storage of raw material
        0 0 90.0 100.0; % manpower
        0 0 40.0 50.0; % equipment
        100.0 199.9 700 800; % budget
        ];
    % right-hand side
    b = [0; 1000; 2000; 800; 100000];
    % solve problem via CVX
    cvx_begin
       variable x(4,1)
       minimize( c'*x )
       subject to
            A*x <= b;
            x >= 0;
    cvx_end
    pstar = cvx_optval; % optimal value of the problem
Solving this problem, we obtain the following optimal value and a corresponding optimal point : Note that both the budget and the balance constraints are active (that is, the production process utilizes the entire budget and the full amount of active agent contained in the raw materials). The solution promises the company a modest, but quite respectable profit of .