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:

mbox{ begin{tabular}{||c||c|c||} hlinehline Parameter&DrugI&DrugII hlinehline begin{tabular}{c} Selling price, $ per 1000 packs end{tabular}&5,500&6,100 hline begin{tabular}{c} Content of agent A, g per 1000 packs end{tabular}&0.500&0.600 hline begin{tabular}{c} Manpower required, hours per 1000 packs end{tabular}&90.0&100.0 hline begin{tabular}{c}Equipment required, hours per 1000 packs end{tabular}&40.0&50.0 hline begin{tabular}{c} Operational costs, $ per 1000 packs end{tabular}&700&800 hlinehline end{tabular} }

Contents of raw materials:

mbox{ begin{tabular}{||c||c||c||} hlinehline Raw material&Purchasing price,& Content of agent A, & $ per kg& g per kg hlinehline RawI& 100.00 &0.01 hline RawII& 199.90& 0.02 hlinehline end{tabular} }

Resources:

mbox{begin{tabular}{||c||c||c||c||} hlinehline begin{tabular}{c}Budget, $ end{tabular} &begin{tabular}{c}Manpower, hours end{tabular} &begin{tabular}{c}Equipment, hours end{tabular} &begin{tabular}{c} Capacity of raw materials storage, kg end{tabular} hline 100,000 &2,000& 800& 1,000 hlinehline end{tabular} }

Linear programming formulation

Variables

Let us denote by $x_{rm Drug I}$ , $x_{rm Drug II}$ the amounts (in 1000 of packs) of Drug I and II produced, while $x_{rm Raw I}$ , $x_{rm Raw II}$ 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 $f_0(x) = f_{rm costs}(x) - f_{rm income}(x)$ , where

$f_{rm costs}(x) = 100 cdot x_{rm Raw I} + 199.90 cdot x_{rm Raw II} + 700 cdot x_{rm Drug I} + 800 cdot x_{rm Drug II}$

represents the purchasing and operational costs, and

$f_{rm income}(x) = 5,500 cdot x_{rm Drug I} + 6100 cdot x_{rm Drug II}$

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:

$0.01 cdot x_{rm Raw I} + 0.02 cdot x_{rm Raw II} - 0.05 cdot x_{rm Drug I} - 0.600 cdot x_{rm Drug II} ge 0 .$

This constraint says that the amount of raw material must be enough to produce the drugs.

Storage constraint:

$x_{rm Raw I} + x_{rm Raw II} le 1000.$

This constraint says that the capacity of storage for the raw materials is limited.

Manpower constraint:

$90.0 cdot x_{rm Drug I} + 100.0 x_{rm Drug II} le 2000,$

which expresses the fact that the resources in manpower are limited, we cannot allocate more than 2,000 hours to the project.

Equipment constraint:

$40.0 cdot x_{rm Drug I} + 50.0 x_{rm Drug II} le 800.$

This says that the resources in equipment are limited.

Budget constraint:

$100.0 cdot x_{rm Raw I} + 199.90 cdot x_{rm Raw II} + 700 cdot x_{rm Drug I} + 800 cdot x_{rm Drug II} le 100,000 .$

This limits the total budget.

Sign constraints:

$x_{rm Raw I} ge 0, ;;x_{rm Raw II} ge 0, ;;x_{rm Drug I} ge 0, ;;x_{rm Drug II} ge 0.$

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 x^ast :

$p^ast = -8819.658, ;; x^ast_{rm Raw I} = 0, ;; x^ast_{rm Raw II} = 438.789, ;;x^ast_{rm Drug I} = 17.552, ;;x^ast_{rm Drug II} = 0.$

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 8.8% .