economical interpretation of the dual solution

Dual SoIution as Dual Prices
First, consider the dual variables u1 and u2. u1 has a value of \$8, which means an hour of time of machine 1 has a value of \$8. How would the optimal solution change if there were an additional hour of machine 1 time available? Since the original solution was to produce 4 units of A (x1 = 4) and none of B, the additional machine 1 time would be used to produce more units of A. Since one unit of product A takes 2 hours on machine 1, an additional hour will enable us to produce one half of a unit of A. The resulting change in profit is one half of \$16, or \$8. (Note that there is sufficient excess machine 2 capacity for this production.)

The actual cost of renting additional capacity of machine 1 may be greater or less than \$8. If it is less (say \$6), then the company should consider renting some additional capacity of machine 1. This is because for each additional hour of available capacity rented, the return is \$8, whereas the cost is \$6, resulting in an incremental gain of \$2.

The value of u2 is zero, which means the "opportunity cost" of an hour of time on machine 2 is zero. This is consistent with the fact that machine 2 has idle hours following the optimum schedule of production and making more time on machine 2 available would not increase profit.

Dual prices are also called shadow prices or marginal values.

Dual prices can be interpreted as the "cost" of a constraint. We can say that a unit of slack of machine l, which has only four hours available, "costs" at a rate of \$8 per hour (u1 has a value of \$8 per unit). It would be worth \$8 per hour in increased profitability to obtain an additional hour on machine 1. Thus, the dual price measures the value or worth of relaxing a constraint by acquiring an additional unit of that factor of production.

Now consider the dual slack variables, u3 and u4. The dual slack variables measure the opportunity loss involved in production of the corresponding primal variable. Since u3 is the slack variable in the first dual constraint, it corresponds to x1 in the primal. Similarly, u4 corresponds to x2. u4 has a value of \$18; this means that the cost of forcing a unit of x2 into the solution is \$18. U3 has a value of zero, which means that there is no opportunity loss involved in producing x1. Thus, x1 is being produced in positive quantity in the optimal solution, whereas x2 is not being produced. If there is a positive opportunity loss associated with a variable, it will have the value of zero in the optimal solution; conversely, if the opportunity loss associated with a variable is zero, the variable will take on some positive value in the optimal solution. We use the term-reduced costs to refer to the dual solution variables interpreted in this fashion.

Note that the values of four for u1 and u4 are the same (except for the sign) as the  values for x3 and x2. This is not a coincidence. The u values of the dual solution are uniquely the  values of corresponding variables in the primal solution. By corresponding variables, it is meant that dual ordinary variables (u1 and u2) are associated with primal slack variables (x3 and x4), and dual slack variables (u3 and u4) are associated with primal ordinary variables (x1 and x2). These associations make sense, since the dual ordinary variable u1 measures the opportunity cost per hour of using machine 1, and the primal slack variable x3 represents the amount of unused capacity of machine 1. Similarly, the  values of the dual give the values of the corresponding variables in the primal solution. Thus, the simplex table for the primal solution provides both the primal solution and (through the  values) the values of the dual variables.