A pictorial representation of a simple linear program with two variables and six inequalities. The linear programming discrete mathematics 7th edition filetype pdf is to find a point on the polyhedron that is on the plane with the highest possible value. If every entry in the first is less-than or equal-to the corresponding entry in the second then it can be said that the first vector is less-than or equal-to the second vector.
Linear programming can be applied to various fields of study. Industries that use linear programming models include transportation, energy, telecommunications, and manufacturing. Hitchcock had died in 1957 and the Nobel prize is not awarded posthumously. Dantzig provided formal proof in an unpublished report “A Theorem on Linear Inequalities” on January 5, 1948. In the post-war years, many industries applied it in their daily planning.
Dantzig’s original example was to find the best assignment of 70 people to 70 jobs. The theory behind linear programming drastically reduces the number of possible solutions that must be checked. Linear programming is a widely used field of optimization for several reasons. A number of algorithms for other types of optimization problems work by solving LP problems as sub-problems. Therefore, many issues can be characterized as linear programming problems. There are two ideas fundamental to duality theory.
Additionally, every feasible solution for a linear program gives a bound on the optimal value of the objective function of its dual. A linear program can also be unbounded or infeasible. Duality theory tells us that if the primal is unbounded then the dual is infeasible by the weak duality theorem. Likewise, if the dual is unbounded, then the primal must be infeasible. However, it is possible for both the dual and the primal to be infeasible.
The primal problem deals with physical quantities. With all inputs available in limited quantities, and assuming the unit prices of all outputs is known, what quantities of outputs to produce so as to maximize total revenue? The dual problem deals with economic values. With floor guarantees on all output unit prices, and assuming the available quantity of all inputs is known, what input unit pricing scheme to set so as to minimize total expenditure? To each variable in the primal space corresponds an inequality to satisfy in the dual space, both indexed by output type. To each inequality to satisfy in the primal space corresponds a variable in the dual space, both indexed by input type.
The coefficients that bound the inequalities in the primal space are used to compute the objective in the dual space, input quantities in this example. The coefficients used to compute the objective in the primal space bound the inequalities in the dual space, output unit prices in this example. Both the primal and the dual problems make use of the same matrix. In the primal space, this matrix expresses the consumption of physical quantities of inputs necessary to produce set quantities of outputs.
In the dual space, it expresses the creation of the economic values associated with the outputs from set input unit prices. Since each inequality can be replaced by an equality and a slack variable, this means each primal variable corresponds to a dual slack variable, and each dual variable corresponds to a primal slack variable. This relation allows us to speak about complementary slackness. Sometimes, one may find it more intuitive to obtain the dual program without looking at the program matrix. Since this is a minimization problem, we would like to obtain a dual program that is a lower bound of the primal. Note that we assume in our calculations steps that the program is in standard form.