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Chapter 4: Problem 47

Explain the need for Phase I in a nonstandard LP problem.

Short Answer

Expert verified
The need for Phase I in a nonstandard LP problem is crucial as it helps find an initial feasible solution by creating an auxiliary problem with artificial variables. It also identifies infeasible problems, provides a starting point for Phase II to find the optimal solution using the simplex method, and simplifies the original problem by converting nonstandard LP problems into standard LP problems, making the optimization process easier in Phase II.

Step by step solution

01

Definition of Nonstandard LP Problem

A nonstandard LP problem is a linear programming problem where the initial feasible solution is not readily available. In the objective function, at least one initial basic variable takes on a non-zero value. To solve such problems, the two-phase simplex method is employed.
02

Two-Phase Simplex Method

The two-phase simplex method is a technique for solving nonstandard LP problems. It consists of two phases: 1. Phase I: The primary objective is to find an initial feasible solution. 2. Phase II: The main objective is to find the optimal solution using the simplex method, typically used for standard LP problems.
03

The Need for Phase I

The role of Phase I in a nonstandard LP problem is crucial due to the following reasons: 1. Find Initial Feasible Solution: In a nonstandard LP problem, we don't have an initial feasible solution that satisfies all the constraints. Phase I helps find this starting solution by creating an auxiliary problem with artificial variables. 2. Identify Infeasibility: If there exists no feasible solution for the problem, Phase I will determine this by having the artificial variables in the solution with any positive value at the end of the process. 3. Basis for Phase II: Phase I provides a feasible solution, which serves as a starting point for Phase II to find the optimal solution using the simplex method. 4. Simplification: Phase I simplifies the original problem by converting nonstandard LP problems into standard LP problems, making the optimization process easier in Phase II. In conclusion, Phase I plays a critical role in solving nonstandard LP problems by finding an initial feasible solution, identifying infeasible problems, providing a basis for Phase II, and simplifying the overall optimization process.

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Most popular questions from this chapter

I Purchasing Cheapskate Electronics Store needs to update its inventory of stereos, TVs, and DVD players. There are three suppliers it can buy from: Nadir offers a bundle consisting of 5 stereos, 10 TVs, and 15 DVD players for \(\$ 3,000\). Blunt offers a bundle consisting of 10 stereos, \(10 \mathrm{TVs}\), and 10 DVD players for \(\$ 4,000\). Sonny offers a bundle consisting of 15 stereos, \(10 \mathrm{TVs}\), and 10 DVD players for \(\$ 5,000\). Cheapskate Electronics needs at least 150 stereos, \(200 \mathrm{TVs}\), and 150 DVD players. How can it update its inventory at the least possible cost? What is the least possible cost?

Your friend Janet is telling everyone that if there are only two constraints in a linear programming problem, then, in any optimal basic solution, at most two unknowns (other than the objective) will be nonzero. Is she correct? Explain.

Solve the LP problems. If no optimal solution exists, indicate whether the feasible region is empty or the objective function is unbounded. \(\nabla\) Maximize \(p=x+y\) subject to \(\begin{aligned} x+2 y & \geq 10 \\ 2 x+2 y & \leq 10 \\ 2 x+y & \geq 10 \\ x \geq 0, y & \geq 0 . \end{aligned}\)

\(\nabla\) \mathrm{\\{} T r a n s p o r t a t i o n ~ S c h e d u l i n g ~ W e ~ r e t u r n ~ t o ~ y o u r ~ e x p l o i t s ~ c o - ~ ordinating distribution for the Tubular Ride Boogie Board Company. \({ }^{36}\) You will recall that the company has manufacturing plants in Tucson, Arizona and Toronto, Ontario, and you have been given the job of coordinating distribution of their latest model, the Gladiator, to their outlets in Honolulu and Venice Beach. The Tucson plant can manufacture up to 620 boards per week, while the Toronto plant, beset by labor disputes, can produce no more than 410 Gladiator boards per week. The outlet in Honolulu orders 500 Gladiator boards per week, while Venice Beach orders 530 boards per week. Transportation costs are as follows: Tucson to Honolulu: \(\$ 10 /\) board; Tucson to Venice Beach: \(\$ 5 /\) board; Toronto to Honolulu: \(\$ 20 /\) board; Toronto to Venice Beach: \(\$ 10 /\) board. Your manager has said that you are to be sure to fill all orders and ship the boogie boards at a minimum total transportation cost. How will you do it?

If a linear programming problem has a bounded, nonempty feasible region, then optimal solutions (A) must exist (B) may or may not exist (C) cannot exist
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