Don Levine Corporation is considering adding an additional plant to its three ex

Question

Don Levine Corporation is considering adding an additional plant to its three existing facilities in Decatur, Minneapolis, and Carbondale. Both St. Louis and East St. Louis are being considered. Evaluating only the transportation costs per unit as shown in the tables below, which site is the best?

From Existing Plants

O Decatus Minneapolis Carbondale Demand

Blue Earh $20 $17 $21 250
Ciro 25 27 20 200
Des Moines 22 25 22 350
Capacity 300 200 150

From Proposed Plants
To East St. Louis St.Louis
Blue Earth $29 $27
Ciro 30 28
Des Moines 30 31
Capacity 150 150

A. Is this a minimization or maximizaion problem? Explain

B. What would the total transportation cost be if the East St. Lous plant were opened?

C. What would the total transportation cost be if the proposed plant in St. Louis were opened?

D. Which plant would you advise Levin Corp. to open and why?

Explanation / Answer

In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete variables is known as a combinatorial optimization problem. In a combinatorial optimization problem, we are looking for an object such as an integer, permutation or graph from a finite (or possibly countable infinite) set. NO THIS IS NOT a minimization or maximizaion problem An NP-optimization problem (NPO) is a combinatorial optimization problem with the following additional conditions.[3] Note that the below referred polynomials are functions of the size of the respective functions' inputs, not the size of some implicit set of input instances. the size of every feasible solution scriptstyle yin f(x) is polynomially bounded in the size of the given instance x, the languages scriptstyle {,x,mid, x in I ,} and scriptstyle {,(x,y), mid, y in f(x) ,} can be recognized in polynomial time, and m is polynomial-time computable. This implies that the corresponding decision problem is in NP. In computer science, interesting optimization problems usually have the above properties and are therefore NPO problems. A problem is additionally called a P-optimization (PO) problem, if there exists an algorithm which finds optimal solutions in polynomial time. Often, when dealing with the class NPO, one is interested in optimization problems for which the decision versions are NP-hard. Note that hardness relations are always with respect to some reduction. Due to the connection between approximation algorithms and computational optimization problems, reductions which preserve approximation in some respect are for this subject preferred than the usual Turing and Karp reductions. An example of such a reduction would be the L-reduction. For this reason, optimization problems with NP-complete decision versions are not necessarily called NPO-complete.[4] NPO is divided into the following subclasses according to their approximability:[3] NPO(I): Equals FPTAS. Contains the Knapsack problem. NPO(II): Equals PTAS. Contains the Makespan scheduling problem. NPO(III): :The class of NPO problems that have polynomial-time algorithms which computes solutions with a cost at most c times the optimal cost (for minimization problems) or a cost at least 1/c of the optimal cost (for maximization problems). In Hromkovic's book, excluded from this class are all NPO(II)-problems save if P=NP. Without the exclusion, equals APX. Contains MAX-SAT and metric TSP. NPO(IV): :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio that is polynomial in a logarithm of the size of the input. In Hromkovic's book, all NPO(III)-problems are excluded from this class unless P=NP. Contains the set cover problem. NPO(V): :The class of NPO problems with polynomial-time algorithms approximating the optimal solution by a ratio bounded by some function on n. In Hromkovic's book, all NPO(IV)-problems are excluded from this class unless P=NP. Contains the TSP and Max Clique problems. Another class of interest is NPOPB, NPO with polynomially bounded cost functions. Problems with this condition have many desirable properties.

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