CS 600 Test 2

November 8, 1999

1. Find an initial basic feasible solution for the following problem by formulating and solving by the simplex method the auxiliary problem. Then solve the original problem by the simplex method.

Max Z = -x₁ - x₂
Subject to:
$-2x_{1} - x_{2} \leq -4$
$-x_{1} - x_{2} \leq -3$
$5x_{1} + 4x_{2} \leq 20$
$x_{1}, x_{2} \geq 0$

2. Formulate the following problem as a LP problem:

A meat packing plant produces 450 hams, 380 pork bellies and 320 picnic hams every day; each of these products can be sold either fresh or smoked. The total number of hams, bellies, and picnics that can be smoked during a normal working day is 450; in addition, up to 230 products can be smoked on overtime at a higher cost. Due to the availability of resources, picnics are to be produced 10% (of the total number of products) more than bellies, and 15% more than hams. The net profits are as follows:

	Fresh	Smoked on	Smoked
		regular time	on overtime
Hams	$16	$28	$22
Bellies	$8	$24	$14
Picnics	$8	$26	$18

3. At North Central University courses are scheduled as follows: First, time slots and instructors are assigned to the courses. Then rooms are assigned. For simplicity assume that any room can be assigned to any course as long as it is available (ignoring size and type of rooms).

Prove that the problem of finding a room to each of the courses without conflicts is in general NP-Complete.

You do NOT have to prove that it is in NP.
Also you may use the fact that the following problems are NP-Complete:
3 Dimensional Matching, Partition, Graph Coloring and Bin Packing.