Problem 1:

Consider the following feasible set:

• $latex x \ge -1 \text{ and } x\le 1$
• $latex y \ge -1 \text{ and } y\le 1$
• $latex z \ge -1 \text{ and } z\le 1$
• $latex w \ge -1 \text{ and } w\le 1$

The vertices of this feasible set have four coordinates, and each coordinate is either equal to positive one or negative one. For example (1,-1,1,1) is a vertex of the feasible set.

Two vertices are neighbors if they agree in all but one coordinate. For example, (1,1,1,1) is a neighbor of (1,-1,1,1).

Starting with the vertex (1,-1,1,1) as $latex v_0$, maximize the following objective function: $latex -2x-y+3z+4w$

Problem 2:

The permutahedron, $latex P_n$, is an example of a feasible set. The vertices of the permutahedron are given by permutations of $latex n$ (we’ll talk more about these in the coming weeks). I have listed the vertices of the permutahedron $latex P_3$ below:

• (1,2,3)
• (2,1,3)
• (1,3,2)
• (2,3,1)
• (3,1,2)
• (3,2,1)

Correction:

The neighbors a vertex are obtained by swapping entries that are adjacent in value.

• For example: (1,2,3) and (2,1,3) are neighbors. But (3,1,2) and (2,1,3) are also neighbors. In the second example, I am looking at the entries whose values are 3 and 2, and swap.

Problem 2 part 1: Show that each vertex of $latex P_3$ has exactly two neighbors. (Thus, this feasible set is a polygon, even though it naturally lives in 3-space.)

Problem: 2 part 2: Use the simplex method, starting at the vertex (1,2,3) to maximize the objective function $latex 2x+4y+z$

From Section 3.3 complete the following problems for practice:

• Numbers 22 (solution: operate refinery I for 4 days and refinery II for 3 days), 23 (see example 1 for a similar problem), 24, 25, 26 (solution: ship 4 cars from Baltimore to Philadelphia, 1 car from Baltimore to Trenton, and 6 cars from NY to Trenton), 31

From Section 3.2:

• 25-31 odds only
• 33, 34 (Solution: Cleveland should operate 20 days, Toledo should operate 50 days), 37, 38 (Solution: 8 hours from smelter A, 11 hours from smelter B), 39, 40 (Solution: 80 homes of the first type, 60 of the second type), 41

For the test you should be able to carry out all of the steps to a linear programing problem.