Chapter 4, Problem 16RP

a.

Explanation of Solution

• Consider the following variables:
  • x₁ = items of type 1
  • x₂ = items of type 2
• Weight of item is a₁ lb and item 2 weighs a₂ lb.
• Therefore, the constraints is a₁x₁+a₂x₂ ≤ b
• Each type 1 item carries a benefit of c₁ units, each type 2 item carries a benefit of c₂ units...

b.

Explanation of Solution

• from the result of part (a), it can be observed that the constraint in linear programming problem is less than or equal to type.
• So, add the slack variable s₁ to the constraint.
• Thus, the standard form of the linear programming problem is,
  • Max z= c₁x₁+c₂x₂
  • Subject to constraints,
  • a₁x₁+a₂x₂+s₁ = b
• The initial simplex table is shown below:

z	x1	x2	s1	rhs	Basic variable
1	-c1	-c2	0	0	z = 0
0	a1	a2	1	b	s1 = b

• Now, there are two negative entries in the row0, suppose the most negative entry is in column 2.
• Therefore, choose entering variable to be x₂.
• The leaving variable is calculated with the use of the following formula,
  • Leaving variable = {(rhs/x₁), x₁ ≥ 0}
• Consider the below table:

z	x1	x2	s1	rhs	Basic variable
1	-c1	-c2	0	0	-
0	a1	a2	1	b	(b/a2)→

• From the above table, the pivot table is intersection of both entering and leaving variables...

c.

Explanation of Solution

• The following are the assumptions that are violated by this formula:
  • Deterministic Nature
  • Additivity
  • Direct Proportionality
  • Fractionally
• The follo...