Chapter 6.4, Problem 7P

Explanation of Solution

Optimal solution of the diet problem:

• The LINDO output is considered for the Dakota problem:
• max 60DSK+30TBLS+20CHRS
• Subject to constraint
  • $8DSK+6TBLS+CHRS\le 48$
  • $4DSK+2TBLS+1.5CHRS\le 20$
  • $2DSK+1.5TBLS+0.5CHRS\le 8$
• End
• LP optimum is found at step 2
• Objective function value: 90.000000

Variable	Value	Reduced cost
DSK	2.000000	0.000000
TBLS	0.000000	5.000000
CHRS	8.000000	0.000000

Row	Slack or Surplus	Dual prices
2	24.000000	0.000000
3	0.000000	10.000000
4	0.000000	10.000000

• Number of iterations= 2
• The ranges in which the basis is unchanged:

Variable	Current coefficient	OBJ coefficient ranges allowable increase	Allowable decrease
DSK	60.000000	20.000000	4.000000
TBLS	30.000000	5.000000	Infinity
CHRS	20.000000	2.500000	5.000000

Row	Current RHS	Right-hand side ranges allowable increase	Allowable decrease
2	48.000000	Infinity	24.000000
3	20.000000	4.000000	4.000000
4	8.000000	2.000000	1.333333

• As brownies and pineapple cheesecake have no zero reduced costs, so case 2 is applied to the states where,
  • “At least one variable whose objective function coefficient is changed has a reduced cost of zero”.
• In case 2 for each variable x_j the ratio r_j is defined as follows:
  • If $\Delta c_j\ge 0$, then $r_j=\frac{\Delta c_j}{I_j}$
  • If $\Delta c_j\le 0$, then $r_j=\frac{-\Delta c_j}{D_j}$
• Here,
  • $\Delta c_j=$ Change in original objective function coefficient (c_j) for x_j
  • $I_j=$ Maximum allowable increase in c_j from LINDO output.
  • $D_j=$ Maximum allowable decrease in c_j from LINDO output.
• The 100% rule for objective function states that,
  • “If $\sum r_j\le 1$, then the current basis remains optimal”
  • Also,
  • “If $\sum r_j>1$, then the current basis may or may not be optimal”.
• If the current basis remains optimal, the value of the decision variables remains unchanged, but the optimal z-value may change.
• Now, the price of desk is increased to $65, the price of the table is decreased to $25 and the price of chair is decreased to $18