Chapter 6, Problem 37RP

Program Plan Intro

Optimal solution:

• Here, consider the LP:
• min z=$\begin{array}{l}400x_1+400x_2+400x_3+400x_4+450y_1+450y_2\\ +450y_3+450y_4+20i_1+20i_2+20i_3+20i_4\end{array}$
• Subject to constraints:
• $x_1,x_2,x_3,x_4\le 40$
• $i_1=10+x_1+y_1-40$, $i_2=i_1+x_2+y_2-60$
• $i_3=i_2+x_3+y_3-75$, $i_4=i_3+x_4+y_4-25$
• $i_t,y_t,x_t\ge 0$
• Where
• $x_t=$ Number of sailboats produced by regular time labor during quarter t.
• $y_t=$ Number of sailboats produced by overtime labor during
• $i_t=$ Number of sailboats on hand at end of quarter t.
• In order to calculate the optimal solution of the primal, use the software Lindo
• Step 1:
  • Type in the Linear Program in the command window. Select yes when asked about the range analysis.
• The output is shown below:
• LP optimum found at step 7
• Objective function value: 78450

Variable	Value	Reduced Cost
$x_1$	40.000000	0.000000
$x_2$	40.000000	0.000000
$x_3$	40.000000	0.000000
$x_4$	25.000000	0.000000
$y_1$	0.000000	20.000000
$y_2$	10.000000	0.000000
$y_3$	35.000000	0.000000
$y_4$	0.000000	50.000000
$i_1$	10.000000	0.000000
$i_2$	0.000000	20.000000
$i_3$	0.000000	70.000000
$i_4$	0.000000	420.000000

Row	Slack or Surplus	Dual prices
2	0.000000	30.000000
3	0.000000	50.000000
4	0.000000	50.000000
5	15.000000	0.000000
6	0.000000	-430.000000
7	0.000000	-450.000000
8	0.000000	-450.000000
9	0.000000	-400.000000

• Number of iterations=7
• Ranges in which the basis is unchanged:

Obj coefficient ranges
Variable	Current coefficient	Allowable increase	Allowable decrease
$x_1$	400.000000	30.000000	Infinity
$x_2$	400.000000	50.000000	Infinity
$x_3$	400.000000	50.000000	Infinity
$x_4$	400.000000	50.000000	400.000000
$y_1$	450.000000	Infinity	20.000000
$y_2$	450.000000	20.000000	20.000000
$y_3$	450.000000	20.000000	50.000000
$y_4$	450.000000	Infinity	50.000000
$i_1$	20.000000	30.000000	20.000000
$i_2$	20.000000	Infinity	20.000000
$i_3$	20.000000	Infinity	70.000000
$i_4$	20.000000	Infinity	420.000000

Righthand side ranges
Row	Current RHS	Allowable increase	Allowable decrease
2	40.000000	10.000000	10.000000
3	40.000000	10.000000	40.000000
4	40.000000	35.000000	40.000000
5	40.000000	Infinity	15.000000
6	30.000000	10.000000	10.000000
7	60.000000	Infinity	10.000000
8	75.000000	Infinity	35.000000
9	25.000000	15.000000	25.000000

• Therefore, the optimal solution of primal is given by
• $z=78450,x_1=x_2=x_3=40,x_4=25,y_1=0,y_2=10,y_3=35,y_4=0,i_1=10,i_2=i_3=i_4=0$

Explanation of Solution

b.

• It can be seen from the Lindo output that the Obj coefficient ranges block is used to determine the amount by which each variable’s objective function coefficient may change before the current basis becomes suboptimal.
• If the objective function coefficient for $c_1$ is increased above $400+30=430$. Now the current basis is no longer optimal.
• Since, $\Delta c_1=20$ and the allowable increase is 30, the current basis remains optimal. The optimal values of the decision variables remain unchanged, that is,
• $x_1=x_2=x_3=40,x_4=25,y_1=0,y_2=10,y_3=35,y_4=0,i_1=10,i_2=i_3=i_4=0$
• The new optimal z-value can be obtained by substituting the substituting the optimal values of the decision variables

Explanation of Solution

c.

• If the demand of the sailboat increases by 1 unit in the first quarter, then the cost would increase by $430.
• As it can be observed from row (6) Dual prices section of the Lindo output.
• Since, the amount offered is less than the rise in the total cost, there is no incentive for Sailco to cater to this new demand...