Question

The following linear programming problem has been solved by LINDO. Use the output to answer the questions. (Scroll down to see all).

LINEAR PROGRAMMING PROBLEM

MAX 41X1+52X2+21X3
S.T.
C.1) 5X1 + 5X2 + 9X3 < 1200
C.2) 11X1 + 14X2 + 5X3 < 1500
END

LP OPTIMUM FOUND AT STEP      1
OBJECTIVE FUNCTION VALUE
1)      5795.049
VARIABLE        VALUE          REDUCED COST
X1 0.000 0.217822
X2         74.247 0.000000
X3 92.079 0.000000
ROW   SLACK OR SURPLUS     DUAL PRICES
C.1)         0.000 0.336
C.2)         0.000 3.594
NO. ITERATIONS=       1
RANGES IN WHICH THE BASIS IS UNCHANGED:
OBJ COEFFICIENT RANGES
VARIABLE         CURRENT        ALLOWABLE        ALLOWABLE
COEF          INCREASE         DECREASE
X1       41.000000         0.217822 INFINITY
X2       52.000000   6.800000   0.297299
X3       21.000000 72.59999 1.466675
RIGHTHAND SIDE RANGES
ROW         CURRENT        ALLOWABLE        ALLOWABLE
RHS          INCREASE         DECREASE
C.1     1200.000000      1500.000000   664.285706
C.2     1500.000000      1140.000000       833.333313

c.	What is the dual price for the first constraint? What interpretation does this have? (5 points)
d.	Over what range can the objective function coefficient of X2 vary before a new solution point becomes optimal? (5 points)
e.	By how much can the amount of resource constraint 2 decrease before the dual price will change? (5 points)
f.	What would happen if the first constraint's right-hand side decreased by 200 and the second's increased by 400? (5 points) Hint: Perform the 100% rule test.

Answer 1

The information provided has some structural issue. Hence, obtained the output again on Lindo. Based on this we will answer the questions posted.

c.

Dual price for first constraint is 0.336

d.

The optimality range for X2 shows to have an allowable increase of 6.8 and allowable decrease of 0.297299. This means the range is from (52-0.297299) to (52+6.8).

Range: From 51.702701 to 58.8

e.

The feasibility range for constraint 2 has allowable increase of 1140 and allowable decrease of 833.33. This means the range is from (1500-833.33) to (1500+1140).

Range: From 666.67 to 2640

Beyond this, the dual prices will change.

f.

Using 100% rule test we see that

200/664.28 + 400/1140 = 0.65

This means that it is within the 100% range and thus the dual prices of the constraints will impact the objective function value.

Total impact will be -200*0.336 + 400*3.594 = 1307.4

The new objective function value will be 1307.4 + 5795.049 = 7165.449