Msc Logistics & Supply Chain Management

...

NO. ITERATIONS= 3

BRANCHES= 0 DETERM.= 1.000E 0

It should be mentioned that taking into account that the “BRANCHES= 0” that was shown in the solution script, the initial command script was altered to the following form, in order to solve it as a usual LP (with no integer values constraint):

max 12x1+5x2+15x3+10x4

st

5x1+x2+9x3+12x4

2x1+3x2+4x3+x4

3x1+2x2+5x3+10x4

x1>=40

x2>=130

x3>=30

x4

end

The resulting solution script clearly shows that we obtain the very same solution, which is obtained by the initial command script (containing integer number constraints for the decision variables).

The resulting solution script is as follows:

LP OPTIMUM FOUND AT STEP 3

OBJECTIVE FUNCTION VALUE

1) 2660.000

VARIABLE VALUE REDUCED COST

X1 130.000000 0.000000

X2 130.000000 0.000000

X3 30.000000 0.000000

X4 0.000000 30.000000

ROW SLACK OR SURPLUS DUAL PRICES

2) 450.000000 0.000000

3) 230.000000 0.000000

4) 0.000000 4.000000

5) 90.000000 0.000000

6) 0.000000 -3.000000

7) 0.000000 -5.000000

8) 10.000000 0.000000

NO. ITERATIONS= 3

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES[pic 2]

VARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE

X1 12.000000 INFINITY 3.000000

X2 5.000000 3.000000 INFINITY

X3 15.000000 5.000000 INFINITY

X4 10.000000 30.000000 INFINITY

RIGHTHAND SIDE RANGES[pic 3]

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

2 1500.000000 INFINITY 450.000000

3 1000.000000 INFINITY 230.000000

4 800.000000 270.000000 270.000000

5 40.000000 90.000000 INFINITY

6 130.000000 135.000000 130.000000

7 30.000000 54.000000 30.000000

8 10.000000 INFINITY 10.000000

Excel QM™ Software Used

The specific software, which is essentially an MS EXCEL add-on can be obtained via the following weblink

http://wps.prenhall.com/bp_taylor_introms_11/220/56508/14466195.cw/content/index.html

Contrary to the LINDO™ interface which requires the typing of specific commands in accordance with the aforementioned tutorial, the Excel QM™ Software is more user friendly with straightforward data input. The solution is obtained by clicking on “Data” tab and subsequently on the “Solver” button. The emerging dialogue allows for adding additional constraints like integer&binary values. In our case (like LINDO™) the Linear Problem was solved once by adding integer value constraint and re-solved by not having the integer value constraint. The resulting solution is exactly the same as LINDO™ and it is represented as follows:[pic 4]

Optimal Solution Results

It is concluded that irrespective of the used software or the existence constraints pertaining to integer values of decision variables, the Optimum Solution for the LP is as follows:

-x1 : the quantity of produced tables = 130

-x2 : the quantity of produced chairs = 130

-x3 : the quantity of produced desks = 30

-x4 : the quantity of produced libraries = 0 (no libraries will be produced)

The maximized profit will be 2,660 monetary units (or 2,660,000 drachmas)

Sensitivity Analysis

The task of Sensitivity Analysis will be conducted in accordance with the resulting script of the LINDO™ Software (denoted above by the title box “Pertains to Sensitivity Analysis”).

The above task constitutes of two (2) separate cases:

- Concerning the values of the coefficients of the Decision Variables in the Objective Function (ci), hence the range of variables of ci for which the current basis remains optimal.

- Concerning the values of the of the Right-Hand Side (RHS) coefficients ( bi ) and specifically the range of values within which the current basis remains optimal.

According to the obtained LINDO script pertaining to Sensitivity Analysis, the following apply:

- Objective Function Coefficient Ranges

12 - 3 ≤ c1 ≤ 12 + (+ ∞) => 9 ≤ c1 ≤ + ∞

5 – (+ ∞) ≤ c2 ≤ 5 + 3 => - ∞ ≤ c2 ≤ 8

15 – (+ ∞) ≤ c3 ≤ 15 + 5 => - ∞ ≤ c3 ≤ 20

10 – (+ ∞) ≤ c4 ≤ 10 + 30 => - ∞ ≤ c4 ≤ 40

- Right-Hand