Question

Please submit your solutions as ONE single workbook with the sheets clearly labelled to indicate the corresponding problem numbers as indicated on this test.

• For each problem, you are required to use a textbox to show the definition of your decision variables. You may include this textbox only with part a) of the problem if the following parts do not require additional variables to reformulate.

• If a problem requires sensitivity analysis, re-order the worksheets so that they appear in the order of the questions.

• Answers to the questions should be clearly described in a separate textbox of the corresponding labelled worksheets. Since you will only be credited with problems that have correct answers, please check carefully your report textbox.

• For each problem formulation, please highlight the decision variables in Yellow, the constriants (Left Hand Side) in Blue, and the Objective function in Red.

4)​The relevant project data are given as follows.

Activity	Predecessor(s)	Normal time (weeks)	Crash time	Normal cost	Crash cost	Possible number of weeks to crash	Cost/week to expedite
A	--	7	6	$7,000	$8,000
B	A	2	1	5,000	7,000
C	A	4	3	9,000	10,200
D	B,C	5	4	3,000	4,500
E	D	2	1	2,000	3,000
F	D	4	2	4,000	7,000
G	E,F	5	4	5,000	8,000

a) ​Draw the AOA (Activity-On-Arc) diagram as shown in Chapter 13's Excel Worksheets.

b) ​Formulate the problem of finding project completion time as a LP problem. [You would only need the first 3 columns of the table and the diagram from a) to do the job!]

c) ​Reformulate the problem when crashing the project completion time of 3 weeks is required. That is, reformulate the problem when the project completion time is required to be shorted by 3 weeks. Obviously, crashing is based on shortening the project completion time by 3 weeks with the minimal additional cost. In order to formulate the problem, you need to fill in the blank columns of the table.]

Answer 1

b) LP formulation to find the project completion time is following:

Let each alphabet denote the earliest completion time of respective activity. and T be the final project completion time.

Minimize T

s.t.

A >= 7

B - A >= 2

C - A >= 4

D - B >= 5

D - C >= 5

E - D >= 2

F - D >= 4

G - E >= 5

G - F >= 5

T - G >= 0

A, B, C, D, E, F, G, T >= 0

c) The completed table is following:

c) LP formulation to find the lowest cost to shorten the project completion time by 3 weeks is following:

Let each alphabet denote the earliest completion time of respective activity. and T be the final project completion time.

and, Xa, Xb, Xc, .... Xg be the number of weeks that respective is crashed by

Minimize 1000Xa+2000Xb+1200Xc+1500Xd+1000Xe+1500Xf+3000Xg

s.t.

A+Xa >= 7

B - A + Xb >= 2

C - A + Xc >= 4

D - B + Xd >= 5

D - C + Xd >= 5

E - D + Xe >= 2

F - D + Xf >= 4

G - E + Xg >= 5

G - F + Xg >= 5

T - G >= 3

Xa <= 1

Xb <= 1

Xc <= 1

Xd <= 1

Xe <= 1

Xf <= 2

Xg <= 1

A, B, C, D, E, F, G, T, Xa, Xb, Xc, Xd, Xe, Xf, Xg >= 0