OUESTION $4[$Total: $22$ marks$]$

Joe has a building project that consists of nine activities. The activities that must be

accomplished in order to complete the project are listed as follows:

a$)$ Construct a network diagram for this problem.

$[6$ marks$]$

b$)$ Determine the expected time, variance, ES$,$ EF$,$ LS$,$ LF and slack for each activity.

$[9$ marks$]$

c$)$ Determine the critical path to complete the building project.

$[1$ marks$]$

d$)$ What is the expected completion time of the building project?

$[1$ marks$]$

e$)$ Calculate the variance of the building project.

$[2$ marks$]$

f$)$ What is the probability that the project can be completed in $60$ weeks or less?

$[3$ marks$]$A manufacturer produces three types of detergents, Ky$,$ Lo and Mi$.$ Each type of detergent

is produced using two machines, machine A and machine $B.$ The following table gives the

information on the usage of the machines.

Due to the cost and manpower relationship, machine A must be used for at most $180$ hours

while the maximum capacity for machine B is $240$ hours. The profit obtained from each

unit of detergent Ky is RM$53,$ from each unit of detergent Lo is RM$62$ and from each unit

of detergent Mi is RM$75.$ The manufacturer wishes to know the quantity of each type of

detergent to be produced in order to maximize its profit.

a$)$ Formulate the problem above as a Linear Programming Model.

b$)$ Set up the initial simplex tableau by including the necessary slack variables. $[$Do

not solve until the final simplex table$]$

The optimal tableau for the above problem is given below, where $x_1$ is Detergent Ky$,x_2$ is

Detergent Lo and $x_3$ is Detergent $Mi,S_1$ and $S_2$ are the slack variables for usage of Machine

A and Machine B$,$ respectively.

c$)$ State the optimal solution and the maximum profit.

d$)$ Which of the constraint$($s$)$ are binding? Why?

e$)$ If the cost required for an additional work hour for Machine B is RM$4\mathrm{.}50$ per hour,

is it worth acquiring additional units of this resource? Explain your answer.