Question One

a$)$ Solve for X & Y using a matrix method of your choice

$5$X $+3$Y $=71$

$4$X $+4$Y $=76[6$ marks$]$

b$)$ The total revenue equation for Mtwapa Ltd is given by: TR $=3,200$X $9$X$2.$ The marginal cost equation is given by: MC $=2$X $600.$ The total fixed costs amount to sh $124,000$ and X represents the number of units

Required

Determine the quantity of output that maximizes profit and the maximum profit $[6$ marks$]$

c$)$ Customers arrive at a sales counter at an average rate of $20$ customers per hour. The arrival rate is random and may be assumed to follow a Poisson distribution. The time required to serve each customer has an exponential distribution with a mean of $2$ minutes.

Required:

$($i$)$ The probability that the server is idle $[2$ marks$]$

$($ii$)$ The average waiting time in the system $[2$ marks$]$

$($iii$)$ The average number of customers in the queue $[2$ marks$]$

d$)$ Solve the game theory problem below & advice the strategies to be adopted by each of the players and the value of the game $[6$ marks$]$

Player a

A$1$ A$2$ A$3$

Player

B B$15-68$

B$2374$

B$39610$

e$)$ There are two competitors, X & Y$.$ In a given quarter, the probability of X losing customers to Y is $0\mathrm{.}3,$ whereas the probability of Y losing customers to X is $0\mathrm{.}2.$

Required

$($i$)$ The transition matrix

$($ii$)$ The long run market share for each of the competitors

$[6$ marks$]$

Question Two

A small project consisting of eight activities has the following characteristics. Time estimates are in weeks:

Activity Predecessor Optimistic $($a$)$ Most likely $($m$)$ Pessimistic $($b$)$

A None $2412$

B None $101226$

C A $8910$

D A $101520$

E A $77\mathrm{.}511$

F B$,$C $999$

G D $33\mathrm{.}57$

H E$,$F$,$G $555$

Required

a$)$ Draw the PERT network for the project, clearly indicating the expected duration of each activity

b$)$ Determine the critical path and the expected project duration

c$)$ If a $30-$ week deadline is imposed, what is the probability that the project will be finished within the time limit$?[15$ marks$]$

Question Three

A product is produced in $4$ plants $($P$1,$ P$2,$ P$3$ & P$4)$ and shipped to three warehouses $($W$1,$ W$2$ & W$3).$ The transportation costs per unit are given in the table below:

Warehouses

W$1$ W$2$ W$3$ Capacity

P$1201624300$

P$211108400$

P$3121810200$

P$491521100$

Demand $290390320$

Required

a$)$ Generate an initial transportation schedule using the Vogel$$s Approximation method and the related costs

b$)$ Evaluate whether the solution in $($a$)$ above is optimal $[15$ marks$]$