QUESTION ONE $[20$ MARKS$]$

A street vendor in Nairobi City sells hot dogs and hamburgers. A hot dog gives a profit of Sh$\mathrm{.}0\mathrm{.}60$ and a hamburger gives a profit of Sh$\mathrm{.}0\mathrm{.}50.$ Each hot dog uses Sh$\mathrm{.}1\mathrm{.}20$ worth of materials and each hamburger uses Sh$\mathrm{.}1\mathrm{.}60$ worth. There is only Sh$\mathrm{.}48$ worth of materials currently on hand to make hot dogs and hamburgers. From previous experience the vendor knows that he should make at least $8$ hamburgers and no more than $20$ hot dogs. He also believes that he should make at most twice as many hot dogs as hamburgers.

Solve the problem graphically, stating optimal values of the decision variables. What is the maximum profit? $[15$ Marks$]$

How low would the profit for hot dogs have to drop before the vendor would stop making them? $[2$ Marks$]$

The vendor remembers that he omitted the constraint that no more than $12$ hamburgers should be produced, all other constraints remaining the same. With this new constraint in the model, what would the optimal solution be$?$ What profit would the vendor make now? $[3$ Marks$]$

QUESTION TWO $[20$ Marks$]$

Manufacturer has two factories, located in Toronto and Winnipeg. Warehouses are located in Moncton, Montreal, and Calgary. Minimum requirements at the three warehouses are for at least $5000,8000,$ and $7000$ units, respectively. The production capacities of the two factories are $9000$ and $13000$ units, respectively. Shipping costs per unit from each factory to each warehouse are given below:

WAREHOUSE

Moncton

Montreal

Calgary

Toronto

Sh$\mathrm{.}20$

Sh$\mathrm{.}16$

Sh$\mathrm{.}24$

FACTORY

Winnipeg

Sh$\mathrm{.}29$

Sh$\mathrm{.}6$

Sh$\mathrm{.}9$

Required

The manufacturer wants to determine the optimal shipping schedule that will minimize transportation costs.

Required

Formulate the linear programming model for this problem. $[4$ Marks$]$

Generate initial solution using Vogels Approximation Method and the Least Cost Method $[6$ Marks$]$

Test the solution for optimality generated using the stepping stone Method $[10$ Marks$]$

QUESTION THREE $[10$ Marks$]$

You are managing a hockey tournament and need a game official for each of the four games. All the games occur at the same time in four different cities. The cities are Migori County $,$ Lamu County $,$ Kakamega and Kericho County $.$ The four game officials live in different parts of Kenya and will need to travel to the games and submit a travel distance in Kilometers according to the table below. The assignment is one official to one game location.

Game Locations

Official

Migori County

Lamu County

Kakamega County

Kericho County

$1$

$70$

$165$

$185$

$215$

$2$

$75$

$200$

$140$

$105$

$3$

$110$

$240$

$130$

$165$

$4$

$55$

$122$

$110$

$79$

Required

Formulate this assignment question as an LP question to minimize cost to send the officials to their games. $[2$ marks$]$

Solve the problem to determine the optimal assignment plan. $[2$ marks$]$

Suppose that Official $4$ Cannot be assigned to Migori County because that is his home county? How will this change the assignment plan. $[2$ marks$]$

Solve the problem to determine the optimal assignment plan. $[2$ marks$]$

Suppose they have recruited a new official $5$ with the following travel Kilometers. How will this change the assignment plan? $[2$ marks$]$

Official

Migori County

Lamu County

Kakamega County

Kericho County

$5$

$170$

$65$

$85$

$200$

QUESTION FOUR

Top Cutz International Barbershop is a popular haircutting and styling salon near the campus of the KCA University, Ruaraka. Four barbers work full$-$time and spend an average of $15$ minutes on each customer. Customers arrive all day long at an average rate of $12$ per hour. When they enter, they take a number to wait for the first available barber. Arrivals tend to follow the Poisson distribution, and service times are exponentially distributed.

REQUIRED

$($a$)$ What is the probability that the shop is empty? $[1$Mark$]$

$($b$)$ What is the average number of customers in the barbershop? $[2$Marks$]$

$($c$)$ What is the average time spent in the shop? $[2$Marks$]$

$($d$)$ What is the average time that a customer spends waiting to be called to the barber chair? $[2$Marks$]$

$($e$)$ What is the average number waiting to be served? $[1$Mark$]$

$($f$)$ What is the shop's utilization factor? $[1$Mark$]$

$($g$)$ Sal's is thinking of adding a fifth barber. How will this affect the utilization rate? $[1$Mark$]$

QUESTION FIVE

The Alfredo Fragrance Company produces only one product, a perfume called Hint of Elegance. Hint of Elegance consists of two secret ingredients blended into an exclusive fragrance, which is marketed in Zurich. An economic expression referred to as the Cobb$-$Douglas function describes the production of Hint of Elegance as follows:

Where X is the amount of perfume produced.

The company operates at a level where ingredient $1$ is set daily at $25$ units and ingredient $2$ at $36$ units. Although the price Alfredo pays for ingredient $1$ is fixed at $$50$ per unit, the cost of ingredient $2$ and the selling price for the final perfume are both probabilistic. The sales price for Hint of Elegance follows this distribution:

SALES PRICE $($$$)$

PROBABILITY

$300$

$0\mathrm{.}2$

$350$

$0\mathrm{.}5$

$400$

$0\mathrm{.}3$

The cost for ingredient $2$ is

Ingredient $2$ Cost $($$$)$

Probability

$35$

$0\mathrm{.}1$

$40$

$0\mathrm{.}6$

$45$

$0\mathrm{.}3$

$($a$)$ What is the profit equation for Alfredo Fragrance Company?

$($b$)$ What is the expected profit to the firm?

$($c$)$ Simulate the firm's profit for a period of nine days, using these random numbers from Table $14\mathrm{.}5$:

$52,06,50,88,88,53,30,10,47,99,37,66,91,35,32,00,84,57,07.$

$($d$)$ What is the expected daily profit as simulated in part $($c$)?[10$ Marks $]$