QUESTION $1$

A production scheduler must develop an aggregate plan for the first two quarters of next year. The

highly automated plant produces graphics terminals for the computer products market. The

company estimates that $700$ terminals will need to be shipped to customers in the first quarter and

$3200$ in the second quarter. It takes an average of $5$ hours of labour to produce each terminal and

only $9000$ hours of straight labour are available. Overtime can be used, but the company has a

policy of limiting the amount of overtime in each quarter to $10$ percent of the straight time labour

available. Labour costs R$120$ per hour at the straight$-$line rate and R$180$ per hour at the overtime

rate. If a terminal is produced in one quarter and shipped in the next quarter, a carrying cost of

R$500$ is incurred. The objective is to determine how many terminals should be produced on

straight$-$line and overtime in each of first and second quarter to minimise straight$-$time labour,

overtime labour and carrying costs.

If the decision variables are defined as follows:Let

$*$ X$1=$ terminals made on straight time in first quarter and shipped in first quarter.

$*$ X$2=$ terminals made on overtime in first quarter and shipped in the first quarter.

$*$ X$3=$ terminals made on straight time in first quarter and shipped in the second

quarter.

$*$ X$4=$ terminals made on overtime in first quarter and shipped in the second

quarter.

$*$ X$5=$ terminals made on straight time in second quarter and shipped in the second

quarter.

$*$ X$6=$ terminals made on overtime in second quarter and shipped in the second

quarter.

$*$ Z $=$ total cost of the plan

$*$ Q$1=$ refers to first quarter and Q$2=$ refers to second quarter.a$)$ Formulate this aggregate planning problem as a linear program. $(15$ marks$)$

b$)$ Solve this model using SOLVER. $(10$ marks$)$

c$)$ Write down the optimal solution and the associated total costs. $(7$ marks$)$

Use ONLY the initial printout of the optimal solution $($the ANSWER and SENSITIVITY reports$)$ to

answer the following questions. $($This means that you may not change the relevant parameters in

the model and do re$-$runs.$)$ Explain how you arrive at your answers.

d$)$ Determine the optimal solution and total costs if the amount of straight labour time available

in quarter $2$ is

$($i$)8750$ hours; $(2$ marks$)$

$($ii$)8500$ hours; $(2$ marks$)$

e$)$ What would the total costs be if amount of straight labour costs in Quarter $1$ is R$130?$

$(2$ marks$)$

f$)$ Under what circumstances will it be possible to use all the available straight$-$ labour time for

the second quarter.

$(2$ marks$)$