Zales Jeweller uses rubies and sapphires to produce two types of rings. A type $1$ ring requires two

rubies, three sapphires and one hour of jeweller$$s labour time. A type $2$ ring requires three rubies,

two sapphires and two hours of jeweller$$s labour time. Each type $1$ ring sells for R$1200$ and each

type $2$ ring sells for R$1500.$ All rings produced by Zales can be sold. At present, Zales has $100$

rubies, $120$ sapphires and $70$ hours of jeweller$$s labour time. Extra rubies can be bought at R$300$

per ruby. Market demand requires at least $20$ type $1$ rings and $24$ type $2$ rings. The linear programming $($LP$)$ model representing this problem is formulated as follows:

where the decision variables used are defined as follows:

o RING$1=$ the number of rings of type $1$ to make;

o RING$2=$ the number of rings of type $2$ to make;

o RUBY $=$ the number of extra rubies to buy.

Max PROFIT $=1200$ RING$1+1500$ RING$2300$ RUBY

subject to

$($RUBIES$)2$ RING$1+3$ RING$2$ RUBY $100$

$($SAPPHIRE$)3$ RING$1+2$ RING$2120$

$($LABOUR$)$ RING$1+2$ RING$270$

$($DEM$1)$ RING$120$

$($DEM$2)$ RING$224$

and all variables $0$

PBA$4804$

OCT$/$NOV $2023$ EXAMINATION

$5$

The LP model is solved using SOLVER and the ANSWER $($Figure $1)$ and SENSITIVITY $($Figure $2)$

Reports are provided below:

Microsoft Excel $16\mathrm{.}0$ Answer Report

Objective Cell $($Max$)$

Cell Name

Original

Value Final Value

$I$$13$ PROFIT RING $1057600$

Variable Cells

Cell Name

Original

Value Final Value Integer

$I$$6$ VALUES RING $1022$ Contin

$J$$6$ VALUES RING $2024$ Contin

$K$$6$ VALUES RUBY $016$ Contin

Constraints

Cell Name Cell Value Formula Status Slack

$I$$10$ MARKET CONDITION RING $122$ $I$$10>=$$B$$10$

Not

Binding $2$

$J$$10$ MARKET CONDITION RING $224$ $J$$10>=$$C$$10$ Binding $0$

$L$$7$ RUBIES UTILISATION $100$ $L$$7<=$$E$$7$ Binding $0$

$L$$8$ SAPHIRES UTILISATION $114$ $L$$8<=$$E$$8$

Not

Binding $6$

$L$$9$ LABOUR UTILISATION $70$ $L$$9<=$$E$$9$ Binding $0$

Figure $1$: ANSWER ReporAnswer the following questions by using only the ANSWER and SENSITIVITY reports provided.

$($This means that you may not change the values of the relevant parameters in the model and do

reruns with SOLVER.$)$ Explain how you arrive at your answers.

$($a$)$ Write down the optimal solution. $(5)$

$($b$)$ Suppose that instead of R$300,$ each ruby costs R$750.$ Will Zales still buy extra rubies? If

so$,$ will the production plan and profit change?

$(3)$

$($c$)$ Suppose that Zales was required to produce at least $23$ type $2$ rings. What would Zales$$

profit now be$?$

$(5)$

$($d$)$ What is the most that Zales will be willing to pay for another hour of jeweller$$s labour time?

How many hours may be obtained at this price?