A company produces tools at two plants and sells them to three customers. The cost of producing

a hundred tools at a plant and shipping them in boxes of hundred to a customer is given in the

following table:

Customer $1$ Customer $2$ Customer $3$

Plant $16003001600$

Plant $213007001700$

Customers $1$ and $3$ pay R$2000$ per hundred tools and Customer $2$ pays R$1500$ per hundred tools.

They need $200$ hours of labour time to produce $100$ tools at Plant $1,$ while $300$ hours are needed

at Plant $2.$ A total of $5500$ hours of labour are available at the two plants. Plant $1$ can produce

up to $10000$ tools and Plant $2$ can produce up to $12000$ tools.

The LP model corresponding to this problem is:

Max Profit

$=1400$T$11+1200$T$12+400$T$13+700$T$21+800$T$22+300$T$23$

subject to

T$11+$ T$12+$ T$13<=10$

T$21+$ T$22+$ T$23<=12$

$200$T$11+200$T$12+200$T$13+300$T$21+300$T$22+300$T$23<=5500$

All variables $>=0.$ Where

Tij denotes the number of tools produced at Plant i and shipped to Customer j$,$ i $=1,2$ and

j $=1,2,3.$

The LINDO printout of the solution to this model is given in Printout. Use the printout to answer

the following questions:

LP OPTIMUM FOUND AT STEP $2$

OBJECTIVE FUNCTION VALUE

$1)23333\mathrm{.}33$

VARIABLE VALUE REDUCED COST

T$1110\mathrm{.}0000000\mathrm{.}000000$

T$120\mathrm{.}000000200\mathrm{.}000000$

T$130\mathrm{.}0000001000\mathrm{.}000000$ T$210\mathrm{.}000000100\mathrm{.}000000$

T$2211\mathrm{.}6666670\mathrm{.}000000$

T$230\mathrm{.}000000500\mathrm{.}000000$

ROW SLACK OR SURPLUS DUAL PRICES

$2)0\mathrm{.}000000866\mathrm{.}666687$

$3)0\mathrm{.}3333330\mathrm{.}000000$

$4)0\mathrm{.}0000002\mathrm{.}666667$

NO$.$ ITERATIONS$=2$

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE

T$111400\mathrm{.}000000$ INFINITY $200\mathrm{.}000061$

T$121200\mathrm{.}000000200\mathrm{.}000031$ INFINITY

T$13400\mathrm{.}0000001000\mathrm{.}000061$ INFINITY

T$21700\mathrm{.}000000100\mathrm{.}000023$ INFINITY

T$22800\mathrm{.}0000001300\mathrm{.}000000100\mathrm{.}000000$

T$23300\mathrm{.}000000500\mathrm{.}000031$ INFINITY RIGHTHAND SIDE RANGES

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

$210\mathrm{.}00000017\mathrm{.}5000000\mathrm{.}500000$

$312\mathrm{.}000000$ INFINITY $0\mathrm{.}333333$

$45500\mathrm{.}000000100\mathrm{.}0000003500\mathrm{.}000000$

$(4\mathrm{.}1)$ Suppose the cost of producing $100$ tools at Plant $1$ and shipping them to Customer $1$ now

changes to R$700.$ What would the new solution to the problem be$?$ Substantiate your answer.

$(5)$

$(4\mathrm{.}2)$ A consultant offers to increase the capacity of Plant $1$ by $5000$ tools at a cost of R$4000.$

Should the company accept this offer? Substantiate your answer. $(4)$

$(4\mathrm{.}3)$ Suppose the profit contribution of sending tools from Plant $2$ to Customer $1$ increases to

R$850.$ What would the influence be on the solution. $(3)$

$(4\mathrm{.}4)$ Give the range in which the labour hours can move without changing the basis of the optimal

solution.