The joe family has owned a $640-$acre farm for several generations. However, ownership has recently passed to the next generation of Dave and his wife Ruth, who feel there is a need for a more scientific approach to manage their farm. They have called you in as a business analyst to advise them on the best way of running the farm.The joe family can produce a total of $4,000$ person$-$hours$$ worth of labour during the winter and spring months and $4,500$ person$-$hours during the summer and autumn months. If they do not need all this time, they can work on neighbouring farms for $5$ per hour during the winter and spring months and $5\mathrm{.}50$ during the summer and autumn months.The farm supports two types of livestock, dairy cows, and hens and three types of cash crops $$ rape, corn, and wheat. All three are cash crops, but the corn also is a feed crop for the cows and the wheat also is used for chicken feed. The crops are harvested during the late summer and autumn.During the winter months, they need to decide what mix of livestock and crops they should aim for in the coming year. Their last harvest, which was very successful, has provided them with a sum of $21,000$ which they feel they can use to buy more livestock. The cost of seed is sufficiently small to be considered part of the routine expenses. The farm at present has $30$ cows valued at $35,000$ and $2,000$ hens valued at $5,000.$They wish to keep all this livestock and perhaps purchase more. Each cow bought would cost $2,000$ and each hen would cost $3.$ Over a year the value of a herd of cows decreases by about $10\%$ and the value of a flock of hens will decrease by about $25\%$ due to aging. Each cow requires $2$ acres of land for grazing and $8$ person$-$hours of work per month while producing an annual net cash income of $700$ for the family.Hens do not require any acreage but need $0\mathrm{.}05$ person$-$hours a month and produce an annual net cash income of $4\mathrm{.}50.$ The chicken house can accommodate a maximum of $5,000$ hens and the milking parlour puts a limit of $42$ on the number of cows. For each acre of crops planted, the following table gives the number of person hours of work required in the first and second halves of the year as well as an estimate of the crops net value in ither income or savings in livestock feed.Rape Corn Wheat

Winter$/$spring person hours $1\mathrm{.}00\mathrm{.}90\mathrm{.}6$

Summer$/$autumn person hours $1\mathrm{.}41\mathrm{.}20\mathrm{.}7$

Net value $706040$

To provide feed, there should be at least $1$ acre of corn for each cow and at least $0\mathrm{.}05$ acres of

wheat for each hen in the next year$$s flock. The Smith family estimates that they need $40,000$ each year for their living expenses and the basic running of the farm. You have been hired as a business analyst to prepare a report on several issues they are seeking advice on:

a$)$ Advise the joe family on how many livestock they should buy and what acreage of the different crops they should plant to maximise their net wealth at the end of the next year.

b$)$ They are interested in how much they should be willing to pay for any farmland that may become available to rent over the coming winter.

c$)$ They are concerned that the price of cows may not stay at $2,000$ but may vary between $1,000$ and $3,000.$ What are the strategies they need to adopt if the prices change in this range and what affect will it have on the net wealth?

d$)$ It is felt that Phil Bowman, the patriarch of the family cannot go on working much longer. This would cut their available labour to $3,000$ and $3,500$ hours in the winter$/$spring and summer$/$autumn$,$ respectively. What effect will this have on their plans, and should they seek to employ someone in his place? If so$,$ what sort of hours and wage levels should they offer?

e$)$ A bank is offering to lend them money at a special rate of $10\%$ to purchase more livestock beyond the $21,000$ they had set aside. Should they take advantage of this?

f$)$ They could build an extension to the milking parlour at a cost of $3,000$ for each extra pair of cows it can take. How large an extension if any extension at all should be built?

g$)$ The details on the above table for crops grown reflect the situation if there are no large$-$scale

frosts that delay the spring sowing. If there are late frosts, the resultant data would be:

Rape Corn Wheat

Winter$/$spring person hours $1\mathrm{.}21\mathrm{.}10\mathrm{.}8$

Summer$/$autumn person hours $1\mathrm{.}31\mathrm{.}10\mathrm{.}7$

Net value $505030$

The joe family are concerned about this alternative scenario. They want to know what they should be doing if this is the situation that will arise to maximise their net wealth. They also want to know what their net wealth would be if they continued using the policy suggested in a$)$ in this scenario. EXPLICITLY SHOW THE LP MODEL STATING THE DECISION VARIABLE, OBJECTIVE FUNCTION, AND CONSTRAINTS AND SOLVE ON EXCEL USING SOLVER.ALSO EXPLAIN IT APPROPRIATELY