1. A mineral exploitation company is considering the purchase of a parcel of land for $3M. The land
may contain deposits of copper, silver, or gold. Only one of the metals is ever present at any site. If
the company buys the land it will spend $1M on exploration for the metals. In the present state of
knowledge the exploration has a 1% chance of nding copper, a 0.2% chance of nding silver, and a
0.05% of nding gold.
The company plans to sell the land on. If copper is found the selling price would be $30M, if silver is
found the price would be $150M, and if gold is found the price would be $250M.
Alternatively, before purchase the company can buy access to the parcel for three days at a cost
$750,000. In these three days the company will spend $250,000 on exploratory drilling. The drilling
has a 50% chance of indicating the presence of signi cant metal deposits and a 50% chance of a null
result. When the test indicates signi cant deposits the probabilities of the presence of the metals
increase to copper 3%, silver 1%, and gold 2%. Otherwise the chances of nding the metals reduce to
copper 0.75%, silver 0.175%, and gold 0.04%.
Construct a decision tree for this problem and advise the company which is the best course of action.
2. A bank makes four kinds of loans to its personal customers and these loans yield the following annual
interest rates to the bank:
First mortgage 14%;
Second mortgage 20%;
Home improvement 20%;
Personal overdraft 10%.
The bank has a maximum lending capability of $250 million and is further constrained by the policies:
rst mortgages must be at least 55% of all mortgages issued;
rst mortgages must be at least 25% of all loans issued;
second mortgages cannot exceed 25% of all loans issued;
legal requirements impose a maximum average interest on the whole loan portfolio of 15%.
Formulate the bank's loan problem as a linear programming problem so as to maximize interest income
whilst satisfying the policy limitations.
Solve the linear programming problem. (There are several solutions all of which give the same value.)
3.5. Consider the primal problem below
maximize z =18 y1 + 30 y2 + 27 y3 :
such that 3 y1 + y2 + 2 y3 小于等于 11
5 y1 + 9 y2 + 7 y3 小于等于44
Construct the dual problem and solve it graphically.
By considering which constraints are active use the duality theorem to determine the a solution of the
primal problem and con rm the optimality of the solutions.
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