Question 1 (10 marks)
A bank is in the process of devising a loan policy that involves a maximum of 120 million rands. The information in Table 1 provides the pertinent data about available loans.
• Bad debts are unrecoverable and produce no interest revenue.
Competition with other financial institutions dictates the allocation of at least 40% of the funds to farm and commercial loans.
To assist the housing industry in the country, home loans must equal at least 50% of the personal, car and home loans.
• The bank limits the overall ratio of bad debts on all loans to at most 4%.
Table 1: Information on different types of loans
Type of loan Interest rate Bad-debt ratio
Personal 0.140 0.10
Car 0.130 0.07
Home 0.120 0.07
Farm 0.125 0.05
Commercial 0.100 0.02
a) Formulate the Linear program (LP) to help find an optimal solution to the problem.. [5]
b) Use LINDO to determine an optimum solution to the problem. (You must submit the LINDO output in conjunction with a brief explanation of what the output entails.) [5]]
Question 2 (10 marks)
Suppose The Department of Home affairs has summarized the minimum numbers of immigration officers needed at Beit Bridge (BB) Border post over each 4-hour period everyday in March in Table 2
Each immigration officer works a shift that lasts 8 hours and the shifts start at 12 midnight, 4am, 8am, 12 noon, 4pm and 8pm.
Table 2: Numbers of immigration officers needed at BB in March
Time period Minimum number of officers required
12midnight-4am 80
4am-8am 70
8am-12noon 40
12noon-4pm 30
4pm-8pm 40
8pm-12 midnight 90
a) Formulate the Linear program (LP) to help find an optimal solution to the problem. [5]
b) Use LINDO to determine an optimum solution to the problem. (You must submit the LINDO output in conjunction with a brief explanation of what the output entails.) [5]]
Question 3 (10 marks)
A cell-phone manufacturing company must determine how many cellphones should be produced over the next three years. They anticipate that demand in the first year will be 30000 units, in the second year it will be 70000 units and in the third year it will be 40000 units. The company must meet demand in time. At the beginning of the first quarter, they have 10000 units in stock. The company can produce a maximum of 40000 units per annum in regular time at a total cost of R1,000,000 per every batch of 1000 cellphones. With the provision of workers working overtime, the company can produce a batch of 1000 cellphones at a cost of R1,200,000.
At the end of each year (after production has occurred and the current year’s demand has been satisfied), a carrying or holding cost of R50,000 per batch of 1000 cellphones is incurred.
a) Formulate the Linear program (LP) to help find an optimal solution to the problem.. [5]
b) Use MATHEMATICA to determine an optimum solution to the problem. (You must submit the LINDO output in conjunction with a brief explanation of what the output entails.) [5]
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