1. For the Reddy Mikks model, construct each of the following constraints, and express it with a linear left-hand side and a constant right-hand side:
*(a) The daily demand for interior paint exceeds that of exterior paint by at least 1 ton. (b) The daily usage of raw material M2 in tons is at most 6 and at least 3.
*(c) The demand for interior paint cannot be less than the demand for exterior paint. (d) The minimum quantity that should be produced of both the interior and the exterior paint is 3 tons.
*(e) The proportion of interior paint to the total production of both interior and 'exterior paints must not exceed .5.
2. Suppose that Reddy Mikks sells its exterior paint to a single wholesaler at a quantity discount. The profit per ton is $5000 if the contractor buys no more than 2 tons daily and $4500 otherwise. Express the objective function mathematically. Is the resulting function linear?
3. Determine the feasible space for each of the following independent constraints, given that xl, x2 ≥ 0.
*(a) -3x, + x2 ≥ 6
(b) x1 - 2x2 ≥ 5
(c) 2x1- 3x2 ≤ 12
(d) xl - x2 ≤ 0
*(e) -xi + x2 ≥ 0
4. Identify the direction of increase in z in each of the following cases:
*(a) Maximize z = x1 - x2
(b) Maximize z = -5x1 - 6x2
(c) Maximize z = - x1 + 2x2
*(d) Maximize z =-3x1+ x2
5. In the Ma-and-Pa grocery store, shelf space is limited and must be used effectively to increase profit. Two cereal items, Grano and Wheatie, compete for a total shelf space of 60 ft2. A box of Grano occupies .2 ft2 and a box of Wheatie needs .4 ft2. The maximum daily demands of Grano and Wheatie are 200 and 120 boxes, respectively. A box of Grano nets $1.00 in profit and a box of Wheatie $1.35. Ma-and-Pa thinks that because the unit profit of Wheatie is 35% higher than that of Grano, Wheatie should be allocated 35% more space than Grano, which amounts to allocating about 57% to Wheatie and 43% to Grano. What do you think?
6. Show & Sell can advertise its products on local radio and television (TV). The advertising budget is limited to $10,000 a month. Each minute of radio advertising costs $15 and each minute of TV commercials $300. Show & Sell likes to advertise on radio at least twice as much as on TV. In the meantime, it is not practical to use more than 400 minutes of radio advertising a month. From past experience, advertising on TV is estimated to be 25 times as effective as on radio. Determine the optimum allocation of the budget to radio and TV advertising.
7. Investco wants to invest money that would generate an annual yield of at least $20,000. Two stock groups are available: bank stocks and tech stocks, with average annual yields of 15% and 25%, respectively. Though tech-stock provide a higher yield, they are more risky, and Investco wants to limit the amount invested in these stocks to no more than 75% of the total investment. What is the minimum amount of Investco should invest in each stock group to accomplish this investment goal?