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2. A manufacturer of homogencous hotel art produces four types of paintings: a lone horse, a horse with a rider, a lonely mountain-scape, and a

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2. A manufacturer of homogencous hotel art produces four types of paintings: a lone horse, a horse with a rider, a lonely mountain-scape, and a mountains-cape with a horse. Painting type Horse alone Horse and rider Paint required per piece (in pints) Red Blue Yellow White Retail Price 0.3 0. 0.5 0.1 0.4 0.3 0.3 0.2 0.2 0.4 0.4 S3.50 S5.50 S5.00 Mountai with horse 0.6 0.3 If the manufacturer has only 10 pints of each color of paint, what is the maximum number of paintings that can be produced? a) b) Assuming paint is paid for in advance of production (i.e, a fixed cost) and all other inputs (e.g, cardboard canvases and labor) are free, what is the maximum profit the manufacturer can earn if the costs of paint are S3 per pint for red, blue, and yellow, but S2 per pint for white? [Hint: You will need a new objective function.] How would your answer to part (b) change if the manufacturer had to produce at least two of each type of painting? [Hint: You will need new constraints.] c) A small trucking company is determining the composition of its next trucking job. The load master has her choice of seven different types of cargo, which may be loaded in full or in part. The specifications of the cargo types are shown in the following table. Cargo Type Freight value Volume per pound Pounds available S8.00 S6.00 $3.50 S5.75 9.50 3.0 2.7 210 150 8.4 5.5 120 130 S5.25 60 4.9 340 The goal is to maximize the amount of freight, in terms of dollars, for the trip. The truck can hold up to 900 pounds of cargo in a 2,500-cubic-foot space. a) What cargo should be loaded, and what will be the total freight charged? b) Now consider that two additional smaller trucks can be added: one that can carry 400 pounds in a 500-cubic-foot space and another that can carry 800 pounds in a 1,500-cubic- foot space. The first additional truck costs $440 to hire and the second truck costs $850 to hire. Is it a profitable decision to hire either or both of the additional trucks? Why or why not?Hint: This will require additional models.] 4. Palouse Grains & Legumes, Inc. (PGL) is trying to determine its crop rotation schedule for the next four years. Despite the corporate-sounding name, PGL is a 160-acre family farm. The crops available to plant are chickpeas, lentils, spring and winter wheat (combined for convenience), and alfalfa. The following table provides average yields and prices Average yield (units/acre 1,750 lbs. 1,500 lbs. 129 bushels 170 bushels Average price (per unit $0.38/lb. S0.39/lb $4.38/bushel $3.90/bushel ro Chickpeas Lentils Wheat, combined Alfalfa Under the most simplistic interpretation, PGL would just choose the most profitable crop and plant t every year. However, agricultural decisions are far more complicated than that. In order to ensure long-term soil quality, etc., farms must allow each acre to go fallow (unplanted) every few years (in PGL's case, every fourth year for each acre). In other words, both crop types and fallow years must be included in the crop rotation schedule. PGL must rotate crops so that the same crop is never planted in two consecutive years on a given acre Additionally to reduce risk and smooth consumption, PGL would like to earn a similar profit each year (so they can't let all acres go fallow in a single year, even if that would be the most profitable long-run option). There is some uncertainty associated with each crop's production, but we will ignore that for now. Using the information provided, answer the owing questi a) What are the decision variables in this problem? Explain them carefully. [Hint: There can be more than 20 depending on how you set up the problem, but you may be able to explain them in groups.] b) Write out the objective function. c) Explain the constraints you would need (in words, at least) d) Can we solve this problem using the linear programming approach? Explain your

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