Problem 3. [30 points] LIMECO owns a lime factory and sells six grades of lime (

Question

Problem 3. [30 points] LIMECO owns a lime factory and sells six grades of lime (grades 1 through 6). The sales price per pound is given in the table below Lime is produced by kilns. Each time a kiln is run, it produces multiple grades of lime (2 lb of grade 1 and 3 lb of grade 2 and). It costs $150 each time a kiln is run Lime that is produced by the kiln may be reprocessed by using any one of the five processes described in the table below. For example, at a cost of $1/lb, a pound of grade 4 lime may be transformed into 0.5 lb of grade 5 lime and 0.5 lb of grade 6 lime Each day the factory believes it can sell up to the amounts (in pounds) of lime given below Any extra lime leftover at the end of each day must be disposed of, with the disposal costs (per pound) given below Formulate an LP whose solution will tell LIMECO how to maximize their daily profit. Clearly define in words what your decision variables represent. In the LP, label each constraint with a keyword for what the constraint corresponds to. You do not need to solve the LP. Do not worry about integrality. Do not worry about the time required for production/reprocessing. Grade 12 141018 20 25 Amount produced 2312 23 Max demand | 20 | 30 | 40 | 35 | 25 | 50 Disposal cost3232 42 Sales price Input (1 Ib Cost ( per pound of input) Output 3 lb Grade 3 2 lb Grade 4 3 lb Grade 5 2 lb Grade 6 Grade 1 Grade 2 1 lb Grade 6 Grade 3 8 lb Grade 4 5 lb Grade 5 5 lb Grade 6 Grade 4 Grade 5 9 lb Grade 6

Explanation / Answer

LP model is following:

Decision variables:

Let N be the number of times the kiln is run, and

D1, D2, D3, D4, D5, D6 be the quantity (lbs) of each of the six grades to be disposed, and

S1, S2, S3, S4, S5, S6 be the total quantity (lbs) of each of the six grades to be sold, and

R1, R2, R3, R4, R5 be the quantity (lbs) of each of the five grades to be reprocessed.

Objective function: MAX 12S1+14S2+10S3+18S4+20S5+25S6-(150N+3D1+2D2+3D3+2D4+4D5+2D6+2R1+R2+R3+R4+2R5)

Constraints:

S1 <= 20

S2 <= 30

S3 <= 40

S4 <= 35

S5 <= 25

S6 <= 50

S1+R1+D1 = 2N

S2+R2+D2 = 3N

S3+R3+D3 = 1N+0.3R1

S4+R4+D4 = 2N+0.2R1+0.8R3

S5+R5+D5 = 2N+0.3R1+0.5R4

S6+D6 = 3N+0.2R1+1R2+0.5R4+0.9R5

All variables >= 0

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