Sunkissed Farms sells bags of fresh oranges and cartons of orange juice. Fruits produced in the farm are graded on a scale of 0 (poor) to 10 (excellent). Currently, the farm has 50,000 kilos of grade 9 oranges and 60,000 kilos of grade 6 oranges. The average quality of oranges to be sold fresh in bags must be at least 7 and the average quality of oranges used to produce orange juice must be at least grade 8. Each kilo of oranges used for juice yields sales of P75 with variable cost of P50. Each kilo of oranges sold in bags yields revenues of P25 with a variable cost of P10. Formulate the LP model that will help Sunkissed Farms maximize its profits.

MAX Z= P25X₁₁+P15X₁₂+P25X₂₁+P₁₅X₂₂

x₁₁+x₁₂<= 50,000(grade 9 oranges restrictions)

x₂₁+x₂₂<= 60,000(grade 6 oranges restrictions)

2x₁₁-x₂₁>= 0(bags quality restriction)

x₁₂-2x₂₂>= 0(juice quality restriction)

where x_ij>= 0 ; i= 1,2 ; j = 1,2

Step-by-step explanation

X_ij>=0

where i=1,2

            j= 1,2

let x11= grade 9 oranges sold in bags

let x₁₂ be grade 9 juice sold in cartons

let x₂₁= grade 6 orange sold in bags

let x₂₂= grade 6 oranges sold in cartons

therefore=>

total amount of oranges sold in bags is

x₁₁+x₁₂<=50,000(for grade 9 oranges)

x₂₁+x₂₂<=60,000(for grade 6 oranges)

the average quality of oranges sold in bags must be at least 7

therefore we formulate

(9x₁₁+6x₁₂)/(x₁₁+x₂₁)>=7

9x₁₁+6x₁₂>=7x₁₁+7x₁₂

=> 2x₁₁-x₂₁>=0

the average quality of oranges used in juice to at least 8

(9x₁₁+6x₁₂)/x₁₁+x₁₂>=8

9x₁₁+6x₁₂>=8x11+8x12

=> x₁₁-2x₂₁>=0

total amount of oranges sold= x₁₁+x₂₁

total amount of juice sold = x₁₂+x₂₂

profit=> revenue-cost

orange sold= p75-p50=p25

juice sold= p25-p10= p15

therefore orange sold = p25(x₁₁+x₂₁)= p25x₁₁+px₂₁

juice sold= p15(x₁₂+x₂₂)= p15x₁₂+p15x₂₂

hence max z= p25x₁₁+p15x₁₂+p25x₂₁+p15x₂₂