Question

3) The seasonal yield of olives in a Piraeus, Greece, vineyard is greatly influenced by a process of branch pruning. If olive trees are pruned every two weeks, output is increased. The pruning process, however, requires considerably more labor than permitting the olives to grow on their own and results in a smaller size olive. It also permits olive trees to be spaced closer together. The yield of 1 barrel of olives by pruning requires 5 hours of labor and 1 acre of land. The production of a barrel of olives by the normal process requires only 2 labor hours but takes two acres of land. An olive grower has 250 hours of labor available and a total of 150 acres for growing. Because of the olive size difference, a barrel of olives produced on pruned trees sells for $20, whereas a barrel of regular olives has a market price of $30. The grower has determined that because of uncertain demand, no more than 40 barrels of pruned olives should be produced. Use graphical LP (corner point) to find a) The maximum possible profit. b) The best combination of barrels of pruned and regular olives. c) The number of acres that the olive grower should devote to each growing process.
please write clearly n show work

Answer 1

ANSWER

As per the given question,

(a) Using X to denote the number of barrels of olives by pruning, and X to denote the number of barrels of olives by normal process Inserting the values, then the LP problem is given by Maximum profit $20X+$30X Subject to SX+2X2250 X+2 150 40 Хх, 20 Using isoprofit line graphical method, first we derive the graph with constraints and feasible region. The blue line denotes the isoprofit line, the intersection point's D,E results by D- 5X 40x5 5X,+2X2-250 2X2-50 х, -40,X, -25 Е- X,+2X, %3150 5X,+2X2-250 4X1 100 х,-25,X, -62.5 As denoted in the graph, we have five corner points, they are (0,0), (0,75), (40,0), (40,25) and (25,62.5)

120 5X1 2X2 250 100 X1 40 80 Isoprofit Line 60 (25, 62.5) Profit $20X+$30X2 40- XI2X2 150 Feasible Region (40, 25) 20 C 40 120- 0 60 80 100 140 co

2 which is less than 3 1 but larger than According to the isocost line, the slope of that is 5 , then we find that the maximum profit happens when X 2 25 and X 625 The maximum profit is profit(25,625) $20x25 +$30 x62 5 $2,375

(b) Hence, the best combination is to produce 25 barrels of pruned olives, and 62.5 barrels of normal olives (c) The number of acres that the olives grower should be used during the process is number of acres X+2X 25+2x625 150

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