Question

2) Buster Sod operates a 1200-acre irrigated farm in the Red River Valley of Arizona. Sod's principal activities are raising wheat, alfalfa, and beef. The Red Valley Water Authority has just given its water allotments for next year (Sod was allotted 2000 acre-feet) and Sod is busy preparing his production plan for the next year. It figures that beef prices will hold at around $600 per ton and wheat will sell at $1.60 per bushel. Best guesses are that it will sell alfalfa at $34 per ton, but if more alfalfa is needed to feed the beef than it can raise, it will have to pay $36 per ton to get the alfalfa to its feedlot. Some technological features of Sod's operation are as follows: wheat yield, 50 bushels per acre; alfalfa yield, 3 tons per acre. Other features are given below. Decision variables: Data for W = wheat raised and sold (acres) Buster A = alfalfa raised (tons) Land Alfalfa B = beef raised and sold (tons) Problem Labor, Machinery Water Reqd. Reqd. Reqd. Az = alfalfa bought (tons) Activity and other Costs (acre-ft.) (acres) (tons) As = alfalfa sold (tons) 1 acre of wheat $8 1. 51 1 acre of alfalfa 30 1 ton of beef 40 40 0.1 0.05 4 Sod 2.5 The LP model along with the Lindo output are provided below. a) Explain the last constraint. b) What is the optimal solution? Does Sod buy or sell alfalfa? c) Which variables are your non-basic variables? d) How much should Sod pay to acquire another acre of land? e) Interpret the dual price on row 3. f) What happens to the optimal planting policy if the price of wheat triples? g) How much can the cost of buying alfalfa decrease before the current optimal planting policy will change? h) What happens to the optimal value of the objective function if the cost of alfalfa purchased increases from $36 to $37? i) Show that complementary slackness conditions hold j) Show that strong duality theorem holds.

Answer 1

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a) Last constraint represents the requirement that 1 ton of bee;f requires 4 tons of Afalfa. So balance of Alfalfa raised plus Alfalfa bought minus Alfalfa sold is equal to 4 times bee;f raised and sold.

b) Optimal solution is:

Bee;f raised and sold = 20000 tons

Alfalfa bought = 80000 tons

Rest of the variables = 0

c) Variables W, A and A5 are non-basic

d) Dual price of land constraint (row 2) is 0 . This is because it is not fully used. There is slack of 200. So Sod should pay to acquire another acre of land = 0

e) Row 3 represents constraint of water allotment (acre-feet) . Dual price represents that each additional acre-feet of water allotted will increase the total profit by 4160

f) This is within allowable increase (6168) . So optimal solution remains unchanged.

g) Allowable decrease for variable A3 is 100.21 . Therefore, cost of buying Alfalfa can decrease by 100.21, (which means it can practically become 0) before the optimal planting policy will change.

h) Allowable increase for A3 is 2. Therefore increase from 36 to 37 is within allowable range. Therefore optimal policy remains unchanged.

Value of objective function = 8320000 - 1*80000 = $ 8,240,000