6.8 Linear Inequalities in Two Variables and Systems of Linear Inequalities 371 Did You Know? Linear programming is a powerful tool for determining the maximum and minimum values of an objective function. Using the fundamental principle of linear programming, we are quickly able to determine the maximum and minimum values of an objective function by using just a few of the infinitely many points in the feasible region. Example 7 illustrates how the fundamental principle of linear programming is used to solve a linear programming problem. The Logistics of D-Day m Winston Churchill called the invasion of Normandy “the most difficult and complicated operation that has ever taken place.” Linear programming was first used to deal with the ageold military problem of logistics: obtaining, maintaining, and transporting military equipment and personnel. George B. Dantzig developed the simplex method for the Allies of World War II to do just that. Consider the logistics of the Allied invasion of Normandy. Meteorologic experts had settled on three possible dates in June 1944. It had to be a day when low tide and first light would coincide, when the winds should not exceed 8 to 13 mph, and when visibility was not less than 3 miles. A force of 170,000 assault troops was to be assembled and moved to 22 airfields in England where 1200 air transports and 700 gliders would then take them to the coast of France to converge with 5000 ships of the D-Day armada. The code name for the invasion was Operation Overlord, but it is known to most as D-Day. Example 7 Using the Fundamental Principle of Linear Programming The Down Home Chair company makes two types of rocking chairs, a plain chair and a fancy chair. Each rocking chair must be assembled and then finished. The plain chair takes 4 hours to assemble and 4 hours to finish. The fancy chair takes 8 hours to assemble and 12 hours to finish. The company can provide at most 160 worker-hours of assembling and 180 worker-hours of finishing a day. If the profit on a plain chair is $40 and the profit on a fancy chair is $65, how many rocking chairs of each type should the company make per day to maximize profits? What is the maximum profit? Solution From the information given, we know the following facts. Assembly Time (hr) Finishing Time (hr) Profit ( )$ Plain chair 4 4 40.00 Fancy chair 8 12 65.00 Let = = = = = x y x y P the number of plain chairs made per day the number of fancy chairs made per day 40 profit on the plain chairs 65 profit on the fancy chairs the total profit The total profit is the sum of the profit on the plain chairs and the profit on the fancy chairs. Since x 40 is the profit on the plain chairs and y 65 is the profit on the fancy chairs, the profit formula is P x y 40 65 . = + The maximum profit, P, is dependent on several conditions, or constraints . The number of chairs manufactured each day cannot be a negative amount. This condition gives us the constraints x 0 ≥ and y 0. ≥ Another constraint is determined by the total number of hours allocated for assembling. Four hours are required to assemble the plain chair, so the total number of hours needed to assemble x plain chairs is x4 . Eight hours are required to assemble a fancy chair, so the total number of hours needed to assemble y fancy chairs is y8 . The maximum number of hours allocated for assembling is 160 per day. Thus, the third constraint is x y 4 8 160. + ≤ The final constraint is determined by the number of hours allotted for finishing. Finishing a plain chair takes 4 hours, or x4 hours to finish x plain chairs. Finishing a fancy chair takes 12 hours, or y 12 hours to finish y fancy chairs. The total number of hours allotted for finishing is 180 per day. Therefore, the fourth constraint is x y 4 12 180. + ≤ Thus, the four constraints are ≥ ≥ + ≤ + ≤ x y x y x y 0 0 4 8 160 4 12 180 Keith Tarrier/Shutterstock
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