844 CHAPTER 11 Systems of Equations and Inequalities Maximizing Profit At the end of every month, after filling orders for its regular customers, a coffee company has some pure Colombian coffee and some special-blend coffee remaining.The practice of the company has been to package a mixture of the two coffees into 1-pound (lb) packages as follows: a low-grade mixture containing 4 ounces (oz) of Colombian coffee and 12 oz of special-blend coffee, and a high-grade mixture containing 8 oz of Colombian and 8 oz of special-blend coffee. A profit of $1.25 per package is made on the low-grade mixture, whereas a profit of $1.75 per package is made on the highgrade mixture.This month, 120 lb of special-blend coffee and 100 lb of pure Colombian coffee remain. How many packages of each mixture should be prepared to achieve a maximum profit? Assume that all packages prepared can be sold. Solution EXAMPLE 4 Step 1 Begin by assigning symbols for the two variables. = = x y Number of packages of the low-grade mixture Number of packages of the high-grade mixture The goal is to maximize the profit subject to constraints on x and y. If P denotes the profit, then the objective function is = + P x y $1.25 $1.75 Step 2 Because x and y represent numbers of packages, the only meaningful values for x and y are nonnegative integers. This yields the two constraints ≥ ≥ x y 0 0 Nonnegative constraints There is only so much of each type of coffee available. For example, the total amount of Colombian coffee used in the two mixtures cannot exceed 100 lb, or 1600 oz. Because 4 oz are used in each low-grade package and 8 oz are used in each high-grade package, this leads to the constraint + ≤ x y 4 8 1600 Colombian coffee constraint Similarly, the supply of 120 lb, or 1920 oz, of special-blend coffee leads to the constraint + ≤ x y 12 8 1920 Special-blend coffee constraint The linear programming problem is = + P x y Maximize 1.25 1.75 subject to the constraints ≥ ≥ + ≤ + ≤ x y x y x y 0 0 4 8 1600 12 8 1920 Step 3 The graph of the constraints (the feasible points) is shown in Figure 45 with the corner points labeled. Step 4 Table 2 shows the value of P at each corner point. From the table, the maximum profit of $365 is achieved with 40 packages of the low-grade mixture and 180 packages of the high-grade mixture. Figure 45 x y 240 (0, 0) (0, 200) 4x 1 8y 5 1600 20 60 100 140 180 220 260 300 340 380 140 100 60 20 12x 1 8y 5 1920 (40, 180) (160, 0) Corner Point ( ) , x y Value of Profit = + 1.25 1.75 P x y ( ) 0, 0 = P 0 ( ) 0, 200 = ⋅ + ⋅ = P 1.25 0 1.75 200 $350 ( ) 40, 180 = ⋅ + ⋅ = P 1.25 40 1.75 180 $365 ( ) 160, 0 = ⋅ + ⋅ = P 1.25 160 1.75 0 $200 Table 2 Now Work PROBLEM 19
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