Determining the optimum price for a product is a job that may be asked of a mana
ID: 2845451 • Letter: D
Question
Determining the optimum price for a product is a job that may be asked of a management scientist. The demand for a product, i.e., the number of units that can be sold, generally decreases with the price. When the price is low demand will be large and thus many units can be sold. However the profit per unit will be low and the total profit, i.e., the product of demand and unit profit will be low. If the price is high the profit per unit will be large but only a small number can be sold so the totatl progit will again be small. The optimal price, the price that will yield the highest total profit will be somewhere in between. Denote the unit price as p and the cost as c. The relation between demand and price is denoted D(p). The total profit as a function of price is expressed generally as the nonlinear relation:
Total Profit= (p - c) * D(p)
Using non-linear programming, find the optimum price p and total profit for a product whose cost c= $225. Assume that surveys have found that the demand is a linearly decreasing function of price expressed as:
D(p)= -0.2214P + 211
Use the single constraint p>c
Explanation / Answer
Total Profit= (P - 225) * (-0.2214P + 211)
Differentiate w.r.t P, (P-225)(-0.2214)+(-0.2214P + 211)=0 => -0.4428P+260=0 =>P=$589(approx)
Total profit=(589-225)(-0.2214*589 + 211)=364*81=$29,484
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