Question

2. A department store has 700 pairs of purple Capri stretch pants that it must sell in the next four weeks. The store manager knows that demand by week for the next four weeks will be linear each week, with the following price-response functions:

   Week 1: d1(p1) =1,000 -100p1 Week 2: d2(p2) =800 -100p2 Week 3: d3(p3) = 700 - 100p3 Week 4: d4(p4) =600 -100 p4

3. Assume that the demands in the different weeks are independent, that is, that customers who do not buy in a given week do not come back in subsequent weeks. Assume she can charge a different price each week. What are the optimum prices by week she should charge? What is her corresponding revenue?

Answer 1

Week 1: d1(p1) = 1000 -100*p1

Week 2: d2(p2) = 800 -100*p2

Week 3: d3(p3) = 700 - 100*p3

Week 4: d4(p4) = 600 -100*p4

This can be formulated as an optimization problem as below:

Max d1*p1+d2*p2+d3*p3+d4*p4 or,

Max (1000-100p1)p1+(800-100p2)p2+(700-100p3)p3+(600-100p4)p4 or,

Max 1000p1+800p2+700p3+600p4-100(p1^2+p2^2+p3^3+p4^2)

s.t.

d1+d2+d3+d4 <= 700

p1, p2, p3, p4 >= 0

Optimal solution is determined by Excel Solver as follows:

Create Excel model

Enter Solver parameters :

Click Solve to generate the optimal solution:

Optimal prices to be charged each week:

p1 = 7.125

p2 = 6.125

p3 = 5.625

p4 = 5.125

Total revenue = $ 4418.75