After New Year's Day and before Spring Festival, it is the peak season for shopping malls. Huayuan Shopping Mall decided to reduce the price of a certain brand down jacket from 20 12 to 1 in order to increase revenue and profit. According to the economic law, price reduction will inevitably stimulate demand, but the price reduction is too small, and the increase in sales volume is not ideal, and the expected income cannot be achieved. Excessive price reduction has led to a sharp increase in sales, increased income but unsatisfactory profit increase. Therefore, it is necessary to locate a reasonable sales price, which can not only satisfy consumers' consumption psychology, but also maximize profits.

Sales list of a brand down jacket in Huayuan Shopping Mall in the fourth quarter of 2006 -20 10 (table 1)

year

share

enter

price

sell

sell

price

sell

sell

quantity

2006

300

580

820

2007

270

520

1000

2008

350

680

9 10

2009

380

760

880

20 10

420

900

900

20 1 1

540

1080

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Sales list of a brand down jacket in Huayuan Shopping Mall in the first quarter of 2007 -20 1 1 (Table 2)

year

share

Sales price in the fourth quarter of last year

at present

sell

sell

price

sell

sell

quantity

2007

580

500

1360

2008

520

480

1650

2009

680

580

1600

20 10

760

680

1500

20 1 1

900

820

1480

2

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2. 1

data analysis

According to the table 1, the sales prices in the fourth quarter of 2006 -20 10 were 580 yuan/piece, 520 yuan/piece, 680 yuan/piece, 760 yuan/piece and 900 yuan/piece respectively, and the quarterly sales volume. Comparing the sales table in the fourth quarter of 2006 -20 10 with the sales table in the first quarter of next year (that is, the first quarter of 2007-20 1 18+0), we can see that the price reduction ranges are 13.79%, 7.69% and 7.69% respectively. Sales increased by 39.7 1%, 39.39%, 43. 13%, 4 1.33% and 39. 19% respectively. Market research shows that the sales volume can increase by 40% when the sales price decreases by 10%.

2.2

Restatement of problems

The purchase price of a certain brand of down jacket in Huayuan Shopping Mall at 20 1 1 is 540 yuan/piece, and the sales price is 1080 yuan/piece. It is estimated that the sales volume in the fourth quarter is 1020 pieces. Market research shows that every time the sales price drops by 10%, the sales volume will increase by 40%. The mall decided to reduce the sales price of this brand down jacket from 20 12 to 1 at one time. What kind of maximum profit can the mall get under the condition of setting the sales price? What is the maximum profit? (This question ignores the cost factors such as wages, taxes and fees of mall attendants.)

three

Solution to the problem

3. 1

construct a mathematical model

set up

For the reduced sales price,

This is a profit function. So the sales volume after the price reduction is

Therefore, the profit after the price reduction is

3.2

Solving mathematical model

By profit analysis function, available.

and

Therefore, the extreme value analysis shows that when the price is reduced by 945 yuan/piece, the mall can get the maximum profit, and the maximum profit at this time is

(yuan)

four

conclusion

Huayuan Shopping Mall will implement a one-time price reduction for a brand on 20 12 and 10. According to the extreme value analysis method, the price will be 945 yuan/piece and the maximum profit will be 3098250 yuan.