the price of meals sold on the train

1 Abstract: The service provider on the train is in a monopoly position and has a unique advantage, so it is easy to sell food at a higher market price. The higher the price, the greater the profit, but passengers have an upper limit. If the price continues to increase, more and more passengers will give up buying, and the total profit obtained by the service provider will change accordingly. In this model, we choose the supply and demand sides as a system, and study how to make the selling price multiply the demand to reduce the cost from the supplier's point of view, that is, the problem of maximizing the sales benefit

2. Restatement:

Long-distance trains need to provide some services on board because of the long time. Providing three meals a day is the main service. Due to the high cost of all aspects on the train, the price of food on the train is also slightly higher. Take the K452 train from Chengdu to Urumqi as an example. The breakfast every day is a bowl of porridge, an egg and some pickles, and the price is 11 yuan. Lunch at noon and evening, the price is 15 yuan. Because of the high price, passengers usually bring their own food such as instant noodles and bread. Instant noodles and bread are also sold on the train, but the price is also expensive. For example, instant noodles sold in 3 yuan are generally sold in 5 yuan. Of course, due to the limited capacity of the train, the amount of meals and food provided is limited, and it is normal to raise the price appropriately. However, there should be a limit to the high price, and it should not be too high. If there are 1111 passengers in the car, 511 of them have the requirement to buy food on the car, but the box lunch on the car can only supply 211 people per meal; In addition, the car can also provide instant noodles for 111 people per meal. Please design a price plan according to the actual situation, so that the train can get the most benefit in dining sales.

3 problem analysis:

The increase of price and profit also leads to the decrease of the number of people who want to buy;

lower prices and lower profits also lead to an increase in the number of people who want to buy;

There is an optimal price to make everything on the train sold and get the maximum benefit at the same time;

this is an optimization problem, and the key is to find the optimal price.

4 Question hypothesis:

1. Every time the price of box lunch increases by 1 yuan, 21 people will choose to give up buying, that is, b1 = 21;

2. Every time the price of instant noodles increases by 1 yuan, 36 people will choose to give up buying, that is, b2 = 36;

3. Since 511 people have the requirement to buy meals in the car, suppose that breakfast can be served for 511 people;

4. Every time the price of breakfast increases by 1 yuan, 31 people will choose to give up buying, that is, b3 = 31;

5. The prices in various catering markets are taken as the cost price here, namely q1 = 11 yuan (box lunch), q2 = 3 yuan (instant noodles) and q3 =5 yuan (breakfast);

6. the sales volume x depends on the price p, and x (p) is a decreasing function

7. further assume that x (p) = a–BP, a, b > 1；

5 symbol description:

q: The cost price here is the price in each catering market, that is, the cost price of food;

p: the selling price of food;

a: absolute demand (demand when P is very small), that is, the number of buyers at the lowest price;

b: the decrease of the number of buyers when the price rises in 1 yuan (the sensitivity of demand to price);

I: income; U: profit; C: expenditure;

x: the number of people who need to buy a certain food;

The corresponding subscripts 1, 2 and 3 respectively represent breakfast, box lunch and instant noodles; For example, x1, x2, x3 and x3 respectively represent the number of people who buy box lunch, instant noodles and breakfast;

6 model establishment and solution:

adopt the algorithm of unification first and then separation;

income I (p) = px; Expenditure C (p) = qx; Profit u (p) = I (p)–C (p); Seek p to maximize U (p);

the optimal price p* that maximizes the profit U(p) satisfies

u (p) = I (p)–c (p)

= (p–q) (a–BP).

=-bpp+(a+bq) p-AQ

because

q/2 ~ half the cost;

b ~ the decline of sales volume when the price rises by 1 unit (the sensitivity of demand to price) b p *

a ~ absolute demand (the demand is very small) a p*

7 For box lunch:

It can be known from the hypothesis that q1 = 11, b1 = 21；; Because 511 people have the requirement to buy food in the car, but the box lunch in the car can only supply 211 people per meal; So: a1 = 511; Number of purchasers x1 = 511–21p1;

obtained from p* = q/2+a/2 * b: p * = Q1/2+A1/2 * B1 = 11/2+511/2 * 21 = 17.5;

from 511–21 * 17.5 = 151 <; 211; Formula cannot be used directly at this time;

from 511–21p1 > = 211 to obtain P1 <: = 15;

therefore, p1 = 15 when the maximum profit is obtained;

8 instant noodles can be obtained in the same way:

Q2 = 3, B2 = 36; Because 511 people have the requirement to buy food in the car, but the box lunch in the car can only supply 211 people per meal, and at this time there are still 311 people who need instant noodles; Therefore, a2 = 311, and the number of buyers x = 311 -36p2;

obtained from p* = q/2+a/2 * b: p * = Q2/2+A2/2 * B2 = 3/2+311/2 * 36 = 6.3;

from 311–36 * 6.3 = 73 <; 111; At this time, the formula

from 311–36 * p2 > cannot be directly used. = 111; Get p2 <: = 5.5 from U(p) = U(p)= -bpp+(a+bq)p-aq, it can be known that:

when u (p) takes the maximum value, p2 = 5.5;

9 breakfast can be obtained in the same way:

Q3 = 5, B3 = 31; Because 511 people have the requirement to buy food in the car, a1 = 511; Number of purchasers x1 = 511–31p3;

Because the supply is not constrained, the formula can be directly used:

It can be obtained from p* = q/2+a/2 * b: p * = Q3/2+A3/2 * B3 = 5/2+511/2 * 31 = 11.5;

therefore, p1 = 11.5 when the maximum profit is obtained;

11 Analysis and test of results:

1) The prices of box lunch, instant noodles and breakfast are RMB 15, 5.5 yuan and RMB 11.5 respectively;

through calculation, all the prices are not more than 3 times of the market price, and the provided (except breakfast) can be sold out, and the maximum benefit can be obtained from it;

These calculated results should be acceptable to consumers;

The price calculated by this model is similar to the actual price of the train, which shows that it is reasonable.

the process of modeling and solving in this model has been tested, and the in-depth test needs to be completed by practice;

11 model evaluation:

Advantages: This price optimization problem is solved by a quadratic function, thus obtaining the maximum benefit; Simplify a complicated and changeable problem;

Disadvantages: the sensitivity of demand to price is not accurate only by self-experience, and the sensitivity problem only uses a function once, which is too simplistic and may not show its true model well;

improvement direction: make the sensitivity problem accurate and closer to reality; Function can try to use a higher level of variety;

Popularize new ideas: the establishment of mathematical model must be considered from many aspects, and its ultimate goal is to simplify complex problems and mathematize practical problems; Mathematical problems are life-oriented; But all these have a premise: there must be strict mathematical rules and mathematical theories as the basis;

12 references:

[1] Xiong Wei. Operations Research [M], Wuhan. Wuhan University of Technology

[2] Jiang Qiyuan, etc. Mathematical Model [M], Beijing. Higher Education Press.