2004 China Undergraduates Mathematical Contest in Modeling (CUMCM 2004)

Problem A: Planning Temporary Mini Supermarkets for the Olympic Games

Acknowledgements: Thanks to the Beijing Municipal Administer Committee and the Beijing City Planning Committee for provision of the stadium planning figure.

The construction work of Beijing 2008 Olympic Games has been in planning and implementation processes. During the Olympic Games, temporary Mini Supermarkets should be built around the stadium for supplying food, souvenir and tourist commodities to the spectators, tourists and members of staff, each Mini Supermarkets (MS) consisting of variety shops. For MS around the stadium, their location, size and the amount of sales should satisfy three basic requirements: demand for shopping, reasonable distribution and commercial profit during the Games.

The planning layout of the main stadiums for contest is shown in Fig. 1. For simplicity we only remain in Fig. 2 the related parts and regions, such as streets (white denote pavement), bus and taxi stops, parking area, subway stations and restaurants etc., where the yellow area marked with A1-A10, B1-B6 and C1-C4 denote the prescribed 20 shopping centers consisting of the MSs.

One way to find the patterns of the consumer flux is to send out questionnaire forms to the spectators, they are the principle consumers during the Olympic Games for investigating purchase demand and appetite of tour. Suppose that three games had been held in a stadium and related data are collected and is shown in the Appendix.

Your team is asked to be a consultant for planning MSs for the 20 shopping centers shown in Fig. 2 according to the following steps:

1. Based on the data of questionnaire given in the Appendix find features of the spectators in tour, meals and purchase etc.
2. Suppose each spectator takes collected of twice tour at one day During the Olympic Games, one for get in and out the stadium and another one formeal, and they always adopt the shortest route. Based on the result in 1 please calculate the distribution of consumer flux (in percent) in the 20 shopping centers in Fig. 2.
3. Suppose there are two different sizes of MS, large and small, can be chosen. Please plann the MSs for the 20 shopping centers , i.e. the numbers of different MS in each center such that three requirements are satisfied.
4. Explain that your method is reasonable and the result is practicable.

Remarks :

1. In commerce the "shopping loop" may be used to describe the covering area of shops. The main factor determining the choice of shop location is the buyer flux with their purchase demand in a shopping loop.
2. For simplicity assume that the National Stadium can admit about 100,000 (spectators), the National Gymnasium about 60,000 and the National Swimming Center about 40,000, where each stand admits about 10,000. Assume also that each exit gate faces just one shopping center, all the shopping center have the same area.

Appendix

Send out questionnaire forms to the spectators three times with 33% reply, total about 10,000 replies. Detail data can be found in the attached access database, where the ages divided into four levers: 1) Less than 20, 2) 20-30, 3) 30-50 and 4) more than 50. We may design four kinds of tour ways: by taxi, bus, subway and drive car; three kinds of repasts provided: Chinese meal, Western-style food and in marketplace (fast food ); and six kinds of expenditure (except for repast): 1) 0-100, 2) 100-200, 3) 200-300, 4) 300-400, 5) 400-500 and 6) more than 500 (RMB ).