For the two problems below, please provide a) formulation of the basic problem (no extensions), b) formulations for specified extensions. Good luck & have fun.
Individual work only please.
1. Three railway cars with a carrying capacity of each car is100 quintals each (1 quintal = 100 kg) have been reserved to transport sixteen boxes. The weight of the boxes in quintals is given in the following table. The interest is to assign boxes to railway cars in order to keep to the limits on the maximum carrying capacity and to minimize the heaviest railway car load.
Box 1 2 3 4 5 6 7 8
Weight 34 6 8 17 16 5 13 21
Box 9 10 11 12 13 14 15 16
Weight 25 31 14 13 33 9 25 25
EXTENSIONS:
i) Suppose that in addition to a weight capacity, you are provided with the volume of each box and railway car capacity. How would you extend the above model?
ii) Building upon i) above, suppose there are different types of railway cars (box, flatbed, refrigeration, etc.). What additional information would you need regarding the boxes? How could you now model the problem?
2. The large company Tatayo in the north of Italy has specialized in the construction of cars for more than ten years. The company wishes to expand and has issued internally a call for proposals for expansion projects for a planning period of five years. Among the many, often cranky, propositions the management has retained five projects. Every project has an annual cost and is designed to produce a benefit after five years. The first table below gives a list of the projects with short descriptions and the expected benefit after five years. The forecast annual costs of the projects (in million €) for the next five years are detailed in the second table, together with the funds available. The interest is to determine which project(s) should the management choose now to maximize the total benefit after five years.
Project Description Expected Benefit (in million €)
1 Expand assembly line 10.8
2 Reorganize the main shop 4.8
3 New painting facilities 3.2
4 Research for a new car concept 4.44
5 Reorganize the logistics chain 12.25
Project Year 1 Year 2 Year 3 Year 4 Year 5
1 1.8 2.4 2.4 1.8 1.5
2 1.2 1.8 2.4 .6 .5
3 1.2 1.0 0 .48 0
4 1.4 1.4 1.2 1.2 1.2
5 1.6 2.1 2.5 2.0 1.8
Funds 4.8 6.0 4.8 4.2 3.5
EXTENSIONS:
i) What is the general mathematical model for this problem (i.e., could have 30 project possibilities with varying timelines – some 2 years, some 7 years)? What data structures will you need to implement this model?