单号：1 3 5 选一道题
双号：2 4 5 选一道题
1. Network design problem
The figure that follows shows 5 pipelines under consideration by a natural gas company to move gas from its 2 fields to its 2 storage areas. The numbers on the arcs show the number of miles of line that would have to be constructed at $100 000 per mile. The figure also shows that storage facilities are both already connected to the company’s main terminal through existing lines. An estimated 800 million cubic feet must be shipped each year from field 1 to the terminal, and 600 million from field 2. Variable shipping cost is $2 000 per million cubic feet on each link of the network, and all links have an annual capacity of 1 billion cubic feet. The company wants a minimum total annual cost system for the required shipping.
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2. Multi-Period Product Mix Problem
Consider the first problem from Home Exam Part 1 as a multi-period production planning problem with the extension to the next five month (the total planning horizon is six months from January to June). Now the storage of finished products is allowed and it is necessary to find the quantities of each product manufactured each month together with the quantities sold each month and the quantities held over in storage at the end of each month.
From January to June the following machines will be down for maintenance:
January 1 grinder
February 2 horizontal drills
March 1 borer
April 1 vertical drill
May 1 grinder and 1 vertical drill
June 1 planer and 1 horizontal drill
and marketing limitations on each product are:
It is possible to store up to100 units of each product at a time at a cost of £0.5 per unit per month. There are no stocks at present in the beginning of planning horizon but it is desired to have a stock of 50 unit of each type of product at the end of June. It may be assumed that each month consists of only 24 working days, and the factory works with two shifts of 8 hours each day.
After you find an optimal solution, find out for which product its price increase will have the largest impact on the total profit, and what is the value of acquiring new machines.
3. 课后习题2-6题 A-C问
A company has two factories, one at Liverpool and one at Brighton. Each factory has a monthly capacity which cannot be exceeded: Liverpool 150 000 tons, Brighton 200 000 tons. In addition, the company has 4 depots with storage facilities at Newcastle, Birmingham, London and Exeter. Each depot has a maximum monthly throughput given below which cannot be exceeded: Newcastle 70 000 tons, Birmingham 50 000 tons, London 100 000 tons, and Exeter 40 000 tons. The company sells its products to six customers C1, C2, …, C6. Customers can be supplied either from a depot or from the factory direct. Each customer has a monthly requirement given below which must be met:
The unit distribution costs in £ per ton delivered (which are born by the company) are known and given below. A dash indicates the impossibility of certain suppliers to serve certain depots or customers.
Find a distribution pattern that would minimize total cost and show it on picture.
5．Aggregated Production Planning Problem
Precision Transfer Incorporated makes more than 300 different products, all precision gears. The operations to make them are all similar, but the time and materials required are somewhat different. An aggregated gear is made as a kind of weighted average of the real gears and is referred to as a standard gear. The forecasted demands for the standard gear for the next 6 months are given in the table below. The number of days in each of the months is also given in the table.
Last year, Precision made 41383 gears of various kinds. There were 260 working days and an average of 40 workers. The production costs, excluding labour, do not change during the time horizon. Inventory holding cost is stipulated to $5 per unit per month. At the beginning of any month new workers can be hired at a cost of $450 per worker and workers can be fired at the cost of $600 per worker at the end of any month. At the end of the previous year the number of workers was 35. Each worker is paid $15 per hour and is paid for 8 hours per day. You can assume that the starting and the ending inventories are zero. No overtime, back-orders or sub-contracting are allowed as a starting point.
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