A building company needs to move earth and level the terrain at three sites: site 1 has a surplus of 300 truckloads of earth; site 2 has a surplus of 1200 truckloads of earth and site 3 has a deficit of 700 truckloads of earth. Earth at surplus locations is loaded via bulldozers onto dumper trucks and then transported to site 3 to rectify the deficit. Any earth in excess of site 3's deficit must be disposed off at a dumpsite. The distances between the relevant sites are shown in the table below and running a dumper truck cost 11 per mile. You may assume that the company has sufficient dumper trucks available. 

        

From/to	Site 3	Dump site
Site 1	4 miles	1 mile
Site 2	8 miles	6 miles

Bulldozers are required to load dumper trucks at the surplus locations (sites 1 & 2) as well as to level the earth at site 3. One bulldozer can load up to 50 dumper trucks per day; one bulldozer can also level the earth at site 3 of up to 100 truckloads per day. Bulldozers are not required at the dump site. The company has 5 bulldozers available, which are pre-positioned at different sites (see below).

Bulldozers can be transported between sites 1, 2 and 3, but this will take a 2 day lead time. When bulldozers are not needed anymore, they are transported to the company's depot, and this takes one full day. Operating a bulldozer at a site costs 2500 per day per bulldozer; transporting a bulldozer between sites costs 700 per bulldozer (lead time is 2 days); transporting (and retiring) a bulldozer to the depot costs 300 per bulldozer (lead time 1 day). The company needs to complete the project within 10 working days. This means that at the end of day 10 (or earlier) all 5 bulldozers have to be back at the depot.

 

The 4 bulldozers are positioned at site 1; 1 bulldozer at site 2 and 0 bulldozers at site 3.

You may assume that all bulldozer transfers are initiated at the beginning of a work day. For example, if a bulldozer is transferred at the start of day t from site 1 to site 3, then this bulldozer cannot be working during days t and t+1, and will be available to work on site 3 at the start of day t+2. When working at a site, the bulldozer cost is 2500 per day, irrespective of whether the bulldozer is working (or not) at its maximum capacity (i.e., loading at most 50 trucks per day at sites 1 & 2, or levelling earth at most at 100 truck loads per day at site 3).

 

The company carrying out the works wants to find a good/optimal terrain levelling & bulldozer allocation plan and has hired you as a Management Science consultant for advice.

Formulate this problem with all relevant constraint.