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Allocation Problem 1 (Single-Period)

Minimize the cost of operating 3 different types of machines while meeting product demand.    
Each machine has a different cost and capacity. There are a certain number of machines    
available for each type.              
                 
Information on machines            
  Initial cost per day Additional cost per product Products per day (Max) Number of machines        
Alpha-1000 $200 $1.50 40 8        
Alpha-2000 $275 $1.80 60 5        
Alpha-3000 $325 $1.90 85 3        
                 
Number of machines to use            
Alpha-1000 1              
Alpha-2000 1              
Alpha-3000 1              
                 
Number of products to make per day          
Alpha-1000 300              
Alpha-2000 300              
Alpha-3000 300              
Total 900              
Demand 750              
                 
Maximum number of products that can be made per day        
Alpha-1000 40              
Alpha-2000 60              
Alpha-3000 85              
                 
Cost $2,360.00              
                 
                 
Problem                
A company has three different types of machines that all make the same product. Each    
machine has a different capacity, start-up cost and cost per product. How should the company    
produce its product with the available machines to meet the daily demand?      
                 
Solution                
1) The variables are the number of machines to use and the number of products to make on    
each machine. In worksheet Alloc1, these are given the names Products_made and    
Machines_used.              
2) First, there are the logical constraints. These are          
  Products_made >= 0 via the Assume Non-Negative option      
  Machines_used >= 0 via the Assume Non-Negative option      
  Machines_used = integer.            
Second, there are the demand and capacity constraints. These are:        
  Machines_used <= Machines_available          
  Products_made <= Maximum_products          
  Total_made >= Demand            
3) The objective is to minimize cost. This is defined on the worksheet as Total_cost.    
                 
Remarks                
Notice that we used an integer constraint to make sure no fractions of machines were used.    
This has the usual drawback; the problem is much more difficult to solve than the 'relaxed'    
version without the integer constraint. When large numbers of machines are involved, the    
integer constraint can often be dropped. It is most often not critical whether 1586 or 1587    
machines are used, for example. This means that a number of 1586.4 would be acceptable.    
However, in this case there are only a few machines and it does make a big difference whether    
2 or 3 machines are used. An answer of 2.5 would not be satisfactory.