Exercise 1.8

1. Define Problem
   
   Determine the number of valves of each type to be ordered from each supplier to meet demand while minimizing cost.
   
   
2. Identify Key Issues
   
   
   • How reliable are the suppliers?
   • Is there any penalty for overstocking? understocking?
   • Is there any difference in quality of valve among the different suppliers?
   • What about transportation costs for the valves?
   • If a shipment is delayed, do the heart valves deteriorate?
   • How long will the current prices hold?
   • Is the percentage of each type of valve in an order from a supplier fixed?
   • Does U.S. labs have to order valves from each supplier, even though one of them may be very expensive?
   
   
   
3. Collect and Assess Information
   
   The table in the problem gives a good bit of the information. The other information is that at most 500 valves can be ordered from each supplier per month and at least 500 large, 300 medium and 300 small valves must be purchased each month.
   
   
4. State Assumptions
   
   
   • the suppliers can each supply at least 500 valves in the percentages expressed in the table.
   • The quality of the valves is the same for each supplier.
   • The demand for valves is constant.
   • The mix of valves for each supplier is constant as given in the table
   • The costs of the valves will remain constant for the life of the problem.
   • There are no hidden costs and penalties.
   • There are no transportation costs charged to U.S. Labs.
   • The valves are all in good condition when they arrive.
   
   
   
5. Break the Problem Apart
   
   
   • Decide how to label the unknowns
   • Find limits on the number of valves to order.
   • Construct the function whose values are to be minimized.
   
   
   
6. Model Subproblems
   
   
   1. To decide how to label the unknowns, ask yourself how many decisions must be made to solve the problem. At first glance it seems that we must make two decisions: the number of valves of each type and the supplier. But from closer inspection we see that the number of valves of each type is set once we know how many valves overall to buy from a given supplier since the table sets the percentages. Thus we need a variable with only one subscript: x_i, where i = 1, 2, 3 depending on which supplier we are referring to. Note that, if we were not given the percentages of valves of various types from each supplier, we would have used two subscripts x_ij, where i represents the supplier and j represents the type of valve.
      
      
   2. The limits on the number of valves to order are as follows:
      

      
           xi <= 500  for i = 1, 2, 3
         .4x1 + .3x2 + .2x3 >= 500  large valve
         .4x1 + .35x2 + .2x3 >= 300 medium valves
         .2x1 + .35x2 + .6x3 >= 300
           xi >= 0  for i = 1, 2, 3
      

   3. The function whose value is to be minimized is:
      f(x) = 5x₁ + 4x₂ + 3x₃

We will get to the other steps in the Problem Solving Methodology later in the course