November 5, 2012 AGEC 352-R. Keeney

 Recall  With 2000 total units (maximum) at harbor and 2000 units (minimum) demanded at assembly plants it is not possible for slack constraints  Supply <=2000  Demand >=2000  Supply = Demand  Total movement of 2000 motors is the only feasible combination, leading all constraints to bind  One binding constraint is trivial

 Transportation problems do not have to be balanced  Real world problems are rarely balanced  If you have an unbalanced model, might want to balance it with other activities  If Supply > Demand introduce a storage destination that takes up the excess  What is the cost of holding excess supply? ▪ Storage costs or waste/spoil  If Demand > Supply introduce a penalty source that deals with the imbalance  What is the cost of shipping less than required? ▪ Lost customers or contract penalties

 If the constraints have integers on RHS the optimal solution will have transport quantities in integers  This can be shown mathematically  Convenient for solving smaller problems by hand  Choose a route to enter the model, then keep adding until you hit the supply or demand constraint  In a balanced problem, one constraint is mathematically redundant  This is the trivial constraint and it is the one with the constraint that binds (LHS=RHS) but has a zero shadow price

 The assignment problem is the mathematical allocation of ‘n’ agents or objects to ‘n’ tasks  The agents or objects are indivisible ▪ Each can be assigned to one task only  Example using Autopower Company:  Auditing the Assembly  Leipzig, Nancy, Liege, Tilburg  A VP is assigned to visit and spend two weeks conducting the audit  VP’s of Finance, Marketing, Operations, Personnel  Considerations…  Expertise to problem areas at plants  Time demands on VP  Language ability

 How do you get those costs?  Clearly when you are talking about opportunity costs and the additional cost of having someone out of their specialty or who is not a native speaker being assigned the problem a solution is heavily dependent on how reliable the opportunity cost information is  Perhaps the cost of having a full-time translator or additional support staff for a VP who is dealing with a lot of problems that are not her specialty  Other ways--think of skill/aptitude tests ▪ ASVAB

 Enumeration is a way of solving a small problem by hand  Enumeration means check all possible combinations…  Combinations for an ‘n’ valued assignment problem are just n factorial (n!) ▪ n = 4  n!=4*3*2*1=24  That’s still a lot to check  There are other tempting methods  Start with the lowest costs and work your way up?

 Tempting and seems logical but does not guarantee you an optimal solution  for a small problem we can find the best solution using tradeoffs  Think of the destinations as demanding VP with the lowest cost VP being the preference  Leipzig prefers Personnel  Nancy prefers Finance  Liege prefers Marketing  Tilburg prefers Finance  Two locations have Finance as a first preference, this is the only thing that makes this problem interesting

 Tradeoff 1: 1000 improvement  Tradeoff 2: 6000 worse  Tradeoff 3: 2000 improvement  Hopefully this convinces you that LP might be easier for solving these types of problems than wrangling all of the potential tradeoffs that occur

 Setup is the same as the Balanced Transportation problem from last week  Destinations are the locations or assignments with >=1 constraints  Sources are the persons or objects to be assigned with <= 1 constraints  What is different?  Number of rows and columns are the same (i.e. square and balanced) ▪ Not the case for transportation problems

 Recall the problem from Monday’s lecture of assigning VP’s to plants to be audited  Objective  We want to minimize the cost of sending Vice Presidents to assembly plants given the per unit costs matrix C(i,j)  Assignees  The sources  For any assignee i, that assignee can be placed in a maximum of one assignment  Assignments  The destinations  For any destination j, the assignment requires that at least one assignee be put in place  Non-negativity  Decision variables must be zero or positive

 Since the problem is balanced and assignments are 1 to 1 (1 person to 1 place)  Decision variables will all have an ending value of either 1 or 0  Recall that balanced transport problems have integer solutions if RHS are integer values  In general, the assignment model can be formulated as a transportation model in which supply at each source and demand at each destination is equal to one