Balanced Transportation and Assignment Model

1. November 5, 2012 AGEC 352-R. Keeney Balanced Transportation and Assignment Model

2. ‘Balanced’ Transportation • 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

3. Balanced Transportation: Trivia • 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

4. Balanced Transportation:Important Facts • 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

5. Assignment Problem • 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 Plants @ • 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

6. Estimated Opportunity Costs(Objective Coefficient Matrix)

7. Assignment Problem-Costs Data • 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

8. Solving a Small Problem • 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?

9. Starting with the Lowest Cost • 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

10. Tradeoffs (Spreadsheet) • 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

11. LP Setup • 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

12. Standard Algebraic Form for an Assignment Problem

13. Interpretation • 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

14. Assignment Problems • 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