Balanced Transportation and Assignment Model

November 5, 2012AGEC 352-R. Keeney

‘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

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

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

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

Estimated Opportunity Costs(Objective Coefficient Matrix)

VP Leipzig Nancy Liege TilburgFinance 24 10 21 11Marketing 14 22 10 15Operations 15 17 20 19Personnel 11 19 14 13

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

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?

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

Tradeoffs (Spreadsheet)

Tradeoff 1: 1000 improvementTradeoff 2: 6000 worseTradeoff 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

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

Standard Algebraic Form for an Assignment Problem

jandiX

jX

iX

ts

XC

jtoassignediofamountthebeXLet

ji

iji

jji

jjiji

i

ji

0 :neg.-Non

1:sAssignment

1 :Assignees

..

min

,

,

,

,,

,

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

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