Carrier Selection Mohawk Case 1

Variations on the shortest path problem Arcs might represent roads, railways, bridges, or links in a communications network. Basic Problem: Find the shortest (fastest) path from node A to node H. Which arc would cause the greatest increase in the length of the shortest path, i.e. which would cause the maximum disruption in transportation/communications? A B C D E G H F

A homeland security problem Consider destroying one of the arcs in this network. Protect the arc that causes the greatest increase in the length of the shortest path ! A B C D E G H F The shortest path may change.

Finding the most reliable route Associated with each arc is the probability that the arc is open/working. Find the most reliable route from A to H  The route with the highest probability that all of its arcs are open (= product of probabilities of arcs along that route).  For example, the reliability of ACDH=0.8*0.8*0.9=0.648  Often not the shortest route! A B C D E G H F

The assignment problem An assignment problem can also be formulated as a network flow problem An assignment problem example:  A company has three workers: Ann, Bob, and Chris.  On a particular day, there are three tasks to perform: 1, 2, and 3.  Each worker can perform a task in a different length of time:  How to assign tasks to workers to minimize total time spent? ABC

Graphical representation A B C (0,1) (0,1) (1,1) XA1 XA2 XA3 Each worker is assigned to at most one task Each task is assigned to one worker Each arc has a binary decision variable: for example, XA3=1 if Ann is assigned to task 1 and 0 otherwise.

Graphical representation A B C (0,1) (0,1) (1,1) What if you want to make sure Chris is not assigned to job 1? Increase the cost of the C1 arc to a large number

The matching problem There are four students {A, B, C, D} and four companies with summer co-op jobs {1, 2, 3, 4} Preferences: Goal:match as many students as possible to jobs they prefer StudentPreferred Jobs (unordered) A1, 3 B2, 3, 4 C1, 3 D3

The graphical representation Arc (A,1) has flow 1 if student A is assigned to job 1. Maximize the overall flow “cost” (= # assignments) A B C D (0,1) 001 Each student is assigned to at most one job. No more than one student is assigned to each job.

The Graphical Representation What if company 3 is willing to hire two students? A B C D (0,1) 001 Change these bounds to (0,2).

Carrier Selection at Mohawk Paper Mills Mill produces paper for customers in 12 cities Each day the load planner determines the loads:  # loads for each destination city  distances from mill to destination cities  interim stops may be required for each destination  each load is designed to fill one truck Atlanta Everett Ephrata

Carrier Selection at Mohawk Paper Mills Mill is served by six carriers  Carriers have different per-mile rates for each destination  Interim stops cost extra  There is a minimum charge per truckload  Carriers have different numbers of available trucks  Some carriers have contracts with minimum requirements Which loads should be assigned to which carriers? Unassigned trucks

Graphical representation 1. ABCT 2. IRST 3. LAST 4. MRST 5. NEST 6. PSST A. Atlanta B. Everett C. Ephrata D. Riverview E. Carson F. Chamblee G. Roseville H. Hanover I. Sparks J. Parsippany K. Effingham L. Kearny (1,4) (7,8) (6,7) (0,7) (0,3) (4,4) (1,1) (3,3) (5,5) (1,1) (2,2) (1,1) (5,5) (7,7) flow limits = (0, ∞) costs = transportation costs for each carrier-destination pair Minimize overall flow cost 6 carrier nodes 12 destination nodes

Graphical representation 1. ABCT 2. IRST 3. LAST 4. MRST 5. NEST 6. PSST A. Atlanta B. Everett C. Ephrata D. Riverview E. Carson F. Chamblee G. Roseville H. Hanover I. Sparks J. Parsippany K. Effingham L. Kearny (1,4) (7,8) (6,7) (0,7) (0,3) (4,4) (1,1) (3,3) (5,5) (1,1) (2,2) (1,1) (5,5) (7,7) flow limits = (0, ∞) costs = transportation costs for each carrier-destination pair Minimum load commitments Number of available trucks Number of loads that must be sent to Atlanta flow = # of Atlanta loads assigned to MRST flow = total # loads assigned to MRST

Decision variables 1. ABCT 2. IRST 3. LAST 4. MRST 5. NEST 6. PSST A. Atlanta B. Everett C. Ephrata D. Riverview E. Carson F. Chamblee G. Roseville H. Hanover I. Sparks J. Parsippany K. Effingham L. Kearny X1 X2 X3 X4 X5 X6 XA XB XC XD XE XF XG XH XI XJ XK XL Xij where i  {1,…,6} and j  {A,…,L}

Input data

Decision variables in the worksheet # of Atlanta loads assigned to MRST Total # loads assigned to MRST Total # loads delivered to Atlanta

Node constraints 1. ABCT 2. IRST 3. LAST 4. MRST 5. NEST 6. PSST A. Atlanta C. Ephrata D. Riverview E. Carson F. Chamblee G. Roseville H. Hanover I. Sparks J. Parsippany K. Effingham L. Kearny XA XB XC XD XE XF XG XH XI XJ XK XL Constraint for 1.ABCT: X1=X1A+X1B+X1C+X1D+X1E+X1F+X1G+X1H+X1I+X1J+X1K+X1L Constraint for B.Everett: X1B+X2B+X3B+X4B+X5B+X6B=XB One constraint for each of the 18 nodes: inflow = outflow X1 X2 X3 X4 X5 X6 B. Everett

Arc constraints 1. ABCT 2. IRST 3. LAST 4. MRST 5. NEST 6. PSST A. Atlanta B. Everett C. Ephrata D. Riverview E. Carson F. Chamblee G. Roseville H. Hanover I. Sparks J. Parsippany K. Effingham L. Kearny XA XB XC XD XE XF XG XH XI XJ XK XL Two constraints for each of the 72 arcs: flow ≤ upper bound flow ≥ lower bound Constraints for arc entering node 1.ABCT: X1≥1, X1≤4 Constraints for arc connecting nodes 1.ABCT and C.Ephrata: X1C≥0, X1C≤∞ Constraints for arc leaving node C.Ephrata: XC≥3, XC≤3 X1 X2 X3 X4 X5 X6