Network Flow Problems Example of Network Flow problems: What shipping plan minimizes cost to ship from m warehouses to n customers? How do you maximize efficiency of a machine shop through the assignment of jobs to a group of machines? What is the maximum flow that can be obtained through a series of pipes? How do you maximize the flow of traffic through a series of one-way streets? What is the shortest route if a truck must make a “milk-run” through a series of stops (TSP)? How do you schedule a series of project activities in order to minimize the total project duration time?  

Network Flow Problems – Transportation Problem a – warehouse capacity b – customer demand 1 b1 c11 1 a1 c12 2 b2 a2 2 3 b3   a3 3 4 b4 Warehouse Customer

Network Flow Problems – Transportation Problem Given: Supply Vector: a = [a1 a2 … am ] Demand Vector: b = [b1 b2 … bn ] Transportation cost matrix: c Objective: Find shipping plan that minimizes transportation cost that meets all customer demands while being constrained by supply capacities.

Network Flow Problems – Transportation Problem Minimize s.t. Total Cost (supply restriction) i = 1…m (demand requirement) j = 1…n

Network Flow Problems – The Assignment Problem Consider the problem of assigning n assignees to n tasks. Only one task can be assigned to an assignee, and each task must be assigned. There is also a cost associated with assigning an assignee i to task j, cij. The objective is to assign all tasks such that the total cost is minimized.  

Network Flow Problems – The Assignment Problem Examples: Assign people to project assignments Assign jobs to machines Assign products to plants Assign tasks to time slots  

Network Flow Problems – The Assignment Problem To fit the assignment problem definition, the following assumptions must be satisfied: The number of assignees and the number of tasks are the same (denoted by n). Each assignee is to be assigned to exactly one task. Each task is to be assigned to exactly one assignee. There is a cost cij associated with assignee i performing task j. The objective is to determine how all n assignments should be made to minimize the total cost.  

Assignment Problem – Flow Diagram a – assignee t – tasks a1 c11 1 1 t1 c12 a2 2 2 t2 a3 3 3 t3   an n n cnn t4 assignees tasks

Assignment Problem – Cost Matrix Let the following represent the standard assignment problem cost matrix, c:

Assignment Problem – Conversion to Standard Cost Matrix Consider following cost matrix, how do you convert to satisfy the standard definition of the assignment problem? Add “big M” to avoid incompatible assignments, and add a dummy assignee (or task) to have equal assignees and tasks.

Assignment Problem – Math Formulation Minimize s.t. Total Cost i j Does this formulation look familiar? Is this a Linear Program?

Network Flow Problems – Maximal Flow Problems Consider the following flow network: k1n ks1 1 n s k13 k21 k3n 3 ks2 2 k23   The objective is to ship the maximum quantity of a commodity from a source node s to some sink node n, through a series of arcs while being constrained by a capacity k on each arc.

Maximal Flow Problems Examples: Maximize the flow through a company’s distribution network from its factories to its customers. Maximize the flow through a company’s supply network from its vendors to its factories. Maximize the flow of oil through a system of pipelines. Maximize the flow of water through a system of aqueducts. Maximize the flow of vehicles through a transportation network.  

Maximal Flow Problems Definitions: Flow network – consists of nodes and arcs Source node – node where flow originates Sink node – node where flow terminate Transshipment points – intermediate nodes Arc/Link – connects two nodes Directed arc – arc with direction of flow indicated Undirected arc – arc where flow can occur in either direction Capacity(kij) – maximum flow possible for arc (i,j) Flow(f ij) – flow in arc (i,j). Forward arc – arcs with flow out of some node Backward arc – arc with flow into some node Path – series of nodes and arcs between some originating and some terminating node Cycle – path whose beginning and ending nodes are the same  

Maximal Flow Problems – LP Formulation 1 n f s 3 2 Objective: Maximize Flow (f) Constraints: 1) The flow on each arc, fij, is less than or equal to the capacity on each arc, kij. 2) Conservation of flow at each node. Flow in = flow out.  

Maximal Flow Problems – LP Formulation 1 n f s 3 Max Z = f st s) fs1 + fs2 = f 1) f13 + f1n = fs1 + f21 2) f21 + f23 = fs2 3) f3n = f13 + f23 n) f = f3n + f1n 0 <= fij <= kij 2 Objective: Maximize Flow (f) Constraints: The flow on each arc, fij, is less than or equal to the capacity on each arc, kij. Conservation of flow at each node. Flow in = flow out.  

Maximal Flow Problems – Conversion to Standard Form What if there are multiple sources and/or multiple sinks? n1 s1 1 n2   3 s2 2

Maximal Flow Problems – Conversion to Standard Form Create a “supersource” and “supersink” with arcs from the supersource to the original sources and from the original sinks to the supersink. What capacity should we assign to these new arcs? n1 f s1 n 1 f s n2     3 s2 2

Maximal Flow Problems – Conversion to Standard Form What if there is an undirected arc (flow can occur in either direction)? See arc (1,2). f 1 n f s k12 3   2

Maximal Flow Problems – Conversion to Standard Form Create two directed arcs with the same capacity. Upon solving the problem and obtaining flows on each arc, replace the two directed arcs with a single arc with flow | fij – fji |, in the direction of the larger of the two flows. f 1 n f s k21 k12 3   2

Project Management - PERT/CPM Let each node represent a project event/milestone (node 1 is start of project, node 11 is end of project). Let each arc represent a project task/job. Each arc is identified by a job letter and duration. Note the dummy jobs indicating precedence that jobs H and I must complete before K or L begins. J,2 D,12 H,5 7 M,5 K,8 A,5 B,8 C,15 G,11 E,10 1 2 3 4 5 6 9 10 11 L,14 N,5 F,8 I,4 8  

Project Management - PERT/CPM What questions might project managers be interested in? How long will the project take? Can I add manpower or tools to reduce the overall project length? To which tasks should I add manpower? What tasks are on the critical path? Is the project on schedule? When should materials and personnel be in place to begin a task? Other?…

Project Management - Examples University Convocation Center Windsor Engine Plant Other major construction projects Large defense contracts NASA projects (space shuttle) Maintenance planning of oil refineries, power plants, etc… other…

Project Management – Minimum Completion Time 1 2 4 5 B,1 D,2 3 LP Solution: Let ti be the time of event i. Min Z = t5 – t1 s.t. t2 – t1 >= 3 t3 – t2 >= 0 t3 – t1 >= 1 t4 – t2 >= 4 t4 – t3 >= 2 t5 – t4 >= 5 ti >= 0 for all i

CPM – Critical Path Method Can normal task times be reduced? Is there an increase in direct costs? Additional manpower Additional machines Overtime, etc… Can there be a reduction in indirect costs? Less overhead costs Less daily rental charges Bonus for early completion Avoid penalties for running late Avoid cost of late startup CPM addresses these cost trade-offs.

CPM – Critical Path Method LP Approach: Let tij – decision variable for time to complete task connecting events i and j. kij – normal completion time of task connecting events i and j. lij – minimum completion time of task connecting events i and j. Cij – incremental cost of reducing task connecting events i and j. Model I: Given project must be complete by some time T, which tasks should be reduced to minimize the total cost? Min s.t. for all jobs (i,j) for all i

CPM – Critical Path Method LP Approach: Model II: Given an additional budget of $B for “crashing” tasks, what minimum project completion time can be obtained while staying within your budget? Min s.t. for all jobs (i,j) for all i