A feasible solution is a setting of variables that satisfies all constraints. It can be shown that an optimal solution will be on one the corner, in other words intersection of some constraints.

The standard algorithm to solve linear programs is Simplex algorithm.

Dual Program

The idea behind is placing a lower bound is that we are founding suitable nonnegative multipliers for the constraints so that we take their sum, the coefficient of each in the sum is dominated by the coefficient in the objective function. In the example below, the constraints are multiplied with 1.

The problem of finding the best lower bound can be formulated as a linear program and called dual program.

LP-duality Theorems

Strong Duality Theorem Primal finite optimum value is equal to the dual finite optimum value.

Weak Duality Theorem

proof:

Complementary Slackness Conditions

if then
if then

A solution is optimum when primal and dual feasible and complementary slackness conditions are satisfied.
By construction every feasible solution to the dual program gives a lower bound on the optimum solution.

Given any acyclic directed graph a minimum path cover of G can be obtained by finding a maximum matching in a properly constructed bipartite graph.

Let’s see on an application.

A transportation problem

A company wants to organize some transports between given locations. Each transport has a specific departure time. The same vehicle can be used for two transports i and j if the following are satisfied:
(1) arrival location of i and departure location of j are the same
(2) arrival time of i is earlier then the departure time of j

The objective is to find the minimum number of vehicles necessary to ensure all the transports.

Solution method

Let us define a bipartite graph G = (U,V,E) where vertex sets U and V represent locations. We put an edge between two vertices u_i and v_j if the transport j can be done by the same vehicle after transport i.

The minimum number of vehicles to make all transports is equal to n-M where M is the size of a maximum matching. Each edge in the matching indicates that the corresponding transports can be done one after the other by the same vehicle. Paths in the cover are obtained by following the edges of the matching and every unmatched vertex corresponds to the end of a path.

An example

Company's transportations

Corresponding bipartite graph and the maximum matching

According to this matching transports (t1,t4,t5) can be done with a single vehicle as well as (t2,t6) and (t3,t7). Therefore the company needs 3 vehicles.

Solution of the path cover problem

What is rescheduling? What is the motivation behind it?

Rescheduling is the process of updating an existing production schedule in response to disruptions or other changes.
In practice, rescheduling is done periodically to plan activities for the next time period based on the state of the system. It is also done occasionally in response to significant disruptions. Because time estimates are incorrect and unexpected events occur, precisely following a schedule becomes more difficult as time passes. In some cases, the system may follow the sequence that the schedule specifies even though the planned start and end times are no longer feasible. Eventually, however a new schedule will be needed.

Types of disturbances:

• machine failures
• processing time delays
• rush orders
• quality problems
• unavailable material

Scheduling point The point of time when a scheduling decision is made.

Scheduling period Time between two consecutive scheduling points.

Scheduling frequency The inverse of the scheduling period and it measures how often rescheduling is performed.

Scheduling stability It measures the number of revisions or changes that a schedule undergoes during execution.

Schedule robustness It measures how much disruptions would degrade the performance of the system as it executes the schedule.

What are the types of rescheduling environments?



Static Finite set of jobs

Deterministic No uncertainty about the future. The specified schedule can be followed without any modification

Stochastic Some variables are uncertain. For instance processing times.

Dynamic Infinite set of jobs. Jobs continue to arrive over an infinite time horizon. Each job requires scheduling before it can be processed.

No arrival variability No uncertainty or variability in the arrival process. The jobs to be processed are known in advance. The production schedule is continuously repeated. If the jobs can be grouped into minimal part set that is continuously repeated then a single scheduling decision is needed. This yields a cyclic scheduling problem.

Arrival variability Uncertainty in job arrivals, but all jobs follow the same route.

Process flow variability There may exist process flow variability along with the variability in job arrivals. Job shops often have this characteristic, since there are many products, but a limited subset of them are being processed at any given time. Thus a specific product’s arrival process has great variability. In some situations, no advance information is available about jobs before thy arrive.

Subcontracting or overtime In some cases there may exist a potential additional capacity using subcontracting or overtime.

How frequent reschedule? Strategies



Dynamic scheduling (Online / reactive scheduling ) Dispatching rules or other heuristics to prioritize jobs waiting for processing.

Dispatching rules When a machine become available it chooses from among the jobs waiting to be processed using a dispatching rule that sorts the jobs by some criteria. Common dispatching rules employ processing times and due dates in simple rules and complex combinations.

Predictive-Reactive Scheduling Two main steps: (i) Generating a production schedule (ii) Update the schedule in response to a disruption.

Periodic In many industrial situations, scheduling is done on a periodic basis, especially in environments where there is no online data acquisition from the shop floor. Unfortunately, following an established schedule in the face of significant changes in the system status may compromise performance. Determining the optimal rescheduling period is also a difficult task when using this policy.

Event-driven

• When a machine failures occur
• (Dynamic scheduling environment) When the total number of job arrivals reaches a threshold
• Every time a new job arrives
• External events create an additional capacity

How to solve rescheduling problems? Methods



Generating robust schedules Simple schedule adjustments (like right shifts) require little effort and easy to implement. However, they may lead to poor system performance. Robust schedule is an attempt to maintaining good system performance with simple schedule adjustments.

Right-shift Postpone each remaining operation by the amount of the time needed to make the schedule feasible.

Partial scheduling Reschedule only operations affected directly / indirectly by the disruption.

Matchup scheduling Repairs a production schedule when a disruption occurs. This procedure uses heuristic ordering rules to resequence all jobs scheduled before a matchup point. If the tardiness cost is too large, the matchup point is increased. If the matchup point becomes too large, the method solves an integer programming.

How to compare different methods? Performance Measures

Schedule efficiency Time based measures: makespan, mean tardiness, mean flow-time, average resource utilization, maximum lateness.

Stability Measuring the deviation from the initial schedule. The starting time deviations between the new schedule and the original schedule. A measure of the sequence difference between the two schedules.

Cost Time-based performance measures do not completely reflect the economic performance of the manufacturing system. So an economic performance measure is needed. Total cost function in terms of due date, completion time, number of jobs, number of operations, operation processing times, job raw material cost, processing cost of operations, job revenue, processing start times, job release time, job tardiness, holding cost rate, tardiness cost rate. Issues such as job profitability, total cost minimization, reduction in WIP and the cost of missed due dates are more important for managers than time based measures.

What are the empirical/analytical findings?

Very high or very low scheduling, reacting to every disturbance policies are not good.

A lower rescheduling frequency lowers the number of setups by grouping similar jobs but increases manufacturing cycle time and WIP.

Matchup algorithm leads to better performance than a simple pushback strategy.

A robust partial schedule leads better system performance than dispatching rules. However as processing time variability increases, dispatching rules lead to better performance.

At tight due date conditions, the rescheduling interval is more significant effect on the performance

What are the future research directions?

More research should be done to compare the performance of manufacturing systems under predictive-reactive rescheduling policies with their performance under dynamic scheduling

More research is needed to understand how the interactions between rescheduling policies and other production planning functions (such as MRP) affect manufacturing system performance

The icons are (almost) chronogically ordered from 1984 to 2009 except Galatasaray which is intensively in my life since 1995 so I decided to put it the center. In both design (Cem and Ece), the color element are removed from the icons, I rather decide to keep them. I know it looks messy but I think without color the icons miss their impact.

I think these designs perfectly define their creators. Maybe it can be seen as an informal CV or it can be used in online dating sites. People upload their designs and a algorithm will match them according to their common interests. It would be a better approach for blind dating, don’t you think?