General
Topic outline
INTRODUCTION
Welcome to Operations Research and Optimisation. This module introduces students to the operational research methodologies and their application to real-life problems. Emphasis will be on the use of operational research approaches to support decision-making in an data extensive environment. The learning outcomes of the module are as follows:
- Comprehend the fundamentals of operations research models.
- Perform the operational research modeling using computer software
- Interpret the output of operational research modeling in solving practical problems in the industry
WEEK 1 - TRANSPORTATION MODELS & LINEAR PROGRAMMING
Our very first topics are transportation models and liar programming. The learning outcomes are as follows:
1. To model transportation, assignment, and transshipment problems in appropriate forms of the linear programming model
2. To present a real-world problem into a linear programming model
3. To interpret the sensitivity report generated by computer software.
Provide an example of a real-life problem where linear programming is suitable as a model to solve. You need to identify the parameter clearly in your description.
Topic 3
Now that we have been exposed to the wonderful application of linear programming. We will delve deeper into goal programming and integer linear programming. The learning outcomes are as follows:
1. To model a real-world problem into a goal programming model with different goals and priority levels.
2. To present a real-world problem as a linear programming model.
3. To interpret the sensitivity report generated by computer software.
Consider the in-class exercise.
M&D chemicals produces two products that are sold as raw materials. Management has specified that the combination production for product 1 and 2 must total at least 350 gallons. Separately, a major customer’s order for 125 gallons of product 1 must also be satisfied. Product 1 requires 2 hours of processing time per gallon while product 2 requires 1 hour of processing time per gallon, and for the coming month, 600 hours of processing time are available. M&D’s objective is to satisfy the above requirements at a minimum total production cost. Production costs are $x per gallon for product 1 and $y per gallon for product 2.
Now set a range of values for x and y; generate a series of sensitivity reports and analyze the trend from your values entered. What are some of the conjectures you would derive?
Topic 4
Hopefully, you are progressing well in the module. Please reach out to me if you have any challenges in keeping up. With many applications of linear programming explained, we now explore non-linear programming as well as network models where applications of linear programming is evident. The learning outcomes are as follows:
1. To apply a non-linear programming model to a real-world problem.
2. To apply appropriate network algorithms to find the shortest path to the destination.
Explain with an example in a real-world context where a minimal spanning tree problem is suitable to solve.
Topic 5
In the process of optimization, we need make certain deicsions in our algorithms. Therefore, the next topic of our module is decision analysis. A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs. The decision alternatives are the different possible strategies the decision maker can employ. • The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive. The learning outcomes of htis topic is:
To apply appropriate decision techniques (without or with probability) to find the best decision for a problem.
With reference to the exercise completed in class:
A company is deciding whether to develop and launch a new product. Research and development costs are expected to be $400,000 and there is a 70% chance that the product will be successful. If it is successful, the expected revenue and the probability of each occurring have been estimated as follows, x, y, and z respectively depending on whether the product’s popularity is high, medium, or low. Set a range of values for x,y, and z and analyze the consequence of its impact on the net profit for each possible scenario and action.
Topic 6
In our previous topic, we witnessed the involvement of probabilistic statements in the occurrence of our events, to further investigate this phenomenon, we study Markov process models. They are useful in studying the evolution of systems over repeated trials or sequential time periods or stages. They have been used to describe the probability in the following scenarios:
- A machine that is functioning in one period will continue to function or break down in the next period.
- A consumer purchasing brand A in one period will purchase brand B in the next period
Further to the example on slide #15: A rat is placed in the maze of the figure. In room 1 is a cat and in room 4 is a piece of cheese. If the rat enters either of these rooms he does not leave it. If he is in one of the other rooms, each time period he chooses at random one of the doors of the room he is in and moves into another room. If the rat starts out in Room 3, estimate the probability he winds up in room 1? Room 4?
Topic 7
Congratulations to you for progressing to our last 3 topics!
We now turn a page to study queuing systems that have vast applications in our daily lives. For instance, we wait in line to pay our bills; we wait to be served by a waiter; we wait for a traffic jam to clear. The learning outcome of this topic is to model and evaluate a waiting line problem.
Further to the example in slide #32: Assume there are Six administrative assistants who use an office copier. The average time between arrivals for each assistant is 40 minutes, which is equivalent to an arrival rate of 0.025 arrivals per minute. The mean time each assistant spends at the copier is 5 minutes, which is equivalent to a service rate of 0.20 minutes. Use M/M/1 model with a finite population to determine probability that the copier is idle. In your computation, what is conjecture of the relationship between the number of administrative assistante and the probability that the copier is idle.
Topic 8
We have seen modeling processes using analytical methods. However, there are practical cases where the process is too complex for analytical approaches. Therefore, Monte Carlo simulation is used instead.
The learning outcome for this topic is to use Monte Carlo simulation to model a complex real-world problem
Further to the example on slide #27, passenger 1 begins being served by the passport control inspector immediately. His service time is 1:30 (90 seconds) at which time he goes immediately to the baggage inspector who waves him through without inspection. Passenger 2 begins service with passport inspector 1:30 minutes (90 seconds) after arriving there (as this is when passenger 1 is finished) and requires 1:00 minute (60 seconds) for passport inspection. He is waved through baggage inspection as well. This process continues in this manner.
Predict a mathematical relationship of a increase of 10 seconds in service and the length of time it will take to clear the first 10 passengers
Topic 9
Time flies and we are at the last topic of the module. we intend to study modeling of project management using Gantt chart; project scheduling with known activity times; project scheduling with uncertain activity times. The learning outcome of this topic is solve a project scheduling problem using PERT/CPM
Further to the example on slide #36, explore the scenario if one of the items, A-J, actually took more than 20% of the pessimistic time to complete, and examine the impact on the likelihood the project can be completed within 20 weeks.