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ECON2326 Operations Research课程简介
Operations Research utlizes mathematical modeling, techniques, and algorithms, to most appropriately allocate resources and meet goals. An alternative title for the course might be “Mathematical Theory of Management, Control and Decision Making” . In general the course is intended for students with interests in applied math, statistics, economics, engineering and science. A second semester of the course devoted to probabilistic techniques will be offered in the spring semester.
PREREQUISITES
Linear programming is a widely-used optimization technique that involves linear objective functions and linear constraints. The simplex algorithm is a widely-used algorithm for solving linear programming problems, and it involves moving from one feasible solution to another in an iterative fashion until an optimal solution is found. Sensitivity analysis involves examining the effects of changes in the objective function coefficients and constraint values on the optimal solution.
Dual problems are closely related to linear programming problems and involve the optimization of a dual objective function subject to dual constraints. The dual problem is used to provide information about the original problem, including bounds on the optimal objective value and information about the shadow prices of the constraints.
Integer programming is a type of linear programming that involves additional constraints that require the variables to take on integer values. Network models involve the modeling of complex systems using networks, such as transportation or communication systems. Dynamic programming is a powerful optimization technique that involves breaking down a problem into smaller subproblems and solving each subproblem in a recursive manner.
Finally, the KKT conditions are a set of necessary conditions for an optimization problem to have an optimal solution. These conditions involve the first-order conditions for optimality and the complementary slackness conditions, and they are used to analyze optimization problems and to derive insights about their solutions.
ECON2326 Operations Research HELP（EXAM HELP， ONLINE TUTOR）
Consider the shortest-path model as discussed in Section 5.2. Suppose that in addition to the costs for passing through the edges, a cost of $2 \frac{1}{2}$ is incurred every time a change of direction occurs in the path. First define the value function with appropriate state variables, and then define a dynamic programming algorithm to calculate the minimum-cost path. Apply the algorithm to the data in Figure 5.1.
Suppose that in the Manhattan network of Figure 5.1, heavily-loaded trucks go from $A$ to $B$. The numbers $R(x, y)$ and $U(x, y)$ along the road segments indicate the total weight in tons the road segment in question can tolerate. Explain that the value function $f(x, y)$ defined as the heaviest weight possible from point $(x, y)$ to the endpoint $B$ satisfies the recursive relation
$$
f(x, y)=\max {\min {R(x, y), f(x+1, y)}, \min {U(x, y), f(x, y+1)}}
$$
and calculate the maximum-load path.
Solve the shortest-path problem from Exercise 3.2 in Chapter 3 with dynamic programming. Define the value function and the recursion relation.
In the Wild West, traveling by stagecoach was risky. A trip from Santa Fé to Fort Thomas took three days. The possible routes are given in Figure 5.6. The numbers along the possible day trips give the probability, as a percentage, of a robbery along the route in question. The beloved country singer Lilian Song from Santa Fé must perform in Fort Thomas. She asks her manager to determine the route the stagecoach should take to maximize the probability of arriving safely in Fort Thomas.
(a) Define the value function and the recursion relation, and then carry out the calculations for the manager.
(b) Once the safest path has been determined, Lilian, capricious as ever, changes her mind and decides to follow the path for which the greatest distance traveled in one day is as small as possible (now, the numbers along the arcs indicate the distances in units of 10 kilometers). Again, define the value function and the recursion relation, and calculate the optimal path.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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