Operational Research: Optimization and Decision Making

Understand the assignment problem, matching each facility to one job to minimize cost or maximize output, using the Hungarian method and a cost matrix for 1-to-1 assignments.

Explore the application areas of the assignment problem across machines, sales, contracts, teachers, and accounts, and learn stepwise methods to solve it using cost matrices, dummy rows, and zero-cover techniques.

The lecture presents a four by four assignment problem, minimizing total cost by applying the Hungarian method with row and column minimization, followed by allocation.

Form a rectangular allocation table, place allocations at zeros, and cover zeros with the fewest lines. Subtract the smallest uncovered value, add at intersections, and repeat.

Demonstrates the final solution of minimization for assigning jobs to people using a balanced sum method, subtracting and updating matrix elements to reach minimum cost.

Balance the unbalanced sum by adding a dummy row when columns exceed rows, then apply column minimization and remove the actual cost; row minimization is skipped.

Apply column minimization to a cost matrix, cover zeros with lines to find an optimal assignment, and allocate tasks to subordinates minimizing total hours in an unbalanced sum.

Explore maximizing profit in an assignment problem by allocating five machines to four jobs, balancing a matrix with a dummy row or column, and subtracting the highest element from entries.

Perform row minimization in a maximization problem, skip column minimization when every column has a zero, then cover zeros with minimum lines, and apply subtract/add to reach the optimum allocation.

Allocate zeros in the matrix to obtain an optimal assignment, ensuring every row and column has a zero. Maximize profit by assigning machines to jobs, yielding 376 lakhs.

Examine the transportation problem, a special linear programming model that minimizes total shipping cost from multiple sources to multiple destinations, balancing supply and demand in two phases.

Learn steps to solve a transportation model: formulating in matrix form, obtaining a basic feasible solution via northwest corner, least cost, or Vogel's method, and applying the Modi optimality test.

Explore initial basic feasible solutions for a transportation problem using the northwest corner, least cost, and Vogel's approximation methods, and perform the optimality test to verify cost minimization.

Use northwest corner method to obtain the initial basic feasible solution by allocating from the top-left, balancing supply and demand, and minimizing transportation cost; total cost is rupees 980.

Apply the least cost method to obtain an initial basic feasible solution for transportation problems by allocating from the lowest cost cells in the matrix.

Apply Vogel's approximation method to determine the initial basic feasible solution by computing row and column penalties, selecting the highest, and allocating supply and demand accordingly.

This lecture continues solving a transportation problem using Vogel's approximation method, computing a basic feasible solution and total cost, and comparing with least-cost and northwest corner results.

In operational research, explore Vogel's approximation method for transportation problems, derive an initial basic feasible solution, compare costs (920 vs 960), and introduce Modi's method for optimality.

Learn to apply the MODI method for optimality testing in a transportation problem, building from a basic feasible solution obtained via northwest corner, least cost, or Vogel's method.

Apply the Modi method to test optimality from the initial basic feasible solution found by Vogel's approximation method, constructing the u_i and v_j for occupied and non-occupied cells.

Calculate opportunity cost for non occupied cells using delta_ij = c_ij − u_i − v_j, identify positive allocations, and determine the minimum total transportation cost via the loop diagram.

Compute opportunity costs, construct a loop diagram from the allocated cells, adjust allocations, and confirm the optimality test to obtain the minimum transportation cost of INR 920.

Explore project network analysis with pert and cpm, focusing on planning, scheduling, and controlling interrelated activities under limited resources, using forward and backward passes to manage large, complex projects.

Explore the components of Pert and CPM networks: activities (arrows) and events (nodes), merge and burst events, preceding and succeeding activities, and dummy activities to avoid errors.

Master constructing project network diagrams by eliminating looping errors, ensuring one start and end, left-to-right flow, unique activities, and proper event numbering per Fulkerson's rule.

Explore critical path analysis within a project network, identifying the longest path and critical activities, and learn to compute earliest and latest times via forward and backward passes.

Explore the practical sum of CPM by constructing a project network, assigning activity durations, and computing earliest and latest start times to determine the critical path.

Compute earliest times with a forward pass by taking the maximum at each event where activities converge, then derive latest times via the backward pass, and identify the critical path.

Compute earliest and latest start/finish times, total float, and the critical path in a CPM network, and illustrate how to identify the critical path for project optimization.

Explore program evaluation and review technique (pert) and its differences from cpm, using three time estimates—optimistic, most likely, and pessimistic—to compute expected time and variance.

Compute pert durations from optimistic, most likely, and pessimistic times, draw the project network, and identify the critical path while assessing completion probabilities for 19 days and 26 days.

Explore practical PERT calculations, including optimistic, most likely, and pessimistic times to derive expected durations and variances, identify the critical path, and assess project completion probabilities.

Compute project completion probabilities using PERT, the critical path, and the most likely time. Apply standard deviation, z-scores, and normal distribution to compare PERT with CPM.

Explore decision theory, decision making under uncertainty and risk, states of nature, payoffs and payoff tables, and criteria such as maximin, maximax, Laplace, and alpha-weighted decisions.

Explore decision theory through a regret table, comparing actions using minimax, Laplace (average payoff), and Hurwicz criteria to identify the best decision under uncertainty.

Apply decision theory under risk with an expected monetary value analysis, using a bakery payoff table to determine the optimal pies to bake (14) for maximum profit of 63.4 INR.

Learn to model sequential decisions with a decision tree for investments across economy states, using probabilities and returns to identify the optimum action for financial planning.

Compute the expected profit for investments A, B, C, and D using the given probabilities and returns in INR, then identify D as the optimum action via a decision tree.

Construct a decision tree to compare plans: small cans with other products (cost 8 lakh) and tv ads (cost 20 lakh), using market response probabilities and revenues to maximize profit.

Evaluate two decision-tree plans for a cold drink, estimate revenue and profit from market responses with probabilities and costs, and select the higher-profit plan (small cans vs tv ads).

Explore queueing theory with service systems, arrivals, single or multiple servers and queues, and queue disciplines from static first-come-first-served to dynamic priority or random service.

The lecture outlines key queuing characteristics: arrival patterns (interarrival time, batch arrivals, bulking, reneging, jockeying), service patterns, and queue discipline, plus their applications.

Examine the limitations and complexity of queueing theory and learn key concepts such as arrival rate (lambda), service rate (mu), traffic intensity, L, L_q, W, W_q, and PN.

This lecture presents a practical single-server queue with Poisson arrivals and exponential service, computing lambda and mu, utilization (0.83), idle probability (0.16), and the probability of four customers (0.115).

Compute the system's expected number of customers as five, with about four waiting in a queue, the non-empty queue length about six, and the average wait time at 25 minutes.

Compute the single-server model metrics: the expected time in the system is 30 minutes and the probability of waiting over 10 minutes is 0.594, with utilization 0.83.

analyze a four-server queue with 12 arrivals per hour and 4 per-hour service rate, computing average in system, average waiting, and probability of waiting.

Compute the practical sum for a four-server queue by evaluating the series terms and idle probability P0, yielding P0 ≈ 0.0377.

Calculate the average number of customers in the system and queue for a four-server model using L and Lq, yielding 4.52 in the system and 1.52 in the queue.

Apply the multiple server model to compute time in the system (22.6 minutes) and queue wait (7.6 minutes) using lambda 12 and mu 4, plus a 0.5 wait probability.