Analytical Decision Making May 2017 Assignment

CUEACMS 321analytical Decision Makingmay Aug 2017 Assignment Answer A

Use the simplex method to solve the following Linear programming model (LP) problem: Max Z = 2x1 + 3x2 + 3x3 Subject to: 3x1 + 2x2 ≤ 60 -x1 + x2 + 4x3 ≤ 10 2x1 – 2x2 + 5x3 ≤ 50 x1, x2, x3 ≥ 0.

A company with three factories (X, Y, Z) and five warehouses (A, B, C, D, E) has transportation costs (in Naira) from factories to warehouses. Factory capacities and warehouse requirements are provided. Determine the optimal transportation solution to minimize costs.

Building a swimming pool consists of nine activities with specified immediate predecessors. The problem involves drawing the project network, identifying critical activities, calculating expected project duration, and determining the probability of completing the project within 25 weeks or fewer days.

Paper For Above instruction

Introduction

Analytical decision-making encompasses mathematical and quantitative methods to optimize outcomes, allocate resources efficiently, and solve complex problems in business and project management. This paper discusses three core areas in analytical decision-making: linear programming, transportation problem optimization, and project scheduling via PERT/CPM techniques. Each area employs specific models and computational methods integral to industrial and managerial decision processes.

Linear Programming Using the Simplex Method

Linear programming (LP) models aim to maximize or minimize a linear objective function subject to a set of linear constraints. The simplex method is a standard technique for solving LP problems efficiently. Consider the LP problem: Maximize Z = 2x1 + 3x2 + 3x3, with constraints:

• 3x1 + 2x2 ≤ 60
• -x1 + x2 + 4x3 ≤ 10
• 2x1 – 2x2 + 5x3 ≤ 50
• x1, x2, x3 ≥ 0

To solve this using the simplex method, the first step involves converting inequalities into equalities by adding slack variables (s1, s2, s3). The transformed model becomes:

• 3x1 + 2x2 + s1 = 60
• -x1 + x2 + 4x3 + s2 = 10
• 2x1 – 2x2 + 5x3 + s3 = 50

Initial basic feasible solution assigns zero to decision variables (x1, x2, x3) and solves for slack variables. Iterative pivot operations aim to improve the objective function value by exchanging variables in and out of the basis until no further improvement is possible. The solution yields optimal values of x1, x2, x3, and the maximum Z.

Transportation Problem Optimization

The transportation problem involves minimizing costs associated with distributing goods from factories to warehouses while satisfying capacity and demand constraints. Given data includes transportation costs, factory capacities, and warehouse requirements:

• Factories: X, Y, Z with capacities 100 units each.
• Warehouses: A, B, C, D, E with demands 500 units total.
• Transportation costs matrix (in Naira) varies between each factory-warehouse pair.

The problem is formulated as a linear optimization model with decision variables indicating shipment quantities from each factory to each warehouse. Constraints ensure capacities are not exceeded and demands met. The objective function sums total transportation costs:

Minimize Z = Σ (cost_{ij} * shipment_{ij}) for all factory-warehouse pairs.

Solving this via methods such as the transportation simplex algorithm yields the optimal shipping plan with minimal total cost, crucial for cost-efficient logistics management.

Project Scheduling: Critical Path Method and Probabilistic Analysis

Building a swimming pool involves activities with specific immediate predecessors, necessitating project scheduling techniques to determine timelines and risks. The nine activities are analyzed through PERT/CPM methodologies, involving:

a) Drawing the Project Network

Each activity is represented as a node, and arrows indicate dependencies based on immediate predecessors. The network identifies the sequence and parallel activities, visually illustrating the project flow.

b) Critical Activities

Critical activities are those on the longest path (critical path) with zero slack time. Identifying them involves calculating earliest start and finish times, and latest start and finish times. Activities with zero slack are critical, affecting the project's total duration.

c) Expected Project Completion Time

In PERT, each activity's expected time (TE) is calculated as (a + 4m + b)/6, where a, m, and b are optimistic, most probable, and pessimistic times. Summing TE values along the critical path provides the project's expected duration.

d) Probability of Completing in 25 Weeks or Fewer

The variance of the project duration is the sum of variances of activities on the critical path. Using the normal distribution approximation, the probability that the project completes within 25 weeks is calculated as the area under the normal curve to the left of 25, considering the mean (expected duration) and standard deviation (square root of variance).

Conclusion

Effective decision-making in operations and project management heavily relies on quantitative methods such as linear programming, transportation models, and project scheduling techniques. The simplex method optimizes resource allocation, transportation models enhance logistics efficiency, and PERT/CPM facilitate project time and risk management. Together, these tools enable managers to make informed, data-driven decisions to maximize efficiency and minimize costs and risks.

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