MOI UNIVERSITY

The three main components of a Linear Programming Problem (LPP) are:

1. Decision Variables: These are the unknowns to be determined. In project planning and management, decision variables often represent quantities to be allocated, such as the amount of resources used or the number of units produced.
2. Objective Function: This is a linear function of the decision variables that needs to be maximized or minimized. In project planning, this could be minimizing cost or time or maximizing profit or efficiency.
3. Constraints: These are the linear inequalities or equations that represent the limitations or restrictions on the decision variables. Constraints in project planning could include resource availability, budget limits, or time restrictions.

Operations Research (OR) originated during World War II to solve complex military logistics and strategy problems. It has since evolved into a discipline widely used in business, engineering, and project management.

Key Historical Milestones:

• 1930s–1940s – World War II Era: OR was developed by military planners to optimize the use of limited resources. Examples include optimizing radar deployment and anti-submarine warfare.
• 1950s – Post-War Industrial Applications: After the war, OR techniques were adopted by industries to improve efficiency in production, scheduling, and inventory control.
• 1960s–1970s – Expansion into Project Management: Techniques like the Critical Path Method (CPM) and Program Evaluation and Review Technique (PERT) were introduced to manage large-scale projects like the Polaris missile program.
• 1980s–1990s – Integration with Computers: With the rise of computers, OR models became more sophisticated and were applied to complex project scheduling, resource allocation, and risk analysis.
• 2000s–Present – Modern Applications: OR is now used in project planning and management to optimize resource utilization, manage supply chains, and support decision-making in dynamic environments using simulation and optimization software.

The four steps in the model-building process in Project Management are:

1. Define the Problem: Clearly identify the problem or decision to be addressed. This includes understanding objectives, constraints, and available data.
2. Develop the Model: Formulate a mathematical or simulation model that represents the real-world situation. This includes defining variables, relationships, and constraints.
3. Test and Validate the Model: Validate the model with historical data or scenarios to ensure it accurately represents the system and produces reliable results.
4. Implement and Monitor the Model: Apply the model in the real world, monitor its performance, and make adjustments as necessary to improve decision-making.

Operations Research helps managers in decision-making by:

1. Providing Quantitative Analysis: OR uses mathematical models and algorithms to provide data-driven insights, enabling more objective and rational decisions.
2. Optimizing Resource Allocation: OR techniques help managers allocate limited resources (time, budget, personnel) in the most efficient way to maximize productivity or minimize costs.
3. Supporting Risk and Uncertainty Management: Through simulation and probabilistic models, OR helps managers assess risks and uncertainties in complex systems, allowing for better contingency planning.

Simulation is widely used in decision-making because:

1. Handles Complex Systems: It allows modeling of complex systems that are difficult to analyze using analytical methods, such as project timelines with multiple interdependent tasks.
2. Evaluates Uncertainty: Simulation incorporates randomness and variability, enabling project managers to assess different scenarios and their probabilities.
3. Cost-Effective Testing: It allows "what-if" analysis without the cost or risk of real-world experimentation, helping managers test strategies before implementation.

There are two main types of transportation problems:

1. Balanced Transportation Problem: Occurs when the total supply equals the total demand. It is solved using methods like the North-West Corner Rule, Least Cost Method, or Vogel’s Approximation Method (VAM) to find an initial feasible solution, followed by the Stepping Stone Method or MODI Method for optimization.
2. Unbalanced Transportation Problem: Occurs when the total supply does not equal the total demand. A dummy row or column is added to balance the problem (with zero cost), and then standard methods are applied to solve it.

1. Re-order Level (ROL):
   
   ROL = Maximum Usage × Maximum Lead Time = 710 × 20 = 14,200
2. Minimum Level:
   
   Average Usage = (560 + 240)/2 = 400
   
   
   Average Lead Time = (15 + 20)/2 = 17.5
   
   
   Minimum Level = ROL − (Average Usage × Average Lead Time) = 14,200 − (400 × 17.5) = 14,200 − 7,000 = 7,200
3. Maximum Level:
   
   Maximum Level = ROL + EOQ − (Minimum Usage × Minimum Lead Time) = 14,200 + 10,000 − (240 × 15) = 24,200 − 3,600 = 20,600

Given:

• Annual demand (D) = 100,000 units
• Cost per order (S) = Kshs. 250
• Holding cost per unit per year (H) = 1% per month × 12 = 12% of cost = 0.12 × 3 = Kshs. 0.36

EOQ Formula: