(ii) The expected maximum profit.

(iii) The expected minimum profit.

ii) Number of people who liked three attributes

iii)Proportion of persons that liked the colour.

iv) Proportion of persons that liked the brand name.

v) Proportion of persons that liked the packaging.

vi) Attribute(s) to be incorporated in the product.

Key Steps in Optimization Using Differential Calculus:

1. Define the Objective Function: Formulate the problem by defining a mathematical function that represents the quantity to be optimized. Let's call it f(x) .
2. Identify Critical Points: Find the critical points of the function by setting its derivative f'(x) equal to zero. Critical points occur where the slope of the function is horizontal (slope = 0).
3. Apply the Second Derivative Test: Determine whether each critical point is a minimum, maximum, or neither. Use the second derivative f''(x) to analyze concavity. If f''(x) > 0 , the critical point is a local minimum. If f''(x) < 0 , it is a local maximum.
4. Check Endpoints (if applicable): If the optimization problem is constrained to a specific interval, check the function's values at the endpoints of the interval to ensure the solution is valid within the given constraints.
5. Interpret the Results: The values of (x) obtained from the critical points and endpoints represent the locations where the function is optimized. Substitute these values back into the original function to find the corresponding optimized quantity f(x).

By following these steps, you can use differential calculus to find the optimal solutions for various problems, such as maximizing profit or minimizing cost in business applications.

ii) The selling Price that will maximise profit

(ii) Average seasonal index for each quarter using the multiplicative model.

(iii) Year 2015 seasonally adjusted import forecasts for each quarter,

2. Critical Path Analysis:

PERT identifies the critical path in a project, which is the sequence of activities that determines the project's minimum duration. This information is crucial for project managers to focus on tasks that directly impact project timelines.

3. Resource Allocation:

By understanding the time requirements for each activity, PERT aids in efficient resource allocation. Project managers can allocate resources optimally, avoiding bottlenecks and ensuring a smooth workflow.

4. Risk Management:

PERT incorporates uncertainty by considering a range of time estimates. This allows project managers to identify high-risk activities and plan contingency measures accordingly, reducing the impact of unforeseen events on project timelines.

5. Communication and Coordination:

PERT provides a visual representation of project activities and their dependencies through network diagrams. This helps in effective communication and coordination among team members, stakeholders, and other project participants.

6. Performance Monitoring:

During project execution, PERT allows for real-time monitoring of activities. Any deviations from the planned schedule can be identified, and corrective actions can be taken promptly to keep the project on track.

7. Optimization of Project Schedule:

PERT facilitates the optimization of project schedules by identifying non-critical activities with float or slack. Project managers can make informed decisions on how to allocate time and resources to non-critical activities without affecting the overall project timeline.

(ii) The product moment correlation coefficient.

(iii) The standard error of estimate.

(iv) A 95 per cent interval estimate of the regression line.

(v) The packaging time interval for 7 units.

(ii) Saddle Point:

A saddle point is a concept in game theory associated with a payoff matrix. In a two-player zero-sum game, a saddle point occurs when the maximum value in a row is the minimum value in its corresponding column. At this point, neither player has an incentive to unilaterally change their strategy because doing so would result in a worse outcome for them. The saddle point represents a stable solution where the players have reached an equilibrium. In such cases, the game is said to have a solution in pure strategies, and the saddle point is a key element in determining optimal strategies for the players.

2. Inventory Management:

Businesses use linear programming to optimize inventory levels. It aids in deciding the optimal quantity to order and when to reorder, considering factors like carrying costs, demand variability, and order processing costs.

3. Financial Portfolio Optimization:

In finance, linear programming is applied to optimize investment portfolios. It helps in allocating assets to maximize returns while adhering to constraints such as risk tolerance, budgetary limitations, and investment guidelines.

4. Marketing Resource Allocation:

Linear programming assists businesses in allocating marketing resources effectively. It helps in determining the optimal distribution of advertising budgets across various channels to maximize reach and impact.

5. Transportation and Logistics:

Linear programming is used in optimizing transportation and logistics networks. It helps in determining the most cost-effective way to transport goods from multiple sources to multiple destinations, considering factors like transportation costs and capacity constraints.

6. Personnel Scheduling:

Businesses utilize linear programming for optimizing employee schedules. It assists in determining the most efficient assignment of staff to shifts while considering factors like labor costs, employee preferences, and legal constraints.

7. Supply Chain Management:

Linear programming is applied to optimize supply chain operations. It helps in coordinating and optimizing various activities within the supply chain, including production, distribution, and inventory management.

Advice:

(ii) The optimal salary plan for Jane Cherop based on the minimax regret criterion.

(iii) The optimal salary plan based on the expected value criterion.

Example: In a financial simulation, the endogenous variable could be the total revenue ( R ) generated based on pricing decisions and market demand.

Parameters are input variables with constant values for a given simulation. They are fixed values that do not change during the simulation run.

Example: In a manufacturing simulation, a parameter could be the cost per unit ( C ) of raw materials, which remains constant throughout the simulation.

Input variables drive the simulation. They can be either controlled (under management control) or uncontrolled (not under management control).

Example: In a project management simulation, a controlled input variable could be the project team size ( T ), while an uncontrolled variable could be external market conditions affecting project timelines.

Status variables reflect the general state of the system at various times during the simulation, capturing overall system behavior.

Example: In a healthcare simulation, a status variable might be the average patient wait time ( W ), indicating the system's efficiency in managing patient flow.

ii) The Mean daily sales of digital boxes

iii) The standard deviation of digital boxes

ii) a 95% confidence level of the mean value of outstanding sales invoices

• X̄ is the sample mean,
• s is the sample standard deviation,
• n is the sample size,
• Z is the Z-score corresponding to the desired confidence level.