Chapter4

Question 1

Front

Answer 1

Back

Question 2

Planning the Product Portfolio

Answer 2

Objective: Distribute available capacity among production lines I, II, and III.

Revenue per Unit:
- Line I: 20
- Line II: 6
- Line III: 8

Resources and Constraints:
- Mill: 200 h/week
- Turnery: 100 h/week
- Grindery: 50 h/week

Technology Matrix:
- Mill: [8, 2, 3]
- Turnery: [2, 3, 0]
- Grindery: [2, 0, 1]

Goal: Optimise resource distribution.

Question 3

Product Portfolio Planning: Linear Program

Answer 3

Objective: Maximise revenue with linear programming.

Formula:
Maximise 20x1 + 6x2 + 8x3

Constraints:
- 8x1 + 2x2 + 3x3 ≤ 200
- 2x1 + 3x2 ≤ 100
- 2x1 + x3 ≤ 50
- x1, x2, x3 ≥ 0

Question 4

Covering Problems

Answer 4

Definition: Problems aimed at meeting requirements with minimal resources, not maximising revenue.

Steps:
- Identify workload
- Define shifts within legal contracts
- Calculate required workers to minimise costs

Question 5

Example of Covering Problem

Answer 5

Objective: Cover demand per time period at minimal cost.

Demand per Time Period:
- 000–0600: 2 workers
- 0600–1000: 8 workers
- 1000–1200: 4 workers
- 1200–1600: 3 workers
- 1600–1800: 6 workers
- 1800–2200: 5 workers
- 2200–2400: 3 workers

Shifts:
- 4h work, 1h break, 4h work
- Possible start times: every hour

Question 6

Dynamic Optimisation Problems

Answer 6

Definition: Time-sequenced decision-making process.

Formulation:
- Use state variables (s_t) and decision variables (x_t)
- Example: stock management - end stock = start stock + production - demand

Question 7

Multi-period Investment Planning

Answer 7

Example: Investment options over multiple time periods with cash flow constraints.

Investments:
- Construction South: costs/gains per period
- Building Complex
- Hotel

Constraints: Ensure cash balance every period with borrowing and saving options.

Question 8

Network Optimisation: Transportation Problem

Answer 8

Objective: Minimise transport costs across a network.

Formulation:
Minimise x₁ + 3x₂ + 2x₃ +…

Constraints: Flow balance at nodes - example: A: x₁ + x₃ = 9

Question 9

Bill of Materials (BOM)

Answer 9

Definition: List of required inputs for each product.

Structure:
- Example matrix format with products (P_i) and materials (P_j)

Question 10

Input-Output Models

Answer 10

Goal: Determine total production demand based on primary demand.

Equation:
x = (I - A)^-1 v

Application: Calculate total required units per product.