Lernzettel: Optimization Strategies for Supply Chain Design

📋 Course Outline

1. Optimization in supply chain and beyond
2. Model-based decision-making support framework
3. Linear programming structure and optimality
4. Linear programming modeling steps and key issues
5. Linear programming solution approach with Excel Solver
6. Knapsack problem as an integer programming example
7. Supplier selection and multi-supplier constraints
8. Network design with opening and transportation costs

📖 1. Optimization in supply chain and beyond

🔑 Key Concepts & Definitions

• Optimization : Optimization is the search for the best possible solution given a defined objective and constraints.
• Predictive analytics inputs : Predictive analytics provide forecasts that become model inputs for optimization decisions.
• Linear programming : Linear programming is an optimization framework where the objective and constraints are linear functions of decision variables.
• Integer programming : Integer programming is optimization where some decision variables must take integer values.

📝 Essential Points

• Optimization is used across supply chain from strategy and tactics down to operations.
• Optimization is also applied beyond supply chain, including sport scheduling, radiotherapy, advertising, system engineering, and organ transplant.
• A weather forecast is an example of predictive analytics input used for optimization under uncertainty.
• The course focus is on decisions under certainty and decisions under uncertainty, with emphasis on LP and integer programming.

💡 Memory Hook

Optimization = “best answer” search, not just “make it better.”

📖 2. Model-based decision-making support framework

🔑 Key Concepts & Definitions

• Formal model : A formal model is a mathematical representation of a situation using variables, constraints, and an objective.
• Decision variables : Decision variables are the quantities the model chooses to determine the decision.
• Constraints : Constraints are mathematical restrictions that feasible decisions must satisfy.
• Objective function : The objective function is the criterion the model maximizes or minimizes.

📝 Essential Points

• The framework starts by building a formal model from what is known, what can be controlled, and what is desired.
• Model validation checks whether the model represents the situation and whether the data are reliable.
• After solving, results must be checked for reasonableness to confirm the model’s usefulness.
• The workflow includes: build model → validate → search for solutions → identify relevant issues → solve with a solver.

💡 Memory Hook

Know → Control → Want → Model → Validate → Solve → Sanity-check.

📖 3. Linear programming structure and optimality

🔑 Key Concepts & Definitions

• Decision : A decision is an assignment of values to all decision variables.
• Feasible decision : A feasible decision is one that satisfies every constraint in the model.
• Optimal decision : An optimal decision is a feasible decision that yields the best objective value among all feasible decisions.
• Non-negativity restriction : A non-negativity restriction forces decision variables to be at least zero.

📝 Essential Points

• LP decision variables can be continuous with $X\ge 0$, integer with $X\in\{1,2,3,\ldots,N\}$, or binary with $X\in\{0,1\}$.
• The objective in an LP is a linear function of decision variables and is written as Max or Min of a linear expression.
• A decision is optimal only if it is feasible and provides the best objective value among all feasible decisions.
• “Solve” an LP means finding the optimal solution, i.e., the best feasible objective value.
• Constraints in LP use relations of the form $\le$, $\ge$, or $=$ between linear expressions.

💡 Memory Hook

Feasible = respects constraints; Optimal = feasible + best objective.

📖 4. Linear programming modeling steps and key issues

🔑 Key Concepts & Definitions

• Decision variable definition : Decision variable definition is the step where each chosen quantity is given a symbol and units.
• Objective function determination : Objective function determination is the step where the criterion to maximize or minimize is expressed in linear form with correct units.
• Constraint determination : Constraint determination is the step where all restrictions are written as linear relations with correct units.
• Units consistency : Units consistency means each variable, objective term, and constraint uses compatible measurement units.

📝 Essential Points

• Key issues include defining decision variables with appropriate units and clear symbols.
• Key issues include determining the objective function with correct units for the criterion being optimized.
• Key issues include determining constraints with correct units so the model is dimensionally consistent.
• Modeling steps include writing the LP structure: decision variables, objective, constraints, and non-negativity restrictions.
• The course emphasizes that modeling choices (variables, objective, constraints) drive whether results make sense.

💡 Memory Hook

LP modeling checklist: Variables (units) → Objective (units) → Constraints (units) → Non-negativity.

📖 5. Linear programming solution approach with Excel Solver

🔑 Key Concepts & Definitions

• Excel Solver : Excel Solver is a Microsoft Excel add-in that finds optimal objective values subject to constraints.
• Decision variable cells : Decision variable cells are the spreadsheet cells Solver is allowed to change to achieve the objective.
• Constraint cells : Constraint cells are spreadsheet cells that impose limits Solver must satisfy.
• Objective cell : The objective cell is the spreadsheet cell containing the value to maximize or minimize.

📝 Essential Points

• Solver adjusts decision variable cells to satisfy constraint limits and produce the desired objective result.
• Solver is used to find an optimal maximum or minimum value for an objective function under constraints.
• In the knapsack example, decision variables are binary: $x_i=1$ if item $i$ is selected and $x_i=0$ otherwise.
• The knapsack objective is to maximize total value $\sum_{i=1}^{n} v_i x_i$.
• The knapsack capacity constraint is $\sum_{i=1}^{n} w_i x_i \le W$.
• The knapsack model enforces $x_i\in\{0,1\}$ to represent selecting or not selecting items.

💡 Memory Hook

Solver = “change variable cells until constraints hold and objective is best.”

📖 6. Knapsack problem as an integer programming example

🔑 Key Concepts & Definitions

• Knapsack problem : The knapsack problem is an integer programming model that selects items to maximize value under a weight capacity.
• Binary decision variable : A binary decision variable indicates whether an item is chosen (1) or not chosen (0).
• Capacity constraint : A capacity constraint limits the total weight of selected items to the bag capacity.
• Total value objective : The total value objective sums the values of selected items to be maximized.

📝 Essential Points

• Each item $i$ has value $v_i$ and weight $w_i$, and the bag has capacity $W$.
• The decision variable is $x_i\in\{0,1\}$ for each item $i$.
• The objective is $\max \sum_{i=1}^{n} v_i x_i$.
• The constraint is $\sum_{i=1}^{n} w_i x_i \le W$.
• The integer nature comes from the binary restriction on $x_i$, making it an integer programming example.

💡 Memory Hook

Knapsack = pick 0/1 items to maximize value without exceeding weight $W$.

📖 7. Supplier selection and multi-supplier constraints

🔑 Key Concepts & Definitions

• Supplier selection : Supplier selection is the optimization task of choosing which supplier(s) to use to meet a supply need.
• Multi-supplier selection : Multi-supplier selection is the extension where the model chooses more than one supplier under additional requirements.
• Energy supply example : The energy supply case frames supplier choice as selecting among multiple suppliers to supply a business.
• Supplier set : A supplier set is the collection of candidate suppliers (e.g., $S1$ to $S5$) considered by the model.

📝 Essential Points

• The supplier selection question asks which supplier is better among $S1$ through $S5$.
• The course presents a scenario where the best single supplier is chosen using data from an Excel file.
• A follow-up scenario asks what changes when selecting 2 or 3 suppliers instead of only one.
• The model can be extended to selecting suppliers for multiple sites, illustrated with two sites labeled A and B.
• For two sites, the supplier choice is made separately for each site (e.g., selecting among $S1$ to $S5$ for A and B).

💡 Memory Hook

Single supplier vs multi-supplier: same idea, but the selection count changes.

📖 8. Network design with opening and transportation costs

🔑 Key Concepts & Definitions

• Network design : Network design is the optimization of a supply network structure and flows to minimize total cost.
• Opening cost : An opening cost is the fixed cost paid when a facility or node is activated in the network.
• Transportation cost : A transportation cost is the variable cost of shipping from an opened facility to a destination.
• Facility activation decision : Facility activation decision determines which nodes are opened in the designed network.

📝 Essential Points

• The network design objective is to minimize the total of opening (use) costs and transportation costs.
• The example network includes candidate suppliers or facilities labeled $S1$ to $S5$ and destinations labeled A and B.
• The course also shows a “same network design” idea when the network is extended to additional nodes.
• A second diagram includes additional suppliers labeled $S6$ and $S7$ along with destinations A and B.
• The network design problem is framed as choosing which facilities to open and how to route shipments to minimize total cost.

💡 Memory Hook

Network design = pay to open + pay to ship; minimize the sum.

📊 Synthesis Tables

LP vs Integer programming (focus of course)

Feature	LP	Integer programming
Decision variable types	Continuous (e.g., $X\ge 0$)	Some variables restricted to integer values (e.g., $X\in\{1,2,\ldots,N\}$ or binary)
Model form in course	Linear objective and linear constraints	Integer restrictions added to the LP structure

⚠️ Common Pitfalls & Confusions

1. Confusing feasibility with optimality: a feasible decision may not be the best objective value.
2. Writing an LP with non-linear objective or constraints: the course’s LP structure requires linearity.
3. Forgetting variable type restrictions: binary/integer requirements are what make knapsack and supplier selection discrete.
4. Mixing up the objective and constraints in Excel Solver: Solver changes decision variable cells to optimize the objective while satisfying constraint cells.
5. Assuming “solve” means any solution: the course defines solving as finding the optimal solution among feasible ones.

✅ Exam Checklist

1. Define optimization in mathematical terms as best possible solution under an objective and constraints.
2. List the elements of the model-based decision-making framework and describe the validation and reasonableness checks.
3. State the LP structure: decision variables, linear objective, and constraints using $\le$, $\ge$, or $=$.
4. Distinguish decision, feasible decision, and optimal decision using the course definitions.
5. Identify allowed variable types in LP: continuous ($X\ge 0$), integer ($X\in\{1,2,\ldots,N\}$), and binary ($X\in\{0,1\}$).
6. Write the LP linear form for objective and constraints in the course’s notation style.
7. Explain the key modeling issues: decision variables, objective function, and constraints with correct units.
8. Describe how Excel Solver works: decision variable cells, constraint cells, and objective cell, and that it finds max/min under constraints.
9. Formulate the knapsack model: binary $x_i$, objective $\max \sum v_i x_i$, and capacity constraint $\sum w_i x_i \le W$.
10. Explain supplier selection as choosing among candidate suppliers and extend it to selecting 2 or 3 suppliers.
11. Explain network design as minimizing total opening (use) and transportation costs, including the idea of opening facilities and routing to destinations A and B.