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• Script Part 5: Master Production Scheduling
• Exercise Questions 1, Exercise Questions 2, Exercise Questions 3

Note on this Note

I did not take lecture notes on Production Management. Below is an AI-generated summary of this chapter, and you’ll want to check out the resources above.

Master Production Scheduling

Master production scheduling (MPS) bridges the gap between tactical and operative planning.

• Task: Development of plant-specific production schedules considering potential capacity bottlenecks. It serves as the basis for material requirements planning.
• Planning Horizon: 3–12 months
• Result: Specific production quantities for final products and key sub-assemblies.

Types of Production Systems

Production systems can be categorized by the customer order decoupling point:

• Make to Stock (MTS): Non-order driven (forecast driven).
• Assemble to Order (ATO): Order driven.
• Make to Order (MTO): Order driven.
• Engineer to Order (ETO): Order driven.

Process Types of Production

• One-of-a-kind production: High product variety, individual processing (e.g., machines, complex projects).
• Batch production: Variable product variety, batchwise processing (e.g., hair dyes, food production).
• Mass production: Homogeneous, continuous processing (e.g., cars).

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Mass Production

Characteristics

• Typically several production sites and distribution centers.
• Integration of suppliers, production, and distribution (Supply Chain Management).
• Make-to-stock or assemble-to-order systems.

Bottleneck Analysis

In the case of a single bottleneck, the optimal criterion for product selection is the profit per unit of bottleneck consumption. Products are accepted in descending order of this metric until the capacity limit is reached.

Multi-Period Linear Optimization Model

For mass production, we minimize inventory holding costs across multiple periods.

Assumptions: Storable products, constant per unit costs, given requirements without shortages, and no extra capacity (e.g., overtime).

Indices & Sets:

• $j\in J$: Products
• $i\in I$: Resources
• $t\in T$: Periods ($t=1,\dots ,T$)

Variables:

• $x_{jt}$: Production quantity of product $j$ in period $t$
• $y_{jt}$: Inventory of product $j$ at the end of period $t$ ($y_{j0}$ is given)

Data:

• $a_{ij}$: Consumption of resource $i$ per unit of product $j$
• $B_{it}$: Available capacity of resource $i$ in period $t$
• $d_{jt}$: Forecasted demand of product $j$ in period $t$
• $h_j$: Inventory holding costs per unit and period for product $j$

Objective Function:
$\min {\sum }_{j\in J}{\sum }_{t\in T}{h}_j\cdot y_{jt}$

Constraints:

• Capacity constraint: ${\sum }_{j\in J}{a}_{ij}\cdot x_{jt}\le B_{it}\forall i\in I,t\in T$
• Inventory balance: $y_{jt}=y_{j,t-1}+x_{jt}-d_{jt}\forall j\in J,t\in T$
• Non-negativity: $x_{jt},y_{jt}\ge 0\forall j\in J,t\in T$

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Batch Production

Characteristics

• Different variants of a basic product type are manufactured using the same production equipment one after another.
• Changeover times and costs apply (e.g., major setups for product families, minor setups for variants).
• Make-to-stock or assemble-to-order.

Multi-Period Linear Optimization Model

Here, we minimize both setup and inventory holding costs, assuming a single bottleneck resource (e.g., a make-and-pack production line).

Additional Data:

• $a_j$: Capacity consumption per unit of product $j$
• $C_t$: Available production capacity in period $t$
• $s_j$: Setup time for product $j$
• $S{C}_j$: Setup costs for product $j$
• $BIG$: A large number (big M), e.g., $M={\max }_{j,t}\frac{C_t-s_j}{a_j}$

Additional Variable:

• $z_{jt}\in \{0,1\}$: $1$ if product $j$ is set up in period $t$, $0$ otherwise.

Objective Function:
$\min {\sum }_{j\in J}{\sum }_{t\in T}(S{C}_j\cdot z_{jt}+h_j\cdot y_{jt})$

Constraints:

• Capacity: ${\sum }_{j\in J}(a_j\cdot x_{jt}+s_j\cdot z_{jt})\le C_t\forall t\in T$
• Product setup: $x_{jt}\le BIG\cdot z_{jt}\forall j\in J,t\in T$
• Inventory balance: $y_{jt}=y_{j,t-1}+x_{jt}-d_{jt}\forall j\in J,t\in T$
• Non-negativity and integrality: $x_{jt},y_{jt}\ge 0$ and $z_{jt}\in \{0,1\}\forall j\in J,t\in T$

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One-of-a-kind Production

Characteristics

• Individual products produced in very small order sizes.
• Often customer-specific adjustment of product design.
• Limited number of production orders at the same time.
• Large work volume, numerous tasks, complex project structure.
• Long throughput times and tight due dates.

Additional planning tasks: Preparation of bids, cost/resource planning, project structuring, and concurrent monitoring.

Critical Path Method (CPM)

Notation:

• $EF{S}_j,EF{E}_j$: Earliest feasible start and end time of task $j$
• $LF{S}_j,LF{E}_j$: Latest feasible start and end time of task $j$
• $d_j$: Duration of task $j$

Forward Calculation:
$EF{S}_0=0$
$EF{E}_j=EF{S}_j+d_j$
$EF{S}_j={\max }_{v\in V(j)}\{EF{E}_v\}$
(where $v\in V(j)$ is a predecessor of $j$)

Backward Calculation:
$LF{E}_J=\text{due date (if given, for final task }J)$
$LF{S}_j=LF{E}_j-d_j$
$LF{E}_j={\min }_{n\in N(j)}\{LF{S}_n\}$
(where $n\in N(j)$ is a successor of $j$)

Total Buffer Time:
$T{B}_j=LF{E}_j-EF{E}_j=LF{S}_j-EF{S}_j$

Critical Path: Connects all tasks that determine the shortest possible duration of the project (tasks with $T{B}_j=0$ when no due date is given, or $T{B}_j=\text{due date}-EF{E}_J$ when a due date is given).