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Resources

• Script Part 7: Lot Sizing and Scheduling
• Exercise Questions 1, Exercise Questions 2

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.

Lot Sizing

Planning Task

The planning task involves the determination of capacity load and production schedules over a short-term horizon (e.g., 1 week).
Scheduling can be performed via:

• Forward scheduling: Starts at “Today” (earliest feasible start date) and works forward to determine the completion time.
• Backward scheduling: Starts at the due date and works backward to find the latest feasible start date.

Lot Sizing in Job Shop Production

In job shop production, products often require frequent changeovers on the same machine, which involves setup operations before production activities can commence. Production is typically batch-wise for stock.

Definition

Lot size is the quantity of a product manufactured without interruption.

The core problem of lot sizing is the trade-off between setup costs and inventory holding costs.

Classic Lot Size Model (Economic Order Quantity - EOQ)

Introduced by F.W. Harris (1913), answering “How many parts to make at once?”.

Simplifying Assumptions

• Constant demand per time unit (static modelling)
• Deterministic demand
• Infinite production or delivery speed
• Only setup and holding costs considered
• No stockouts
• No capacity limits
• Single-product model
• Single-level product

Notation:

• $C(q)$: lot size dependent costs
• $q$: lot size (variable)
• $D$: demand
• $s$: setup costs
• $h$: inventory holding costs per unit and period

Cost function:
The total cost consists of setup costs and holding costs (based on average inventory $\frac{q}{2}$):
$C(q)=\frac{D}{q}\cdot s+\frac{q}{2}\cdot h$

Optimum Lot Size ($q_{opt}$):
Derived by setting the derivative of the cost function to zero:
$\frac{d(C)}{d(q)}=\frac{-D\cdot s}{q^2}+\frac{h}{2}=0$
$q_{opt}=\sqrt{\frac{2\cdot D\cdot s}{h}}$

Dynamic Lot Sizing

When demand fluctuates, dynamic models are used instead of static ones. Dynamic lot sizing heuristics adjust to the discrete timeframe of material requirements planning.

• Basic principle: Generation of lot sizes through the combination of demand from adjacent periods.
• The goal is to determine the range of coverage (the number of periods for which the lot size covers demand).

General Heuristic Algorithm:

1. Start with a range of coverage of one period.
2. Increase the range of coverage stepwise until the underlying cost function reaches the first local minimum.
3. The lot size corresponds to the total demand of all periods supplied by the range of coverage.

Silver/Meal Heuristic

A dynamic lot sizing heuristic that focuses on the minimization of average costs per period.
Inventory holding costs are calculated based on the final inventory of a period.

Notation:

• $\tau$: range of coverage
• $t$: period counter
• $k$: production period
• $s$: setup costs
• $h$: inventory holding costs per unit and period
• $d_t$: demand for period $t$

Formula:
$c_{t,\tau }=\frac{s+h\cdot {\sum }_{t=k}^{k+\tau -1}(t-k)\cdot d_t}{\tau }$

Wagner/Whitin Algorithm

An optimizing procedure that finds the absolute optimum solution to the dynamic lot size problem.

• Assumes no capacity limits and considers only a single product.
• The problem is represented and solved as a shortest path problem, where nodes represent periods and edges represent the costs of producing a lot to cover a specific range of periods.