### Problem

Every service is a small **production**: inputs (people, equipment, organisation) → output.

**Operations** = the part of a firm that turns inputs into outputs. **Productivity** = output ÷ input.

### Sentence of the week

**Productivity is always a ratio, and the interesting question is what goes into the denominator.**

Numerator (output): easy to count. Denominator (input): a choice that decides the answer. No input named → no meaning.

### Formulas

| Measure | In words | Unit, e.g. |
|---|---|---|
| **Productivity** | output ÷ input | guests per person-hour |
| **Single-factor** | output ÷ one input | crates per labor-hour, units per kWh |
| **Multifactor** | output ÷ all inputs, in money | titles per dollar, units per euro |
| **Growth** | (new − old) ÷ old | % |

- Person-hour = labor-hour = one person, one hour
- Multifactor: convert every input to money first (hours + kg + kWh cannot be added)
- Growth: always divide by the **old** value
- Heizer et al. (2023), Ch. 1: *The productivity challenge*, *Productivity measurement*

### Student life

- **Party, 20 guests:** 335 person-minutes = 5.58 person-hours → 20 ÷ 5.58 = **3.58 guests per person-hour**; per euro of food: a second, different productivity
- **Mia's moving day:** output fixed (100 moving units); inputs vary (van size, crew, pizza) → **units per person-hour** vs. **units per euro** can rank the plans differently

### Factory: textbook anchors

- **Collins Title:** 8 titles ÷ 32 labor-hours = **0.25** → new system 14 ÷ 32 = 0.4375, **+75 %**; multifactor 8 ÷ 1,040 $ = 0.0077 → 14 ÷ 1,440 $ = 0.0097, **+26 %**. Same change, two measures, two gains
- **Modern Lumber** (Solved Problems 1.1, 1.2): 240 crates ÷ 300 labor-hours = **0.800** → 260 ÷ 308 = 0.844, **+5.5 %**; multifactor 240 ÷ 4,500 $ = 0.0533 → 260 ÷ 4,580 $ = 0.0568, **+6.4 %**
- **Fisher Technologies** (Table 1.1): contribution 10,500 $; sales +50 % → 18,000 $; finance costs −50 % → 12,750 $; production costs −20 % → **22,500 $**. Operations = strongest lever

### Supply chain

Output = products delivered to customers; input = what **all** firms spend (€) or emit (t CO₂). One firm's gain can be the chain's loss; € and CO₂ can point in opposite directions.

### Productivity variables

- **Labor** — people, skills, health
- **Capital** — equipment, tools
- **Management** — organisation, technology, knowledge (find in the chapter which one contributes most to growth)
- **Services:** output hard to count; quality escapes the count

### Typical errors

1. No unit ("3.6" says nothing)
2. Adding hours, kg and kWh — convert to money first
3. Growth divided by the new value (43 % instead of +75 %)
4. Faster ≠ more productive (more people can lower output per person-hour)
5. Ranking ratios with different denominators against each other

### Practice items (no keys)

1. Copy shop: 1,200 pages, 2 people × 6 h → new printer: 1,500 pages, same staff. Pages per labor-hour before/after? Growth in %?
2. Food truck: 240 wraps; 2 workers × 10 h × 15 €, ingredients 360 €, fuel and gas 40 €, rent 100 €. Wraps per euro? Which input to cut first, and why?
3. Cargo bike 80 parcels/day vs. van 120 parcels/day: two denominators favouring the bike, one favouring the van.