# Process Analysis Concept Card: [Your course or process] > A concept card for process analysis, the first and most used tool of the MBA operations core: draw the process, > find the capacity of every resource, name the bottleneck, compute utilisation, flow rate and flow time, link them with > Little's Law and decide which lever improves the process. It is for MBA students preparing a case discussion or an > exam, and for managers who need a quick, correct reading of a warehouse, a clinic, a call centre or a back office. > One example is fully solved, one is half solved and one is left for you, and the self-test answers sit in a separate > key. Source basis: Little (1961) for Little's Law, Goldratt and Cox (1984) for the bottleneck logic, and the process > view common to the leading operations textbooks, all written up in original words with fictional data (Northfield > Outdoor Co. and Tallow Creek Medical Center are invented companies). > Replace every [bracketed] value and every row marked *Example*. ## Document control | Field | Value | |---|---| | Course or project | [Operations core, session 2 / client project / own team] | | Learner or analyst | [Your name] | | Process analysed | [Northfield Ridgeway fulfilment centre, order picking to shipping (fictional) / your process] | | Flow unit | [Customer order / patient / claim / your unit] | | Data source and period | [Time study of [n] observations, YYYY-MM-DD to YYYY-MM-DD / system log] | | Tool | [Excel / Google Sheets / Python] | | Reviewed by | [Study group / tutor / mentor] | | Last updated | YYYY-MM-DD | | Version | [0.1] | ## When to use it **One-sentence rule:** use process analysis when you need to know how much a process can produce, which step limits it, how long a unit spends inside it and what change would raise output or cut time. Typical questions it answers: - "Can we handle the holiday peak?" (capacity versus demand) - "Where should we add staff or equipment?" (the bottleneck) - "Why do patients spend three hours in the clinic for forty minutes of care?" (flow time versus activity time) - "How much work in progress should we expect if we promise a two-day turnaround?" (Little's Law) - "What does each order cost us in direct labour?" (labour content and idle time) | Use when | Do not use alone when | |---|---| | The process has identifiable steps and a clear flow unit | Demand and service times are highly variable and the question is about waiting (add queueing, mba/mba-queueing-card) | | You can estimate an average processing time per step | The process is a one-off project (use the critical path method, see mba/mba-operations-guide) | | You care about averages over a shift, a day or a season | You need the full distribution of waits or the probability of a long wait | | Resources are dedicated to one step, or you can split their time | Many products share resources with set-ups (use the multi-product workload method in Exercise C, and set-up time adjustments) | ## The idea in plain language A process turns inputs into outputs through a sequence of steps, each carried out by a **resource** (a worker, a machine, a room, a team). Pick one **flow unit** and follow it: an order, a patient, a loan application. Everything else is counting. Each resource can only work so fast. Its **capacity** is how many flow units it can complete per unit of time: a packer who needs 2.8 minutes per order can pack 60 / 2.8 = 21.43 orders an hour, and three such packers can pack 64.29. The **process capacity** is the capacity of the slowest resource, because every unit must pass through every step: a chain is as strong as its weakest link. That slowest resource is the **bottleneck**. What the process actually produces, the **flow rate** or **throughput**, is the smaller of demand and process capacity. If demand is below capacity, the process is **demand constrained** and every resource has some idle time. If demand exceeds capacity, the process is **capacity constrained**: output equals the bottleneck's capacity, and a queue of unserved demand builds up in front of it. **Utilisation** tells you how busy a resource is: flow rate divided by its capacity. The bottleneck has the highest utilisation. When you divide demand (instead of the actual flow rate) by capacity, you get the **implied utilisation**, which can exceed 100%: it tells you by how much a resource is overloaded. **Flow time** is the time a unit spends in the process from entry to exit, including waiting. It is usually much longer than the **activity time** (the time someone actually works on the unit). The gap is waiting, and waiting is where most improvement opportunities hide. **Little's Law** ties the three averages together. The average number of units inside the process (inventory, or work in progress) equals the average flow rate times the average flow time. It needs no assumptions about distributions, only a stable process observed over a long enough period (Little 1961). Measure any two and you get the third: count the orders on the floor and the throughput, and you know the average flow time without timing a single order. ## Formulas and notation For a resource with $m$ parallel servers (people or machines) and processing time $t$ per unit: $$ \text{Capacity}_i = \frac{m_i}{t_i}, \qquad \text{Process capacity} = \min_i \ \text{Capacity}_i $$ $$ R = \min(\text{Demand},\ \text{Process capacity}), \qquad \text{Cycle time} = \frac{1}{R} $$ $$ u_i = \frac{R}{\text{Capacity}_i}, \qquad \text{Implied } u_i = \frac{\text{Demand}}{\text{Capacity}_i} $$ Little's Law (Little 1961), for a stable system in the long run: $$ I = R \times T \qquad \Longleftrightarrow \qquad T = \frac{I}{R} $$ Flow time efficiency and labour measures (for a worker-paced process): $$ \text{Flow time efficiency} = \frac{\text{Activity time}}{T}, \qquad \text{Labour content} = \sum_i t_i, \qquad \text{Average labour utilisation} = \frac{\text{Labour content}}{\text{Labour content} + \text{Total idle time}} $$ $$ \text{Cost of direct labour per unit} = \frac{\text{Total wages per hour}}{R} $$ For several products sharing resources, compute the workload of each resource: $$ \text{Implied } u_i = \frac{\sum_k D_k \, t_{ik}}{\text{Available time}_i} $$ | Symbol | Meaning | Typical unit | |---|---|---| | $m_i$ | Number of parallel servers at resource $i$ | people, machines, rooms | | $t_i$ | Processing time per unit at resource $i$ | minutes per unit | | $\text{Capacity}_i$ | Maximum output rate of resource $i$ | units per hour | | $R$ | Flow rate (throughput) actually achieved | units per hour | | Cycle time | Time between two consecutive units leaving the process | minutes per unit | | $u_i$ | Utilisation of resource $i$ | % | | $I$ | Average inventory (work in progress, units inside the process) | units | | $T$ | Average flow time (time in the process, including waiting) | hours or minutes | | $D_k$, $t_{ik}$ | Demand for product $k$; time product $k$ needs at resource $i$ | units per day; minutes | **Excel set-up.** Put servers in column B, times in column C, then capacity `=B2*60/C2` (units per hour), process capacity `=MIN(D2:D4)`, bottleneck `=INDEX(A2:A4,MATCH(MIN(D2:D4),D2:D4,0))`, flow rate `=MIN(demand,process_capacity)` and utilisation `=flow_rate/D2`. ## Assumptions and limits | Assumption | Why it matters | What to do when it fails | |---|---|---| | Processing times are stable averages | Capacity is computed from averages; real times vary | Collect enough observations; add buffers; use mba/mba-queueing-card for waiting | | Demand is measured over the same period as capacity | A daily average hides an hourly peak | Analyse the peak hour separately (see the peak calculation in Worked example A) | | Each resource works on one step | Shared resources split their time | Use the workload method (Exercise C) | | No set-up or changeover time | Set-ups eat capacity, especially with small batches | Capacity with batches: $B / (S + B\,t)$ for batch size $B$ and set-up time $S$ | | No rework or yield loss | Rework sends units through a step twice | Divide demand by yield at each step, or add the rework flow to the workload | | Little's Law holds over the period | It needs a stable system: inventory not growing without limit | Use it for averages over a full cycle (a day, a season), not for one busy hour that ends with a backlog | | The flow unit is the right one | Mixing orders and order lines gives nonsense rates | Fix one flow unit and convert all times to it | Little's Law itself is robust: it holds for any arrival pattern and any service order, provided the averages exist and the system is stable over the observation window (Little 1961; Little 2011 discusses its uses fifty years on). The other formulas on this card are averages and say nothing about the waits caused by variability, which is why a resource at 93% utilisation can still have long queues. ## How managers use it - **Capacity planning.** Compare process capacity with forecast peak demand, not average demand. A distribution centre at 90% utilisation on an average day can be overloaded on the Monday after a promotion. - **Where to invest.** An extra hour at the bottleneck is an extra hour of output for the whole process; an extra hour at a non-bottleneck buys nothing but idle time. This is the core message of the theory of constraints (Goldratt and Cox 1984), developed in mba/mba-operations-guide. - **Service promises.** Little's Law turns a delivery promise into a work-in-progress cap: to promise a 2-day average turnaround at 300 orders a day, keep no more than 600 orders in the system on average. - **Cost.** Cost of direct labour per unit and labour utilisation feed the cost models in mba/mba-activity-based-costing-card and the break-even analysis in mba/mba-cvp-breakeven-card. - **Working capital.** Inventory turns and days of inventory are Little's Law applied to a balance sheet (see mba/mba-inventory-card and finance/working-capital-review). - **Service design.** In a hospital or bank, flow time is what the customer experiences. Cutting waiting time often matters more than cutting activity time. ## Process flow diagram The diagram shows the fictional Northfield Outdoor Co. Ridgeway fulfilment centre used in Worked example A. Online orders queue in the order pool, are picked, wait for a packer, are packed, labelled and handed to the carrier. ```mermaid flowchart LR A["Online orders arrive (60 per hour)"] --> B[("Order pool")] B --> C["Pick: 6 pickers, 4.5 min per order"] C --> D[("Totes waiting to pack")] D --> E["Pack: 3 packers, 2.8 min per order (bottleneck)"] E --> F["Label and ship: 1 station, 0.8 min per order"] F --> G["Carrier collection"] ``` The decision path for any process question: ```mermaid flowchart TD S["Define the flow unit"] --> P["Draw the process and buffers"] P --> C["Capacity of each resource = servers / time"] C --> B["Bottleneck = lowest capacity"] B --> Q{"Demand above process capacity?"} Q -->|"Yes: capacity constrained"| X["Flow rate = bottleneck capacity, queue builds"] Q -->|"No: demand constrained"| Y["Flow rate = demand, all resources have idle time"] X --> U["Utilisation and implied utilisation"] Y --> U U --> L["Little's Law: inventory, flow rate, flow time"] L --> V{"Goal: more output or less time?"} V -->|"More output"| O["Elevate or offload the bottleneck"] V -->|"Less time"| W["Cut waiting: reduce inventory, smooth arrivals, reduce variability"] ``` ## Worked example A (fully solved): Northfield Ridgeway fulfilment centre **Situation (*Example*, fictional).** Northfield Outdoor Co. ships its online orders from the Ridgeway fulfilment centre. In an average hour of the autumn season 60 orders arrive. A time study gives the processing times below. The operations manager counts on average 45 orders inside the building (in the order pool, in totes and on packing benches) and pays USD 24 an hour per worker. Ten people work the line: 6 pickers, 3 packers and 1 at the label station. **Step 1: capacity of each resource.** | Resource | Servers $m$ | Time $t$ (min per order) | Capacity (orders per hour) | Excel | Utilisation at 60 per hour | Excel | |---|---|---|---|---|---|---| | Pick *(Example)* | 6 | 4.5 | 80.00 | `=6*60/4.5` | 75.0% | `=60/80` | | Pack *(Example)* | 3 | 2.8 | 64.29 | `=3*60/2.8` | 93.3% | `=60/64.29` | | Label and ship *(Example)* | 1 | 0.8 | 75.00 | `=1*60/0.8` | 80.0% | `=60/75` | **Step 2: bottleneck and flow rate.** Process capacity = min(80.00, 64.29, 75.00) = **64.29 orders per hour** (`=MIN(D2:D4)`), set by **packing**. Demand is 60 per hour, below capacity, so the process is demand constrained and the flow rate is $R$ = 60 orders per hour (`=MIN(60,64.29)`). Cycle time = 60 / 60 = 1.0 minute per order. **Step 3: flow time with Little's Law.** $T = I / R = 45 / 60 = 0.75$ hours = **45 minutes** (`=45/60*60`). The activity time is 4.5 + 2.8 + 0.8 = 8.1 minutes, so flow time efficiency = 8.1 / 45 = **18.0%** (`=8.1/45`). An order spends more than four fifths of its time waiting. **Step 4: labour.** Labour content = 8.1 minutes per order. In each 1-minute cycle the 10 workers provide 10 minutes of paid time, so idle time = 10 - 8.1 = 1.9 minutes per order and average labour utilisation = 8.1 / 10 = **81.0%** (`=8.1/(10*1)`). Cost of direct labour per order = 10 x 24 / 60 = **USD 4.00** (`=10*24/60`). **Step 5: the peak.** On Cyber Monday, demand reaches 75 orders per hour for 4 hours. Implied utilisation becomes pick 93.8%, pack 116.7% and label 100.0% (pack: `=75/64.29`). Packing is overloaded: output stays at 64.29 per hour and a backlog of (75 - 64.29) x 4 = **42.9 orders** builds in front of the packing benches (`=(75-64.29)*4`). If demand then falls to 40 an hour, the backlog clears in 42.9 / (64.29 - 40) = **1.76 hours** (`=42.9/(64.29-40)`). **Step 6: improvement levers compared.** | Lever *(Example)* | Pick | Pack | Label | Process capacity | Comment | |---|---|---|---|---|---| | Current | 80.00 | 64.29 | 75.00 | 64.29 | Pack is the bottleneck | | Add a fourth packer | 80.00 | 85.71 | 75.00 | 75.00 | Bottleneck moves to label; covers the 75 per hour peak exactly, with no slack | | Move the 0.3-minute box pre-fold from pack to pick (pick 4.8, pack 2.5) | 75.00 | 72.00 | 75.00 | 72.00 | Free capacity gain from rebalancing; no new staff | | Move 0.3 minutes from pack to label instead (pack 2.5, label 1.1) | 80.00 | 72.00 | 54.55 | 54.55 | Worse: work moved to a single-server step creates a new, tighter bottleneck | | Fourth packer plus a faster label printer (label 0.6) | 80.00 | 85.71 | 100.00 | 80.00 | Pick becomes the bottleneck at 80 per hour | **Interpretation.** On an average day the line copes, but packing runs at 93.3% and has little room for variability. For the peak, the cheapest move is to rebalance the pre-fold to the pickers (72 per hour), which still leaves a small backlog at 75 per hour; adding a fourth packer lifts capacity to exactly 75, so any variability creates a queue at the label station. Before buying anything, the manager should also cut the 36.9 minutes of waiting per order, for example by releasing orders to pickers in smaller waves, which lowers inventory and therefore flow time by Little's Law. **Chart.** The bar chart compares the capacity of each resource with average and peak demand. ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Northfield Ridgeway fulfilment centre: capacity per resource versus demand (fictional, computed)", "width": 360, "height": 200, "layer": [ { "data": {"values": [ {"resource": "Pick", "capacity": 80.0}, {"resource": "Pack", "capacity": 64.29}, {"resource": "Label and ship", "capacity": 75.0} ]}, "mark": {"type": "bar"}, "encoding": { "x": {"field": "resource", "type": "nominal", "sort": null, "title": "Resource"}, "y": {"field": "capacity", "type": "quantitative", "title": "Orders per hour", "scale": {"domain": [0, 90]}} } }, { "data": {"values": [ {"level": "Average demand 60", "orders": 60}, {"level": "Peak demand 75", "orders": 75} ]}, "mark": {"type": "rule", "strokeDash": [6, 4], "size": 2}, "encoding": { "y": {"field": "orders", "type": "quantitative"}, "color": {"field": "level", "type": "nominal", "title": "Demand"} } } ] } ``` ## Little's Law: three quick computed examples | ID | Situation *(Example, fictional)* | Known | Unknown | Result | Excel | |---|---|---|---|---|---| | LL-1 | Tallow Creek walk-in clinic: 18 patients an hour, 40 minutes each on average from door to door | $R$, $T$ | $I$ | 18 x 40 / 60 = 12 patients in the clinic | `=18*40/60` | | LL-2 | Northfield returns desk: 360 parcels waiting or in process, 120 processed a day | $I$, $R$ | $T$ | 360 / 120 = 3 days per return | `=360/120` | | LL-3 | Northfield accessories business: inventory USD 2.4m, cost of goods sold USD 14.4m a year | $I$, $R$ | $T$ | 2.4 / 14.4 = 0.1667 years = 60.8 days | `=2.4/14.4*365` | LL-3 shows that **days of inventory** is Little's Law with money as the flow unit, and that inventory turns (14.4 / 2.4 = 6 per year) are simply $1/T$. ## Worked example B (half solved): Tallow Creek outpatient MRI **Situation (*Example*, fictional).** The Tallow Creek Medical Center outpatient MRI suite runs 10 hours a day. On average 2.5 patients an hour are booked. A nurse audit counts on average 4 patients in the suite (waiting, changing or being scanned). | Resource | Servers | Minutes per patient | Capacity (patients per hour) | Utilisation at 2.5 per hour | |---|---|---|---|---| | Check-in clerks | 2 | 6 | 20.00 | 12.5% | | Screening and changing (nurse) | 1 | 10 | [ ] | [ ] | | MRI scanners | 2 | 40 | [ ] | [ ] | | Image upload and report preparation (technologist) | 1 | 12 | 5.00 | 50.0% | Tasks: 1. Complete the capacity and utilisation columns. Which resource is the bottleneck? Process capacity = [ ] patients per hour = [ ] patients per 10-hour day. 2. Flow time with Little's Law: $T = I / R$ = [ ] / [ ] = [ ] hours = [ ] minutes. 3. Activity time = 6 + 10 + 40 + 12 = [ ] minutes. Flow time efficiency = [ ]%. 4. The referral team wants to book 3.2 patients an hour. Implied utilisation of the scanners = [ ]%. What happens? 5. Name two levers that raise scanner capacity without buying a scanner, and one that lowers flow time without changing capacity. ## Exercise C (unsolved): Northfield bike service workshop, two job types The Northfield flagship store runs a bike service workshop 8 hours a day (480 minutes). It has 1 intake desk worker, 3 mechanics and 1 person who does test rides. Daily demand and minutes per job (*Example*, fictional): | Job type | Daily demand | Intake desk (min) | Mechanic (min) | Test ride (min) | |---|---|---|---|---| | Tune-up | 20 | 6 | 45 | 10 | | Overhaul | 5 | 10 | 120 | 15 | Tasks: 1. Compute the workload (minutes per day) and the implied utilisation of each resource. 2. Which resource is the bottleneck? Can the workshop meet daily demand? 3. If the 5 overhauls are always accepted, how many tune-ups can the mechanics complete per day? 4. Compare two fixes: (a) a part-time fourth mechanic for 4 hours a day, (b) a standard work routine that cuts tune-up mechanic time from 45 to 40 minutes. What is the mechanics' implied utilisation after each fix? Which would you choose, and what risk remains? ## Process improvement levers | Goal | Lever | How it works | Watch out for | |---|---|---|---| | More output | Add capacity at the bottleneck | More servers or faster equipment at the constraint | The bottleneck moves; check the next lowest capacity | | More output | Exploit the bottleneck | Never let it starve or wait: buffer in front of it, staggered breaks, quality checks before it | Do not let it process units that will be scrapped later | | More output | Offload or rebalance | Move tasks from the bottleneck to resources with idle time | Moving work to a single-server step can create a worse bottleneck (Worked example A) | | More output | Reduce set-ups at the bottleneck | Quick changeover methods, larger batches at the bottleneck only | Larger batches raise inventory and flow time | | More output | Cut rework and yield loss before the bottleneck | Quality at the source (mba/mba-quality-card) | Defects found after the bottleneck waste its capacity | | Less time | Reduce inventory in the process | Release work in smaller waves, cap work in progress (kanban) | Too little buffer starves the bottleneck | | Less time | Reduce variability | Standard work, appointment systems, smoothing arrivals | See mba/mba-queueing-card for the size of the effect | | Less time | Run steps in parallel | Do independent tasks at the same time (paperwork while waiting for a scan) | Coordination and hand-off errors | | Less time | Remove non-value-adding steps | Map the value stream, eliminate approvals and transports | Some checks exist for safety or compliance reasons | | Lower cost | Balance the line | Equalise utilisation, reduce idle time per unit | A perfectly balanced line has no slack for variability | ## Connections to other cards - **mba/mba-queueing-card**: why waiting time explodes as utilisation approaches 100%, the effect of variability on the waits this card measures with Little's Law. - **mba/mba-inventory-card**: inventory as a buffer, inventory turns and days of supply from Little's Law, EOQ batch sizes. - **mba/mba-quality-card**: yield, rework and defects reduce effective capacity; statistical process control keeps processing times stable. - **mba/mba-operations-guide**: theory of constraints, lean, the bullwhip effect, the course roadmap and a full mini-case at Tallow Creek Medical Center. - **mba/mba-activity-based-costing-card** and **mba/mba-cvp-breakeven-card**: labour content and capacity feed cost per unit and break-even volume. - **mba/mba-formula-sheet**: one-page formulas for exam day. ## Common mistakes and misconceptions - **Mixing units.** Dividing 2.8 by 3 gives 0.93 minutes per order for the packing team, not a rate; the capacity is 3 / 2.8 = 1.07 orders per minute (64.29 per hour). Write units in every cell: capacity is units per time, processing time is time per unit. - **Forgetting parallel servers.** Picking has the longest time (4.5 minutes) but is not the bottleneck: 6 pickers give 80 orders per hour. Capacity depends on $m/t$, not on $t$ alone. - **Finding the bottleneck from capacities in units per hour when several products share resources.** With a product mix, "units per hour" depends on the mix; compute workload in minutes against available minutes instead (Exercise C). - **Flow rate equal to capacity.** Flow rate is the minimum of demand and capacity. A demand-constrained process does not run at capacity. - **Adding capacity at a non-bottleneck.** Extra pickers in Worked example A raise cost and leave output unchanged. - **Using Little's Law on a growing queue.** During the 4-hour peak the backlog grows, so inventory and flow time are not stable; apply the law over a full day or season. - **Confusing cycle time and flow time.** Cycle time (1 minute per order) is the interval between outputs; flow time (45 minutes) is how long one order stays inside. - **Planning to 100% utilisation.** With variable arrivals and times, a resource planned at 100% builds an unbounded queue; most service operations plan well below that (mba/mba-queueing-card). - **Ignoring yield.** If 10% of packed orders are re-packed, packing needs capacity for 1.1 times demand. - **Analysing averages when the peak decides.** A daily average of 60 orders per hour hides a 75 per hour Monday. ## Self-test 1. A step has 4 workers who each need 12 minutes per unit. What is its capacity per hour? 2. Three steps have capacities of 40, 25 and 30 units per hour and demand is 20 per hour. Name the bottleneck, the flow rate and the utilisation of each step. 3. In question 2, demand rises to 35 per hour. What is the flow rate, and what is the implied utilisation of the bottleneck? 4. A loan office processes 300 applications a day and promises an average turnaround of 2 days. How many applications are in the system on average? 5. Why does adding a fourth packer in Worked example A raise capacity only to 75 orders per hour, not to 85.71? 6. A clinic has 12 patients inside on average and 18 arrive per hour. What is the average time in the clinic? 7. What is the difference between utilisation and implied utilisation, and when can the second exceed 100%? 8. A process has 8.1 minutes of activity time and a flow time of 45 minutes. What is flow time efficiency, and what does a low value suggest? 9. Interleaving: for each question name the right tool (process analysis, queueing, EOQ, newsvendor). (a) How many ski jackets to order for a single season. (b) How many orders per hour a packing line can ship. (c) How long callers wait at a help desk that is 90% busy with random calls. (d) How many headlamps to order each time stock runs low. 10. Why can moving work from the bottleneck to another step make a process worse? ## My notes and next steps | Item | My entry | |---|---| | Confidence with process analysis (1 = lost, 5 = could teach it) | [ ] | | A process I know well (work, daily life) and its bottleneck | [ ] | | Questions for my study group or professor | [ ] | | Mistakes logged in mba/mba-error-log | [Date and short description] | | Next review date (spacing: +2 days, +1 week, +1 month) | YYYY-MM-DD | | Next card to study | [mba/mba-queueing-card / mba/mba-inventory-card] | ## Sources and further reading - Little, J. D. C. (1961). A proof for the queuing formula: L = λW. *Operations Research*, 9(3), 383-387. - Little, J. D. C. (2011). Little's Law as viewed on its 50th anniversary. *Operations Research*, 59(3), 536-549. - Goldratt, E. M., and Cox, J. (1984). *The Goal: A Process of Ongoing Improvement*. North River Press. - Hayes, R. H., and Wheelwright, S. C. (1979). Link manufacturing process and product life cycles. *Harvard Business Review*, January-February 1979, 133-140. - Frei, F. X. (2006). Breaking the trade-off between efficiency and service. *Harvard Business Review*, November 2006. - Textbooks (further reading, by name only): Cachon and Terwiesch, *Matching Supply with Demand: An Introduction to Operations Management*; Anupindi, Chopra, Deshmukh, Van Mieghem and Zemel, *Managing Business Process Flows*; Hopp and Spearman, *Factory Physics*. - Open resource: MIT OpenCourseWare, 15.761 Introduction to Operations Management (Spring 2013), https://ocw.mit.edu/courses/15-761-introduction-to-operations-management-spring-2013/. ## Answer key **Worked example B.** (1) Screening: 1 x 60 / 10 = 6.00 per hour (`=1*60/10`), utilisation 2.5 / 6 = 41.7%. Scanners: 2 x 60 / 40 = 3.00 per hour (`=2*60/40`), utilisation 2.5 / 3 = 83.3%. The scanners are the bottleneck; process capacity = 3.00 patients per hour = 30 per 10-hour day (`=3*10`). (2) $T$ = 4 / 2.5 = 1.6 hours = 96 minutes (`=4/2.5*60`). (3) Activity time = 68 minutes; flow time efficiency = 68 / 96 = 70.8% (`=68/96`). (4) Implied utilisation = 3.2 / 3 = 106.7% (`=3.2/3`): demand exceeds capacity, a backlog of 0.2 patients per hour (2 per day) builds and patients are rebooked or the day runs late. (5) Capacity: move screening questionnaires and changing out of scanner time (patients ready before the scanner frees up), shorten scanner changeover between patients, use protocols that cut scan minutes where clinically appropriate, extend opening hours. Flow time: stagger appointment times so fewer patients wait in the suite at once (lower $I$ at the same $R$). **Exercise C.** (1) Intake: 20 x 6 + 5 x 10 = 170 of 480 minutes, implied utilisation 35.4% (`=(20*6+5*10)/480`). Mechanics: 20 x 45 + 5 x 120 = 1,500 of 3 x 480 = 1,440 minutes, implied utilisation 104.2% (`=1500/1440`). Test ride: 20 x 10 + 5 x 15 = 275 of 480, implied utilisation 57.3%. (2) The mechanics are the bottleneck; the workshop cannot meet demand (60 minutes of mechanic work a day are missing). (3) (1,440 - 5 x 120) / 45 = 18.67, so 18 complete tune-ups a day (`=(1440-5*120)/45`). (4) (a) A 4-hour fourth mechanic adds 240 minutes: 1,500 / 1,680 = 89.3%. (b) Tune-ups at 40 minutes: workload 20 x 40 + 600 = 1,400 minutes, 1,400 / 1,440 = 97.2%. Option (b) is cheaper but leaves almost no slack for variable job times, so queues and late pick-ups remain likely; option (a) gives a buffer. Many managers combine both. **Self-test.** 1. 4 x 60 / 12 = 20 units per hour. 2. Bottleneck is the step at 25 per hour. Flow rate = 20 (demand constrained). Utilisations: 20 / 40 = 50%, 20 / 25 = 80%, 20 / 30 = 66.7%. 3. Flow rate = 25 (capacity constrained); implied utilisation of the bottleneck = 35 / 25 = 140%, and the queue grows by 10 units an hour. 4. $I = R \times T$ = 300 x 2 = 600 applications. 5. With four packers, label and ship (75 per hour) becomes the slowest resource, so process capacity is 75. 6. $T = 12 / 18$ hours = 40 minutes. 7. Utilisation divides the actual flow rate by capacity and cannot exceed 100%; implied utilisation divides demand by capacity and exceeds 100% when a resource is overloaded. 8. 8.1 / 45 = 18%. Most of the flow time is waiting, so reducing inventory in the process or smoothing the flow cuts time more than speeding up the work itself. 9. (a) Newsvendor (mba/mba-inventory-card). (b) Process analysis (this card). (c) Queueing (mba/mba-queueing-card). (d) EOQ and reorder point (mba/mba-inventory-card). 10. If the receiving step has fewer servers or less idle time, the moved work can push its capacity below the old bottleneck's, as with the label station at 54.55 orders per hour in Worked example A. ## Revision history | Version | Date | Author | Change summary | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First draft from the template | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced the Ridgeway example with own time study, peak analysis added | ## Standard and licence note Based on the published operations management literature, chiefly Little (1961) and Goldratt and Cox (1984); all text, data and examples are original and fictional, and textbooks are named as further reading only. This template is not affiliated with or endorsed by any business school, publisher or course. Check your course materials for the notation your professor uses. Template licence: MIT.