Questions & explanations
1. Give a simple example with 3 jobs and their times on two machines.
Consider three jobs A, B, C. Their times on Machine 1 and Machine 2: A: (3,2), B: (5,4), C: (4,1). The smallest time overall is 1 (job C on Machine 2), so put C last. Next smallest is 2 (job A on Machine 2), so put A last among remaining? Wait, careful: after removing C, remaining times: A(3,2), B(5,4). Smallest is 2 (A on Machine 2) -> put A last of remaining (but last is already C? Actually rule: smallest on Machine 2 goes to the right end. Since C is at end, A goes before C. So order: ? Then next smallest is 3 (A on Machine 1) but A already placed. The smallest among remaining times (B:5,4) is 4 on Machine 2 -> puts B before A? Let's apply correctly: The algorithm: find smallest time among all remaining jobs. If smallest on M1, schedule that job next from left. If on M2, schedule from right. For C (M2 time=1) -> schedule last (rightmost). Remaining: A (M1=3, M2=2) and B (M1=5, M2=4). Smallest among all is A on M2 (2) -> schedule that job at the rightmost of remaining positions, which is just before C. So order: (first position) B, then A, then C. Check: smallest among remaining af
2. Give a simple example where serial and parallel schemes produce different schedules for the same tasks and resources.
Suppose three tasks: A (2 days, needs 1 worker), B (2 days, needs 1 worker), C (1 day, needs 1 worker). Only 1 worker exists. Predecessors: A then C, B independent. Using serial with order A, B, C: schedule A at 0-2, then B at 2-4, then C at 4-5. Using parallel: at time 0, eligible tasks are A and B. With rule 'most successors' (none), tie, pick A. At time 2, A finishes, eligible tasks: B and C (since C depends on A done). With rule, pick B? Then C at 4-5 again. But if parallel rule picks C first at time 2, then schedule: A 0-2, C 2-3, B 3-5. So parallel can shorten project from 5 to 5? Actually 5 days in both, but task order different. To show difference: if C had longer duration, say A (2), B (3), C (2). Serial order A,B,C: A 0-2, B 2-5, C 5-7 (7 days). Parallel: at 0 A and B eligible, pick A (shorter), then at 2 A done, eligible B and C, pick C (shorter) -> C 2-4, B 4-7 (7 days) same. Tricky. Let's try: A(2), B(3), C(1) with A->C. Serial A,B,C: A 0-2, B 2-5, C 5-6 (6 days). Parallel: at 0 A and B eligible, pick A, then at 2 eligible B and C, pick C, then B 3-6 (6 days) but note: C
3. Explain how you would implement DBR in a job shop with multiple product types, each having a different routing. How do you identify the drum?
First, identify the bottleneck: the resource with the highest utilization or the one that most often becomes a constraint. In a job shop, different products use different machines, so the bottleneck may shift. You can use load analysis or simulation to find the most commonly overloaded resource. That becomes the drum (schedule for that resource). For products that do not use the bottleneck, they are treated as non-constraints; they can be scheduled based on demand but with controlled release. Products that use the bottleneck are scheduled according to the drum's availability. Protective buffers are placed before the bottleneck operations for all products. The rope then releases each product's raw materials at a time such that they arrive at the bottleneck exactly when scheduled. This requires careful planning of release times for each job.
4. A firm uses a decision tree to plan capacity for two stages: first build a plant, then later add a warehouse. The tree has two decision points. Explain how to solve it backwards (from end to start) and why that is correct.
Solving backwards means starting at the last decision (warehouse) and working to the first (plant). At the warehouse decision, you know the plant size already built and the demand observed. You compare warehouse options (small vs. large) and pick the one with highest net present value at that point. Then move back to the plant decision: you consider all possible warehouse outcomes for each plant choice, and pick the plant option that gives the highest overall expected value. This is correct because future decisions depend on past ones, and you must account for optimal future behavior. By solving backwards, you ensure that at each earlier stage, you assume the best later actions. This method gives the truly optimal strategy, not a shortsighted one.
5. Explain Jackson networks. How do you find the overall arrival rate to each node?
A Jackson network is a set of service stations (queues) where customers may visit multiple stations in a random path. Each station has its own queue and servers. Arrivals from outside follow a Poisson process, and service times are exponential. The key is that the network can be analyzed as independent M/M/c queues if we know the total arrival rate to each station, which includes external arrivals plus arrivals from other stations. To find the overall rate to station j, solve flow balance equations: λ_j = external_j + sum_i λ_i * p_{ij}, where p_{ij} is probability of going from station i to j. Then each station is treated as M/M/c with its own λ_j and service rate. This is used for factory floor layout, where products go through multiple steps.
6. What is a production function and how does it relate to capacity output?
A production function is a mathematical equation that shows the maximum output possible from given amounts of inputs like labor and capital. Capacity output is the highest level of production a firm can sustain with its current resources. The production function describes how inputs are turned into output, and capacity is the output level when all inputs are used fully. For example, a Cobb-Douglas function might say output = (labor^0.3)*(capital^0.7). This shows how changes in labor or capital affect capacity. Managers use production functions to plan how much capacity to build and to find the best mix of inputs. Different functions (Cobb-Douglas, CES, Translog) make different assumptions about how easily inputs can be substituted.
7. How does the concept of elasticity of substitution in a CES production function influence the choice of input mix for a given capacity?
Elasticity of substitution measures how easily one input can replace another. A high elasticity (say, 3) means that a small change in input prices leads to a large change in the input mix. A low elasticity (close to 0) means inputs are used in fixed proportions. With a CES function, capacity can be achieved with different combinations of labor and capital depending on their relative costs. For example, if labor becomes expensive, a firm with high elasticity can switch to more machines without losing much output. This flexibility allows the firm to adjust its capacity cost-effectively. The chosen input mix minimizes cost for a given output level. Capacity planning must consider these substitution possibilities to stay competitive.
8. What is the Drum-Buffer-Rope (DBR) method? Explain each part with a simple example.
DBR is a scheduling method from the Theory of Constraints. The 'drum' is the schedule of the bottleneck resource (the slowest machine) that sets the rhythm. The 'buffer' is a time buffer before the bottleneck to protect it from disruptions. The 'rope' is a signal that releases raw materials only as fast as the bottleneck can process, based on drum consumption. Example: In a factory, the bottleneck is a painting machine that processes 100 units per day. The drum sets a schedule for painting. A buffer of 2 days of work is placed before painting. The rope says: release raw materials only when the buffer has free space, so that materials don't pile up. This ensures the bottleneck is always busy and work-in-process is controlled.
9. What is throughput accounting? Define the three key measures: throughput, inventory, and operating expense.
Throughput accounting is a performance measurement method from the Theory of Constraints. It focuses on three measures: Throughput (T) is the rate at which the system generates money through sales (selling price minus truly variable costs). Inventory (I) is all the money invested in things the system intends to sell (raw materials, work-in-process, finished goods). Operating Expense (OE) is all the money spent to convert inventory into throughput (wages, rent, utilities, etc., but not material). The goal is to increase throughput, reduce inventory, and minimize operating expense. It differs from traditional cost accounting because it treats only material as variable cost and sees all other costs as operating expenses.
10. A company can build a factory with flexible capacity that costs more but can be adjusted up or down. Compare this to a decision tree approach for fixed capacity. Why might the flexible option be better?
Flexible capacity lets you change output quickly based on demand, but it costs more to build. A decision tree for fixed capacity gives a single choice now, while flexible capacity is like having multiple future branches. For example, flexible plant costs $12M and can produce either 10,000 or 20,000 units. Fixed small costs $6M for 10,000 units, fixed large costs $10M for 20,000 units. If demand is uncertain, flexible capacity avoids the risk of being too small or too large. The decision tree would show that flexible capacity often has higher expected profit when demand swings are big, even though its base cost is higher. The tree helps you compare the 'expected value of flexibility' against the extra upfront cost.
11. Explain how 'input modeling' is done for simulation. What steps do you take to model the arrival process?
Input modeling means choosing probability distributions that match real-world data for arrival times, service times, and other random variables. Steps: 1) Collect a sample of actual data (e.g., time between arrivals). 2) Summarize with histogram and statistics (mean, variance). 3) Hypothesize a distribution family (e.g., exponential, Weibull, lognormal). 4) Estimate parameters using method of moments or maximum likelihood. 5) Test goodness-of-fit with chi-square or Kolmogorov-Smirnov test. For arrivals, often Poisson (exponential interarrival times) fits well, but if there is seasonality, use non-stationary Poisson or empirical distributions. Proper input modeling is critical, because garbage in gives garbage out.
12. Explain how throughput accounting can be used to measure the performance of a manager. How is it different from using profit or cost variances?
Throughput accounting measures a manager by how they affect global throughput, inventory, and operating expense. For example, a production manager is evaluated on increasing bottleneck throughput and reducing inventory, not on absorbing overhead. Traditional cost variance analysis might punish a manager for idle labor (unfavorable labor variance), pushing them to overproduce. In throughput accounting, producing more than what the bottleneck can process just adds inventory and does not improve throughput. So a manager's performance is tied to real profitability: sell more, protect bottleneck, cut operating expenses wisely. This aligns local decisions with company profit, avoiding the pitfalls of cost accounting.