Operations and Resource-Planning Models

Introduction

Operations decisions determine how an organisation converts resources into products and services while meeting cost, quality, speed, dependability and flexibility objectives. In a manufacturing firm, these decisions include how much inventory to hold, how much production capacity to install and how to schedule interdependent activities. In a hospital, they include staffing, bed capacity and patient queues. In a consulting firm, they include assigning specialists to projects and managing uncertain completion times.

Operations and resource-planning models provide disciplined ways to analyse these choices. They do not replace managerial judgement; rather, they make assumptions explicit, quantify trade-offs and reveal bottlenecks. This chapter examines four closely related families of models: inventory models, queuing models, capacity models, and PERT/CPM project-scheduling models.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Explain how inventory policies balance ordering, holding and shortage costs.
2. Calculate economic order quantities, reorder points and safety stock under stated assumptions.
3. Analyse waiting lines using basic queuing measures and recognise when the assumptions are inappropriate.
4. Distinguish design capacity, effective capacity and actual output, and diagnose bottlenecks.
5. Construct and interpret CPM networks and calculate project duration, critical paths and slack.
6. Use PERT to incorporate uncertainty in activity times.
7. Integrate these models into resource-allocation and service-level decisions.
8. Evaluate the ethical, technological and governance implications of quantitative operational decisions.

1. Operations Models as Decision Instruments

An operations model is a simplified representation of a system. A useful model identifies decision variables, uncontrollable parameters, constraints, an objective and performance measures. For example, an inventory model may choose order quantity while demand and lead time are treated as uncertain parameters. A queuing model may estimate waiting time from arrival and service rates.

Models are useful only within their applicability conditions. A result from a constant-demand inventory model should not be presented as a precise answer when demand is seasonal, suppliers are unreliable or quantity discounts materially affect cost. Graduate-level analysis therefore requires both calculation and model criticism.

A useful performance perspective is the balanced scorecard of operations:

Improving one dimension can damage another. More safety stock may improve availability but increase carrying cost and obsolescence. More servers may reduce queues but increase labour expense. The appropriate objective is therefore not simply “minimise cost,” but optimise performance subject to strategic and operational constraints.

2. Inventory Planning

2.1 Inventory functions and classifications

Inventory decouples supply from demand. It can protect against uncertainty, support economies of scale, permit production smoothing and buffer different stages of a process. Common categories include:

• Cycle stock: inventory created by ordering or producing in batches.
• Safety stock: inventory held against demand or supply uncertainty.
• Pipeline inventory: goods in transit or otherwise committed but not yet available.
• Anticipation inventory: stock accumulated for predictable peaks or disruptions.
• Decoupling inventory: stock separating stages with different operating rates.

Inventory creates costs. Ordering or setup costs occur when replenishment is initiated. Holding costs include capital, storage, insurance, deterioration and obsolescence. Shortage costs include lost sales, expediting, penalties and reputational damage. The manager seeks a policy that balances these costs and the required service level.

2.2 The basic economic order quantity model

The classical economic order quantity (EOQ) model assumes constant and known demand, instantaneous replenishment, no shortages, a constant unit price and independent orders. Let D be annual demand in units, S ordering or setup cost per order, and H annual holding cost per unit. The annual relevant cost is:

Here, Q is order quantity. The first term is annual ordering cost: the number of orders, D/Q, multiplied by cost per order. The second is annual holding cost, because average cycle stock is Q/2. Purchase cost is omitted when unit price is constant because it does not change with Q.

Minimising total relevant cost gives:

The formula is dimensionally consistent: annual units × currency/order ÷ currency/unit/year produces units², whose square root is units. The number of orders per year is D/Q*, and cycle time between orders is approximately working-days/(D/Q*) working days.

Worked example. A distributor sells 12,000 units annually. Ordering cost is $50 per order and annual holding cost is $2 per unit. Then:

Q* = √((2 × 12,000 × 50)/2) = √600,000 ≈ 775 units.

The firm would place approximately 12,000 ÷ 775 = 15.5 orders per year. Average cycle stock is approximately 775 ÷ 2 = 387.5 units. Annual ordering cost is approximately (12,000 ÷ 775) × $50 = $774.19; annual holding cost is approximately 387.5 × $2 = $775. The small difference is rounding; at the exact optimum, the two costs are equal.

EOQ is not automatically an order quantity. Minimum order constraints, supplier pack sizes, warehouse capacity, cash restrictions, quantity discounts and strategic supplier agreements may require adjustment.

2.3 Reorder point and safety stock

The reorder point specifies when replenishment should be initiated. With constant demand and known lead time:

where d is demand per time unit and L is lead time in the same time unit. Unit consistency is essential: daily demand must be paired with lead time in days.

When demand varies but lead time is constant, a common safety-stock expression is:

and therefore:

Here, σ_d is the standard deviation of demand per period and z is the standard-normal value associated with the desired cycle service level. This formulation assumes independent demand across periods and a fixed lead time.

If both demand and lead time are variable and independent, the standard deviation of lead-time demand can be approximated by:

where σ_L is the standard deviation of lead time. The reorder point is then:

This approximation should not be used blindly when demand and lead time are correlated, demand is highly intermittent or the distributions are strongly non-normal. In such cases, simulation or an empirical demand-lead-time distribution is preferable.

The cycle service level is the probability of avoiding a stockout during a replenishment cycle. It differs from the fill rate, which is the proportion of demand fulfilled immediately from stock. Confusing these measures can lead to inappropriate safety-stock decisions.

2.4 Periodic review and ABC analysis

Under a periodic-review policy, inventory is checked at intervals and replenishment raises the stock position to a target level. If review interval is R, lead time is L, average demand per period is d, and standard deviation per period is σ_d, a normal-demand target may be expressed as:

This assumes independent demand and fixed review and lead-time periods. The order quantity is the target stock level minus the inventory position, where inventory position includes on-hand inventory plus on-order inventory minus backorders.

ABC analysis classifies items by annual dollar usage, usually calculated as annual demand multiplied by unit cost. “A” items receive close control, accurate records and frequent review; “C” items may use simpler controls. ABC classification is a prioritisation device, not a substitute for risk analysis. A low-value but critical medical component may deserve tighter control than its dollar usage suggests.

2.5 Inventory implications of lean systems

Lean operations seek to expose and remove causes of inventory rather than merely eliminate inventory buffers. Smaller lots, reliable suppliers, quick changeovers, preventive maintenance and quality at source make lower inventory feasible. However, reducing stock without improving process capability can convert visible inventory into hidden shortages and lost service. Resilience analysis should consider supplier concentration, geopolitical exposure, cyber risk and recovery time.

3. Queuing Analysis and Service Levels

3.1 The logic of waiting lines

A queue forms when demand for service temporarily exceeds available service capacity. Its components are an arrival process, a waiting discipline, one or more servers and a service-time distribution. Important measures include:

• λ: average arrival rate, customers per time unit.
• μ: service rate per server, customers per time unit.
• s: number of parallel servers.
• ρ: utilisation.
• L_q: average number waiting.
• W_q: average waiting time.
• L: average number in the system.
• W: average time in the system.

For a stable single-server system, the utilisation is:

For s identical parallel servers, utilisation per server is:

The stability condition is necessary but does not guarantee acceptable service. At utilisation close to 1, small variability can produce very long waits.

3.2 Little’s Law

Little’s Law applies broadly to stable systems over a sufficiently long observation period:

Similarly, for the waiting line:

The arrival rate and time must use compatible units. If 30 customers arrive per hour and the average system time is 0.2 hours, the average number in the system is 30 × 0.2 = 6 customers. Little’s Law is an identity linking flow, inventory and time; it does not by itself explain the cause of waiting.

3.3 M/M/1 model

The M/M/1 model assumes Poisson arrivals, exponentially distributed service times, one server, a first-come-first-served discipline and an unlimited queue. For a stable system:

Worked example. A service desk receives 18 customers per hour and serves 24 per hour. Thus ρ = 18/24 = 0.75. The expected queue length is:

L_q = 0.75²/(1 − 0.75) = 2.25 customers.

The expected waiting time is W_q = 2.25/18 = 0.125 hours, or 7.5 minutes. Expected total time is W = 1/(24 × 0.25) = 1/6 hour, or 10 minutes. The difference, 2.5 minutes, is the expected service time.

These are long-run averages, not guarantees. They may be inappropriate for batch arrivals, appointments, finite populations, priority classes, balking customers or non-exponential service times.

3.4 Managerial use and limitations

Managers can reduce waiting by increasing service rate, adding servers, smoothing arrivals, separating high- and low-complexity work, using appointments, digitising transactions or redesigning the process. A decision should compare the marginal cost of capacity with the value of shorter waits and improved retention.

Queue discipline also has ethical implications. A first-come-first-served rule appears fair, but emergency departments may rationally prioritise clinical urgency. Priority systems should be transparent, clinically or operationally justified and monitored for disparate effects. Automated triage requires auditability and human escalation for exceptional cases.

4. Capacity Planning and Bottlenecks

4.1 Capacity concepts

Design capacity is the theoretical maximum output under ideal conditions. Effective capacity is the maximum realistic output under normal product mix, maintenance, quality and scheduling conditions. Actual output is what the operation produces. Capacity utilisation is:

Efficiency relative to effective capacity is:

A plant producing 7,200 units against design capacity of 10,000 has 72% utilisation. If effective capacity is 8,000, efficiency is 90%. These are different managerial signals: low utilisation may reflect a realistic protective capacity cushion, while low efficiency suggests operational losses.

4.2 Capacity requirements and bottleneck analysis

If a product requires t_j minutes at work centre j and forecast demand is D, required processing time is D t_j. With available time per resource of A_j minutes and efficiency e_j, the approximate number of resources required is:

The ceiling function rounds up to a whole resource. This formula assumes a stable product mix and that efficiency appropriately represents downtime, performance losses and quality losses.

A bottleneck is a constraint limiting system throughput. Adding capacity away from the bottleneck may increase local utilisation but not total output. The Theory of Constraints recommends identifying the constraint, exploiting it, subordinating other activities, elevating the constraint and repeating the process after the constraint moves.

For a process with sequential stages, throughput is constrained by the stage with the lowest effective rate. If stage rates are 40, 30 and 36 units per hour, nominal process capacity is 30 units per hour at stage 2. In practice, buffers and variability mean realised throughput may be lower.

4.3 Capacity cushion and aggregate planning

Capacity cushion can be expressed as:

A service organisation may maintain a larger cushion because waiting is immediately visible and demand is uncertain. A capital-intensive manufacturer may accept lower cushion but use overtime, subcontracting or inventory to absorb peaks.

Aggregate planning balances forecast demand and resources over a medium-term horizon. Strategies include level capacity, chase demand, overtime, subcontracting, inventory accumulation and backorders. The appropriate plan depends on labour regulations, cash flow, service commitments, workforce skills and reversibility of decisions.

5. CPM and PERT Project Scheduling

5.1 Network logic

Projects consist of activities with precedence relationships. In an activity-on-node network, each node represents an activity and arrows represent dependencies. A project begins when all required predecessor activities are complete. The critical path is the longest-duration path through the network; it determines the earliest possible completion time.

For each activity, the forward pass calculates:

where ES is earliest start, EF earliest finish and t activity duration. For an activity with several predecessors, its earliest start is the maximum predecessor earliest finish:

The backward pass calculates:

where LS is latest start and LF latest finish. For an activity with several successors, latest finish is the minimum successor latest start:

Total slack is:

Activities with zero total slack are critical under the specified network and durations. If two paths have nearly equal duration, the project has near-critical-path risk even if only one path has exactly zero slack.

5.2 CPM worked example

Consider the following activities:

The paths are A–B–D–F = 3 + 4 + 5 + 2 = 14 days and A–C–E–F = 3 + 2 + 6 + 2 = 13 days. Therefore, the project duration is 14 days and A–B–D–F is critical. The second path has one day of slack, assuming the stated deterministic durations. Delaying activity E by one day would not delay completion, but a two-day delay would.

CPM supports resource allocation by identifying activities where delay directly affects completion. It does not, however, automatically solve resource conflicts. If two activities require the same scarce specialist, resource levelling may delay a nominally non-critical activity and create a new critical path.

5.3 PERT under uncertainty

PERT uses three time estimates: optimistic a, most likely m and pessimistic b. The expected activity time is:

The commonly used activity variance is:

These expressions are based on a beta-like approximation and are most defensible when the three estimates reflect genuine uncertainty rather than arbitrary optimism or pessimism. For a selected critical path, the expected project duration is the sum of expected activity times, and—assuming independent activity times—the path variance is the sum of activity variances:

A normal approximation for the probability of completing by deadline D is:

where Φ is the standard-normal cumulative distribution function. This approximation becomes questionable when the project has substantial path switching, correlated activities or a highly skewed completion distribution. Monte Carlo simulation is then preferable.

PERT example. An activity has estimates a = 2, m = 5 and b = 8 days. Expected time is (2 + 4×5 + 8) ÷ 6 = 5 days. Variance is [(8 − 2) ÷ 6]² = 1 day², so standard deviation is 1 day. For a path containing independent activities with expected times 5, 4 and 6 days and variances 1, 0.25 and 0.36, path mean is 15 days and variance is 1.61 day²; standard deviation is approximately 1.27 days. For a 17-day deadline, the standardised value is (17 − 15) ÷ 1.27 ≈ 1.57, giving an approximate completion probability of 0.94 under the normal approximation.

6. Integrated Case: Regional Diagnostic Centre

The following is an illustrative case, not a report of an actual organisation.

Background and problem

A regional diagnostic centre performs laboratory tests and promises results within 24 hours. Demand has grown, patients complain about waiting, and managers frequently expedite reagent orders. The centre has one sample-receiving desk, two analysers and a specialist validation team.

Analysis

Daily reagent demand is 80 kits. Ordering cost is $120 per order and annual holding cost is $6 per kit. With 250 operating days, annual demand is 20,000 kits. EOQ is:

Q* = √((2 × 20,000 × 120)/6) = √800,000 ≈ 894 kits.

The calculated quantity is reviewed against reagent shelf life and storage limits; a lower quantity may be necessary if expiry risk is material.

The receiving desk receives 24 samples per hour and processes 30 per hour. Under an M/M/1 approximation, utilisation is 0.80 and:

L_q = 0.8²/(1 − 0.8) = 3.2 samples.

W_q = 3.2/24 = 0.1333 hours, or about 8 minutes. The desk is stable, but variability makes the queue sensitive to disruptions. Appointment staggering and a second cross-trained employee during morning peaks are considered.

The validation process has effective capacity of 180 tests per day against demand of 170, leaving a capacity cushion of approximately 5.6%. A planned instrument upgrade would increase analyser capacity, but analysis shows validation—not analysis—is the current constraint. The upgrade is therefore unlikely to solve the service problem unless validation capacity is also expanded.

Finally, a CPM schedule for a software-enabled workflow redesign identifies configuration, data migration, staff training and validation activities. The critical path is used to protect the go-live date, while near-critical training activities receive contingency because staff availability is uncertain.

Approach and solution

Management adopts a segmented inventory policy: high-risk reagents receive service-level-based safety stock, while stable, low-cost consumables use periodic review. It redesigns the receiving process, adds peak-period capacity and measures queue time by hour rather than relying only on daily averages. The project team simulates alternative staffing plans and tracks the critical path weekly.

Outcome and learning points

The central learning is that the models must be used together. EOQ alone cannot guarantee reagent availability; a queue model alone cannot identify validation as the system constraint; CPM alone cannot resolve staffing capacity. Integrated analysis links stock availability, process flow, scarce expertise and implementation timing.

7. Technology, Governance and Contemporary Practice

Enterprise resource-planning systems can automate reorder points, supplier lead-time records, capacity requirements and project baselines. Internet-of-Things sensors may provide real-time inventory and equipment-status data. Digital twins and discrete-event simulation can test staffing, layout and disruption scenarios without changing the live operation.

Artificial intelligence can forecast demand, estimate no-show probabilities, detect bottleneck patterns and recommend schedules. Nevertheless, automated recommendations require data-quality controls, versioned assumptions, exception handling and human accountability. Historical data may encode discriminatory access patterns; an algorithm that reduces service to low-volume communities may be financially efficient but socially unacceptable.

Good governance includes documented model assumptions, separation of forecast and decision approval, audit trails, sensitivity analysis, access controls and periodic back-testing. Managers should report uncertainty rather than presenting a single forecast as fact. Regulatory requirements may apply to healthcare, employment, safety, privacy and financial reporting contexts. Operational data should be minimised, protected and retained only as long as justified.

8. Limitations and Best Practices

The most common modelling failures are:

1. Treating averages as certainty and ignoring variability.
2. Using inconsistent time units in demand, service or lead-time calculations.
3. Applying EOQ when quantity discounts, perishability or shortages dominate.
4. Estimating queue performance with M/M/1 assumptions when arrivals are scheduled or service times are highly variable.
5. Optimising a non-bottleneck while leaving the constraint unchanged.
6. Treating the first CPM critical path as permanent despite resource conflicts and uncertainty.
7. Ignoring behavioural responses, such as customers abandoning a queue or employees changing work pace.
8. Failing to test sensitivity to demand, lead time, processing time and capacity assumptions.

Best practice is to define the decision and service objective first, validate data, state assumptions, calculate a baseline, test alternative scenarios and monitor actual performance after implementation. Sensitivity analysis is especially important. If a small change in demand changes the recommended staffing or inventory policy, management should retain flexibility rather than over-commit to a precise point estimate.

Chapter Summary

Inventory models balance ordering, holding and shortage considerations. EOQ gives the least-cost lot size under restrictive assumptions; reorder points and safety stock connect demand and lead-time uncertainty to service levels. Queuing models quantify utilisation, waiting and system time, while Little’s Law provides a general relationship between flow, time and work in process. Capacity models distinguish design capacity, effective capacity and actual output and direct managerial attention to bottlenecks. CPM identifies project duration, critical activities and slack. PERT incorporates uncertain activity times but depends on approximation and independence assumptions. In practice, these models should be integrated, validated with operational data, tested through sensitivity or simulation, and governed with attention to fairness, privacy, safety and accountability.

Key Terms

EOQ: Economic order quantity that minimises relevant ordering and holding costs under stated assumptions.

Safety stock: Inventory held to protect against uncertainty.

Reorder point: Stock position at which replenishment is initiated.

Cycle service level: Probability of avoiding a stockout during a replenishment cycle.

Fill rate: Proportion of demand fulfilled immediately from available stock.

ABC analysis: Classification of inventory items by relative annual dollar usage or importance.

Queue: Waiting line formed when demand temporarily exceeds service capacity.

Arrival rate: Average number of customers or jobs arriving per unit of time.

Service rate: Average number of customers or jobs served by one server per unit of time.

Utilisation: Proportion of available service or production capacity being used.

Little’s Law: Relationship L = λW linking average system content, flow rate and time.

Capacity cushion: Capacity maintained above expected demand.

Bottleneck: Constraint that limits total system throughput.

Design capacity: Maximum theoretical output under ideal conditions.

Effective capacity: Realistic maximum output under normal operating conditions.

CPM: Critical Path Method for deterministic project scheduling.

Critical path: Longest-duration path determining earliest project completion.

Slack: Time an activity can be delayed without delaying project completion.

PERT: Project Evaluation and Review Technique using uncertain activity-time estimates.

Monte Carlo simulation: Repeated random sampling used to estimate distributions of possible outcomes.

Review Questions

Short Answer Questions

1. What costs are included in the basic EOQ model, and why is purchase cost usually omitted?
2. Distinguish cycle service level from fill rate.
3. State Little’s Law and explain the units of its variables.
4. Differentiate design capacity, effective capacity and actual output.
5. What makes an activity critical in a CPM network?

Long Answer Questions

1. Explain the assumptions of the EOQ model and discuss circumstances in which a manager should not use its result without modification.
2. Describe how demand variability and lead-time variability affect reorder points and safety stock.
3. Explain why high utilisation can produce disproportionately long waiting times in a queue.
4. Discuss how bottleneck analysis can improve resource allocation in a multi-stage operation.
5. Compare CPM and PERT, including their assumptions, outputs and limitations.

Analytical / Application Questions

1. A retailer has annual demand of 18,000 units, ordering cost of $75 and annual holding cost of $3 per unit. Calculate EOQ, annual ordering cost and annual holding cost at EOQ. State the assumptions required.
2. A single service desk receives 20 customers per hour and serves 25 per hour. Using the M/M/1 model, calculate utilisation, average queue length and average waiting time. Interpret the operational risk.
3. A project contains two paths: A–B–D with durations 4, 6 and 5 days, and A–C–D with durations 4, 8 and 5 days. Identify the critical path and project duration, then explain how a two-day delay in activity B affects completion under the stated deterministic assumptions.

Multiple Choice Questions with Answers

Question 1

In the basic EOQ model, which change increases the optimal order quantity?

A. An increase in annual holding cost

B. An increase in ordering cost

C. A decrease in annual demand

D. A decrease in purchase price when price is constant for all quantities

Correct Answer: B. An increase in ordering cost

Question 2

Daily demand is 40 units and lead time is 5 days. With no safety stock, the reorder point is:

A. 8 units

B. 45 units

C. 200 units

D. 400 units

Correct Answer: C. 200 units

Question 3

A queue has an arrival rate of 15 customers per hour and a service rate of 20 customers per hour at one server. Its utilisation is:

A. 0.25

B. 0.75

C. 1.33

D. 5.00

Correct Answer: B. 0.75

Question 4

Which relationship is Little’s Law?

A. L = λW

B. L = λ/W

C. W = Lλ

D. λ = LW

Correct Answer: A. L = λW

Question 5

If actual output is 7,200 units and design capacity is 9,000 units, utilisation is:

A. 20%

B. 72%

C. 80%

D. 125%

Correct Answer: B. 72%

Question 6

A process has stage capacities of 50, 42 and 60 units per hour. Which stage is the bottleneck?

A. The first stage

B. The second stage

C. The third stage

D. All stages equally

Correct Answer: B. The second stage

Question 7

The critical path in a CPM network is the path that:

A. Has the fewest activities

B. Has the lowest resource cost

C. Has the longest total duration

D. Contains every activity

Correct Answer: C. Has the longest total duration

Question 8

For PERT estimates a = 2, m = 5 and b = 8 days, the expected activity time is:

A. 2 days

B. 4 days

C. 5 days

D. 8 days

Correct Answer: C. 5 days

Question 9

An activity has ES = 6 days and duration 4 days. Its earliest finish is:

A. 2 days

B. 6 days

C. 10 days

D. 24 days

Correct Answer: C. 10 days

Question 10

Which is the strongest reason to use simulation rather than a simple deterministic capacity calculation?

A. Simulation removes the need for data

B. Simulation can represent variability and alternative scenarios

C. Simulation guarantees the optimal decision

D. Simulation makes assumptions unnecessary

Correct Answer: B. Simulation can represent variability and alternative scenarios