Mathematical Optimization Models

Introduction

Aggregate production planning (APP) determines how an organization will match forecast demand with available production, workforce, inventory, subcontracting, and capacity over a medium-term horizon, usually three to eighteen months. The central managerial problem is not to schedule each machine or individual job, but to decide the aggregate quantities and resource levels that provide an economically and operationally feasible plan.

Mathematical optimization converts this decision problem into a formal model. A model identifies decision variables, an objective function, constraints, parameters, and assumptions. The resulting solution is not automatically a managerial decision; it is a logically consistent recommendation that must be interpreted in relation to labour agreements, service requirements, cash availability, strategic priorities, and uncertainty.

Four model families are particularly important in aggregate planning:

1. linear programming (LP);
2. transportation and transshipment models;
3. mixed-integer programming (MIP); and
4. goal programming.

These models differ mainly in the types of relationships and decisions they can represent. LP is appropriate when costs and capacities are linear and decisions may be fractional. Transportation models exploit the flow structure of production and inventory. MIP represents indivisible or fixed-charge decisions, such as opening overtime capacity or hiring a crew. Goal programming handles situations in which several objectives must be satisfied simultaneously.

Learning Objectives

After studying this chapter, the reader should be able to:

• formulate an aggregate production plan as an LP;
• represent inventory, workforce, overtime, subcontracting, and backorders correctly;
• explain the relationship between APP and transportation models;
• formulate and interpret mixed-integer and goal-programming models;
• assess feasibility, optimality, and sensitivity information;
• evaluate model assumptions and communicate results to managers.

1. Structure of an Aggregate Planning Model

Let periods be indexed by t = 1,2,…,T. Demand in period t is D_t. A typical plan may include regular production, overtime production, subcontracting, inventory, workforce changes, and backorders.

A decision variable is a controllable quantity whose value is determined by the optimization model. A parameter is known or estimated before the model is solved. For example, P_t may denote regular-time production in period t, whereas c^P_t is its unit cost.

The standard material balance is:

where:

• I_t = ending inventory in period t (units);
• B_t = ending backorders in period t (units);
• P_t = regular-time production (units);
• O_t = overtime production (units);
• S_t = subcontracted production (units); and
• D_t = demand (units).

The equation states that beginning inventory, current production, and beginning backorders available for fulfilment must equal current demand, ending inventory, and ending backorders. An equivalent formulation uses net inventory N_t = I_t - B_t:

The separate-inventory formulation is usually preferable when positive inventory and backorders have different costs or when simultaneous inventory and backorders must be prohibited.

Typical objective function

A cost-minimizing aggregate plan may be written as:

Here, H_t and F_t are hiring and firing quantities; the corresponding costs are c^H_t and c^F_t. All cost coefficients must be expressed in compatible monetary units, such as currency per unit produced, currency per unit-period held, or currency per employee hired.

This objective assumes additive and linear costs. It does not automatically include revenue, quality, employee morale, environmental effects, or strategic risk. Such considerations must be represented through additional costs, constraints, or separate decision analysis.

Capacity and workforce constraints

If one employee provides a_t regular production hours in period t, and each unit requires r hours, then regular capacity can be represented by:

where W_t is the workforce level. Workforce continuity is represented by:

If overtime hours per employee are limited to o_t, then:

These constraints are dimensionally consistent: unit requirements per unit multiplied by units produced gives hours, while available hours per employee multiplied by employees also gives hours.

Common boundary conditions include:

A terminal inventory requirement is important. Without it, the model may deliberately consume all inventory or build backorders in the final period because no later consequence is represented. Similarly, if the workforce must return to a specified level, a terminal workforce constraint is required.

2. Linear Programming for Aggregate Planning

An LP has a linear objective and linear constraints. A general minimization form is:

The vector x contains decision variables, c contains objective coefficients, A contains technological coefficients, and b contains resource limits or requirements. LP assumes divisibility, known parameters, proportionality, additivity, and certainty over the planning horizon.

Illustrative LP formulation

A manufacturer plans two periods. Demand is 800 units in period 1 and 1,000 units in period 2. Initial inventory is 100 units and the desired ending inventory is 100 units. Regular production costs 20 currency units per unit, overtime costs 28, inventory holding costs 2 per unit-period, and subcontracting costs 35. Regular capacity is 700 units per period; overtime capacity is 200 units per period; subcontracting is unrestricted for modelling purposes. Backorders are not permitted.

Decision variables are P_t, O_t, S_t, and I_t for t = 1,2. The model is:

subject to:

Because regular production is cheaper than overtime and subcontracting, the model uses regular capacity first. Period 1 requires a net production of 700 units to finish with 100 units of inventory, so P₁ = 700 and I₁ = 0 are feasible. In period 2, 1,000 units of demand plus 100 units of ending inventory must be supplied. Regular production contributes 700 units and overtime contributes 200 units; the remaining 200 units are subcontracted. Thus P₂ = 700, O₂ = 200, and S₂ = 200. The total cost is:

The calculation illustrates an important modelling point: a cheaper activity may not be sufficient if capacity prevents it from meeting demand. The optimum is shaped jointly by unit costs, timing, inventory, and capacity.

LP interpretation

The solution reports more than production quantities. It identifies which constraints are binding. A capacity constraint is binding when its left-hand side equals its limit. In the example, period-2 regular and overtime capacities are binding. A nonbinding constraint has unused capacity.

The dual value, or shadow price, associated with a binding constraint estimates the marginal change in the optimal objective value caused by a one-unit relaxation of that constraint, within the allowable sensitivity range and subject to model assumptions. For a minimization model, the interpretation must be stated carefully: a positive shadow price for an additional required resource commonly indicates the value of relaxing a restrictive requirement, while a capacity constraint may have a negative marginal cost effect when increased. Solver conventions vary, so the sign and units must be read from the solver’s report.

3. Transportation Models and Network Representations

The transportation model allocates supply from sources to destinations at minimum cost. In APP, a source may represent production capacity in a particular period, while a destination represents demand in a future period. Inventory carryover is represented by a shipment from an earlier production period to a later demand period.

Let x_ij be units produced at source period i and used to satisfy demand in destination period j. Let c_ij be the cost per unit shipped from i to j, including production and any inventory carrying cost. The classical model is:

subject to supply limits:

and demand requirements:

with x_ij ≥ 0. Here S_i is available supply and D_j is demand. If total supply equals total demand, the table is balanced. If not, a dummy source or destination can be added, with a carefully selected penalty or zero cost. A dummy source must not be interpreted as free physical production; it represents unmet demand or an artificial balancing device.

For a production period i serving a later demand period j, a common cost is:

where p_i is production cost and h(i,j) is the applicable inventory cost. If inventory is charged per unit per period at h, then, for j > i, h(i,j) = h(j-i). The formula applies only when storage is feasible and production can serve the destination period.

Transportation structure is computationally attractive because specialized algorithms exploit its network form. However, the model becomes inadequate when workforce transitions, setup decisions, minimum batch sizes, nonlinear costs, or multiple interacting resources are central. In such cases, a general LP or MIP is more appropriate.

4. Mixed-Integer Programming

Many aggregate decisions are discrete. A plant may either activate a night shift or not; a subcontracting contract may involve a fixed charge; a workforce may consist of whole employees; and a production line may require a minimum batch. These features require integer or binary variables.

A mixed-integer linear program (MILP) has linear relationships but requires some variables to be integer. A binary variable y_t ∈ {0,1} can indicate whether an action occurs. For example, if overtime capacity of 300 units is available only when a fixed activation cost is paid, use:

and add fixed cost F y_t to the objective. If y_t = 0, overtime must be zero; if y_t = 1, overtime may be used up to 300 units. To impose a minimum overtime amount L whenever activated:

where U is the maximum overtime quantity.

For whole-number workers:

A model may use binary variables for hiring decisions and integer variables for workforce levels. Such models are generally harder to solve than LPs because the feasible region is no longer a continuous polyhedron. Solvers commonly use branch-and-bound or branch-and-cut. The solver may report an incumbent solution, a best bound, and an optimality gap. If the gap is not zero, the solution is feasible but may not be proven optimal.

MIP should be used selectively. Excessive binary variables, weak big-M coefficients, and unrealistic detail can produce long solution times and poor managerial transparency. A valid big-M value should be as small as possible while remaining mathematically sufficient; an unnecessarily large value can weaken computational bounds and cause numerical instability.

5. Goal Programming for Multiple Objectives

Managers rarely minimize cost alone. They may wish to maintain employment, achieve a service target, avoid overtime, and remain within a budget. These goals can conflict. Goal programming converts deviations from desired targets into variables and minimizes a weighted or lexicographic combination of those deviations.

For goal aᵀx = b, introduce nonnegative underachievement and overachievement variables d^- and d^+:

If the goal is to keep overtime at or below a target, only d^+ may be penalized. If the goal is to achieve minimum employment, only d^- may be penalized. Penalizing both deviations is appropriate when exact closeness to a target is desired.

A weighted goal-programming objective is:

where w_k^- and w_k^+ are penalty weights. Weights must be meaningful. Assigning a weight of 10 to service deviation and 2 to cost deviation does not have an automatic economic interpretation unless the deviations have been normalized or expressed in comparable value units.

In pre-emptive, or lexicographic, goal programming, priorities are ranked. The model first minimizes the highest-priority deviation, then minimizes the next priority without worsening the earlier result. This approach is appropriate when service is non-negotiable and cost is secondary. Weighted and lexicographic approaches can produce different solutions; the choice is a managerial value judgment, not merely a technical detail.

6. Feasibility, Validation, and Sensitivity Analysis

A model is feasible if at least one assignment satisfies every constraint. Infeasibility often results from inconsistent demand, insufficient capacity, excessive terminal inventory, or contradictory workforce restrictions. An infeasibility diagnosis should not be treated as a solver failure. It is evidence that assumptions or requirements conflict.

Useful remedies include:

• adding a controlled backorder or emergency-supply variable with a very high penalty;
• relaxing a terminal inventory requirement;
• allowing subcontracting or capacity expansion;
• reviewing demand or resource data; and
• using an irreducible infeasible subsystem to locate conflicting constraints.

Validation has three levels. First, dimensional validation checks units. Second, logical validation checks whether the equations represent the business rules. Third, managerial validation asks whether the solution is operationally and strategically credible.

Sensitivity analysis examines how the result changes when parameters change. Important questions include:

• How much can a unit cost change before the production pattern changes?
• What is the value of one additional unit of regular capacity?
• How much can demand increase before the current plan becomes infeasible?
• Is the plan dependent on a narrow and uncertain parameter estimate?

Classical LP sensitivity reports include reduced costs, shadow prices, and allowable ranges. A reduced cost for a variable at zero indicates the improvement in its objective coefficient required before that variable can become attractive, subject to the report’s assumptions. These interpretations are local; they should not be extrapolated beyond the allowable range or applied unchanged to integer models.

Scenario analysis is often more informative than a single sensitivity report. For example, planners may solve the model under base demand, demand increased by 10%, delayed supply, higher overtime cost, and reduced subcontracting capacity. Stochastic programming and robust optimization extend this logic by explicitly modelling uncertainty, but they require probability distributions, scenario structures, or uncertainty sets that must be justified.

7. Illustrative Case: Northstar Appliances

Background

Northstar Appliances, an illustrative manufacturer, produces a standard countertop appliance over a four-month horizon. Forecast demand is 2,400, 2,800, 3,100, and 2,700 units. Initial inventory is 300 units, and management requires at least 200 units at the end of month 4. Regular capacity is 2,500 units per month. Overtime capacity is 400 units per month, but overtime is available only if a binary decision activates a second shift. Activating the shift costs 12,000 currency units per month. Regular production costs 42 per unit, overtime costs 55, and subcontracting costs 68. Inventory costs 3 per unit per month. Backorders are allowed only temporarily and cost 25 per unit per month, in addition to the eventual production or subcontracting cost.

Problem

The operations director wants a plan that controls cost while protecting customer service. The production manager prefers to avoid subcontracting, whereas finance is concerned about the fixed second-shift cost.

Model approach

Let P_t, O_t, S_t, I_t, and B_t represent regular production, overtime, subcontracting, inventory, and backorders. Let y_t be a binary second-shift activation variable. Material balance is:

The capacity constraints are:

The objective includes production, inventory, backorder, subcontracting, and activation costs:

with I₀ = 300, I₄ ≥ 200, and all physical quantities nonnegative. If the organization does not permit simultaneous inventory and backorders, additional binary logic is required; otherwise, the formulation should be checked for economically undesirable simultaneous values.

Analysis and managerial interpretation

Total demand is 11,000 units. Initial and required ending inventory contribute 500 units, so net production or subcontracting required over the horizon is 10,500 units. Regular capacity provides at most 10,000 units. Therefore, at least 500 units must come from overtime or subcontracting, unless backorders remain at the horizon—which is ruled out by the service requirement.

The second shift is economically attractive only if its variable and fixed costs are lower than the avoided alternative. Five hundred overtime units would incur 27,500 currency units in variable cost plus 12,000 activation cost, totalling 39,500. Five hundred subcontracted units cost 34,000. On this simplified comparison, subcontracting is cheaper. However, timing and capacity matter: subcontracting may be needed in a peak month, while overtime can be used in other months and inventory can transfer supply. The MILP must therefore be solved rather than decided from total quantities alone.

A practical policy is to compare at least three solutions: a cost-minimizing plan, a no-subcontracting plan, and a service-prioritized plan. Management should examine not only total cost but also shift activation, inventory peaks, supplier dependency, labour impact, and sensitivity to demand error.

Learning points

The case demonstrates that aggregate planning decisions are interdependent. A fixed-charge decision changes the marginal economics of overtime. Inventory links decisions across months. A terminal condition prevents the model from postponing the problem indefinitely. Finally, a mathematically optimal plan can be strategically unacceptable if it creates excessive supplier dependence or violates workforce agreements.

8. Implementation and Governance

A reliable optimization process begins with a clearly defined planning calendar, consistent units, and traceable data sources. Demand forecasts should be version-controlled, and cost parameters should distinguish accounting cost from relevant incremental cost. Planners should document whether inventory cost includes capital, warehousing, insurance, obsolescence, and handling.

Model governance should include independent review of equations, test cases with known answers, and automated checks for conservation of flow. A model should be tested under extreme cases: zero demand, zero capacity, abundant capacity, and prohibitively high subcontracting cost. The solution should be reproducible using recorded data, solver version, tolerances, and configuration.

Advanced analytics and AI can support demand forecasting, anomaly detection, scenario generation, and parameter estimation. They should not silently replace managerial assumptions. Forecast uncertainty, bias, and data drift must be monitored. Human decision-makers remain responsible for approving constraints, interpreting trade-offs, and ensuring that optimization does not encourage unsafe overtime, discriminatory workforce treatment, or misleading service commitments.

9. Limitations and Contemporary Developments

Optimization models simplify reality. Linear costs may ignore quantity discounts, learning curves, congestion, and nonlinear fatigue. Forecasts are uncertain. Aggregate products may conceal component bottlenecks. A plan that is feasible in aggregate may be infeasible at a particular work centre. Furthermore, optimizing expected cost may produce a fragile plan when disruption risk is high.

Contemporary practice increasingly combines LP or MIP with rolling-horizon planning, digital supply-chain platforms, robust optimization, stochastic programming, and simulation. A rolling horizon solves the model repeatedly as new demand and capacity information becomes available. Simulation can test whether an aggregate plan survives variability in arrivals, processing times, and supply. These tools complement rather than eliminate the need for sound model formulation.

Chapter Summary

Mathematical optimization gives aggregate production planning a disciplined structure. Linear programming represents divisible production, inventory, workforce, overtime, and subcontracting decisions under linear cost and capacity assumptions. The material-balance equation is the foundation of temporal feasibility. Transportation models exploit the flow of supply from production periods to demand periods and are especially useful when inventory carryover is the central issue. Mixed-integer programming represents indivisible workers, setup choices, fixed charges, minimum batches, and activation decisions. Goal programming manages multiple, potentially conflicting objectives by penalizing deviations from targets according to weighted or lexicographic priorities.

A valid model requires correctly defined variables, consistent units, realistic boundary conditions, and explicit assumptions. Feasibility analysis identifies conflicts among demand, capacity, and policy requirements. Sensitivity analysis reveals the value and fragility of resources, while scenario analysis examines uncertainty more broadly. In practice, the best plan is not simply the one with the smallest calculated cost; it is a solution that is feasible, robust, explainable, ethically acceptable, and aligned with organizational strategy.

Key Terms

Decision variable: A controllable quantity whose value is determined by the optimization model.

Parameter: A known or estimated input, such as demand or unit cost.

Material balance: An equation equating available supply, demand, inventory, and backorders.

Linear programming: Optimization with a linear objective and linear constraints.

Feasible solution: A set of variable values satisfying every model constraint.

Optimal solution: A feasible solution with the best objective value under the model.

Transportation model: A network allocation model assigning supply sources to demand destinations.

Inventory carryover: The use of production from an earlier period to satisfy later demand.

Backorder: Demand postponed for fulfilment in a later period.

Shadow price: The marginal value of relaxing a constraint within a valid sensitivity range.

Reduced cost: The objective-coefficient improvement needed for a currently zero variable to become attractive, under LP assumptions.

Mixed-integer programming: Linear optimization containing integer or binary variables.

Binary variable: A variable restricted to 0 or 1.

Fixed charge: A cost incurred when an activity is activated, independent of its volume.

Big-M constraint: A logical modelling constraint using a sufficiently large coefficient.

Goal programming: Multi-objective programming based on deviations from target goals.

Lexicographic priority: Sequential objective priorities in which higher-priority goals dominate lower-priority goals.

Rolling horizon: Repeated planning in which the horizon moves forward as new information arrives.

Robust optimization: Optimization designed to perform acceptably across specified uncertainty sets.

Review Questions

Short Answer Questions

1. What is the purpose of the material-balance equation in aggregate planning?
2. State two assumptions of a linear programming model.
3. How does a transportation model represent inventory carryover?
4. What is the difference between a continuous and a binary decision variable?
5. Why is a terminal inventory condition often necessary?

Long Answer Questions

1. Formulate a linear aggregate planning model containing regular production, overtime, inventory, and subcontracting. Define all variables and explain each constraint.
2. Compare LP, transportation, MIP, and goal programming in terms of decision types, strengths, and limitations.
3. Explain shadow prices, reduced costs, and allowable sensitivity ranges. Why should their interpretation be restricted to local changes?
4. Discuss how workforce continuity and fixed-charge overtime can be represented in a mixed-integer model.
5. Evaluate the ethical and managerial risks of implementing an optimization model without validating its assumptions.

Analytical / Application Questions

1. A firm has initial inventory of 120 units, demand of 500 units in period 1, regular capacity of 300 units, and overtime capacity of 250 units. Formulate the period-1 balance and determine the minimum overtime required if ending inventory must be zero and subcontracting is unavailable.
2. Design a two-period transportation representation in which period-1 production may satisfy either period-1 or period-2 demand. Explain how production cost and inventory cost enter the shipping costs.
3. A manager wants to minimize cost, keep overtime below 100 units, and ensure at least 95% of demand is served on time. Recommend whether weighted goal programming, lexicographic goal programming, or a hard-constraint LP is most appropriate, and justify the choice.

Multiple Choice Questions with Answers

Question 1

Which equation correctly represents net inventory in period t?

A. N_t = N_t-1 + P_t + O_t + S_t - D_t

B. N_t = N_t-1 + D_t - P_t

C. N_t = P_t - O_t - S_t

D. N_t = D_t + I_t + B_t

Correct Answer: A. N_t = N_t-1 + P_t + O_t + S_t - D_t

Question 2

What does a binary variable most commonly represent in an aggregate planning MILP?

A. A fractional inventory quantity

B. Whether an activity is activated

C. A unit production cost

D. A demand forecast

Correct Answer: B. Whether an activity is activated

Question 3

In a transportation model, what does a shipment from an earlier production period to a later demand period usually represent?

A. Hiring

B. Backorder penalty only

C. Inventory carryover

D. Machine breakdown

Correct Answer: C. Inventory carryover

Question 4

If regular production costs 20 per unit and overtime costs 28 per unit, what is the variable production cost of 100 regular units and 50 overtime units?

A. 2,800

B. 3,400

C. 4,000

D. 3,600

Correct Answer: B. 3,400

Question 5

Which condition is necessary for a solution to be feasible?

A. It must use every available resource

B. It must have zero inventory

C. It must satisfy every model constraint

D. It must contain at least one binary variable

Correct Answer: C. It must satisfy every model constraint

Question 6

What is the purpose of a terminal inventory constraint?

A. To eliminate all production costs

B. To prevent the model from ending with an unintended inventory position

C. To force overtime in every period

D. To make all costs nonlinear

Correct Answer: B. To prevent the model from ending with an unintended inventory position

Question 7

In goal programming, what do deviation variables measure?

A. Machine speed

B. The difference between achieved and targeted goal levels

C. Forecast probability only

D. The number of suppliers

Correct Answer: B. The difference between achieved and targeted goal levels

Question 8

A capacity constraint has a shadow price of 6 currency units per additional unit of capacity within its allowable range. What does this generally indicate?

A. Capacity has no economic value

B. The objective may improve by approximately 6 currency units per added capacity unit, subject to sign convention and range

C. Demand will fall by 6 units

D. The constraint must be removed

Correct Answer: B. The objective may improve by approximately 6 currency units per added capacity unit, subject to sign convention and range

Question 9

Why can an unnecessarily large big-M value be problematic?

A. It always makes the model infeasible

B. It may weaken computational performance and numerical stability

C. It converts all variables to binary

D. It eliminates fixed costs

Correct Answer: B. It may weaken computational performance and numerical stability

Question 10

Which method is most suitable when workers and activation decisions must be integral?

A. Pure unconstrained calculus

B. Mixed-integer programming

C. A demand average alone

D. A transportation model without constraints

Correct Answer: B. Mixed-integer programming