Composite inventory items, defined here as SKUs with both price-elastic demand and a holding cost that depends on more than one state variable, are now specified using separable holding-cost forms D(p) coupled with h(p,t) or h(p,I) rather than the classic constant h assumption [S3].
The fit matters because price-elastic SKUs in composites, food, and engineered plastics typically lose 8 to 18 percent of theoretical optimal profit when the standard EOQ formula h = i·c is substituted in place of a separable nonlinear h(p, t) [S3]. Researchers close the gap by solving ∂(profit)/∂p = 0 jointly with ∂(profit)/∂T = 0, then reporting both the optimal cycle length T* and the optimal price p* in the same sensitivity table [S3] [S8].
Why a single h assumption fails for composite SKUs
Composite SKUs whose demand falls as price rises, and whose on-hand cost per unit per period grows with either time in storage or current inventory level, violate the constant-h premise of the original Wilson formula Q* = √(2DS/H) [S7]. The first Tripathi model (2025) builds demand as a multiplicative decreasing function of selling price, while holding cost is taken as a nonlinear multiplicative function of selling price and time, which together produce a closed-form interior optimum under standard regularity conditions [S3].
Two technical points from the literature. First, when h depends on the current inventory level I rather than t, the integrand for average holding cost shifts from h(p,t) to h(p,I), and the optimality condition changes accordingly; Cárdenas-Barrón et al. (2020) document this for the stock-level case with 209 citations, confirming broad uptake in operations research [S8]. Second, when h is a function of time only and demand is stock-dependent, the cycle-time condition still collapses to a single equation but the gradient with respect to p is no longer separable, which is what forces the joint p, T optimisation route [S2] [S4].
The two separable forms in use: h(p,t) and h(p,I)
Two separable forms dominate published work. Form A, the time-based variant, writes h(p, t) = α·p^β·t^γ, and pairs it with a price-dependent demand D(p) = a·p^(-b) to obtain a jointly concave profit function whose first-order conditions yield unique p* and T* [S3]. Form B, the inventory-level variant, writes h(p, I) = α·p^β·I^γ and the same demand, producing a different optimality condition because the carrying-cost integral now runs over the inventory trajectory rather than over calendar time [S3].
Selection rule from peer-reviewed work. Choose h(p, t) when deterioration, freshness decay, or storage-time-driven obsolescence dominates (perishables, dated chemicals, time-sensitive composites), and choose h(p, I) when the carrying cost is driven by floor space, insurance tiers, or working capital committed to stock-on-hand, where the marginal cost of one extra unit on the shelf is the controlling signal [S1] [S3]. A direct comparison of the two forms, with demand = 1000·p^(-1.5) and α = 0.05, shows Form A reports a 4 to 6 percent larger optimal order quantity than Form B for the same composite SKU, because the time-weighted integral penalises late-cycle units less aggressively than the level-weighted one [S3].
Decision criteria: which composite SKU fits which model
Match the SKU to the model on four criteria, drawn from the Tripathi and Alfares lines of work. Criterion 1, demand elasticity to price: if |b| > 1 in D(p) = a·p^(-b), joint p, T optimisation is mandatory; if |b| ≤ 0.3, the standard Wilson Q* = √(2DS/H) is within 3 percent of the joint optimum and the extra algebra is not justified [S3] [S7]. Criterion 2, holding-cost driver: time-driven h (h = α·p^β·t^γ) for perishables and dated composites, stock-level-driven h (h = α·p^β·I^γ) for slow-moving capital goods and oversized parts [S2] [S3] [S4]. Criterion 3, deterioration rate: when deterioration is non-trivial, the form D(p, t) with an explicit exponential decay term is required, and a rework branch has to be added if imperfect-quality items are accepted, as in Cárdenas-Barrón et al. (2021) [S9]. Criterion 4, complementarity or substitution: when SKUs are consumed together or substitute for one another, demand becomes a vector function D(p₁, p₂, p_c, p_s, t) and a multiple-item formulation is required, as demonstrated in the Mahlangu dissertation at the University of Pretoria [S5].
Cost drivers and total-cost-of-ownership for composite-SKU EOQ
For a price-elastic composite SKU, total landed cost decomposes into four moving parts, and each one responds differently to a price change. Part 1, unit purchase cost c: rises as p* rises, but the optimal p* is a function of the demand elasticity b and is not a free variable. Part 2, ordering cost S per cycle: fixed per order, so it falls on a per-unit basis as Q* rises, and the rise in Q* is roughly proportional to (S/h)^0.5 in the standard form, with the same 0.5 exponent preserved in the separable nonlinear h cases [S3] [S7]. Part 3, holding cost over the cycle: ∫₀^T h(p, x)·I(x) dx, which under h(p, t) = α·p^β·t^γ is dominated by late-cycle units because t^γ weights them more heavily; this is the channel through which the nonlinear h penalises long cycles more than a constant h does [S3]. Part 4, shortage and rework cost: when imperfect items are accepted, distinct holding rates for good and defective stock must be tracked, and the joint cost of rework plus lost sales must be netted against the savings from a longer cycle, as in the imperfect-quality model with nonlinear price-dependent demand [S9].
Quoted result, from the worked example in the Tripathi paper: for a = 1000, b = 1.5, α = 0.05, β = 0.4, γ = 0.6, the joint optimum lands at T* = 0.42 year and p* = 12.6 currency units, against T = 0.38 year and p = 11.9 for the constant-h case; the 10.5 percent longer cycle and 5.9 percent higher price together raise profit per cycle by 11.3 percent [S3]. Operations teams that price-elastic source for composite material lead times should treat this 11.3 percent delta as the realistic ceiling for what a properly separable h buys you over Wilson, before adding deterioration and rework penalties.
Real use cases and the substitutions and complements problem
The published use cases cluster in three groups. Group 1, dated composites and engineered plastics: holding cost is dominated by the time-to-expiry clock, and the Form A h(p, t) specification is the right call; engineered plastics buyers dealing with regional stock versus offshore orders run into the same time-decay penalty when offshore lead time exceeds 60 days, and the right way to size the safety stock is through a price-dependent, time-weighted h rather than a flat carrying charge [S1] [S3]. Group 3, complementary and substitutable items in a bill of materials: when one SKU is consumed alongside another (complement) or can replace another (substitute), a multiple-item formulation with a vector demand D(p₁, p₂, p_c, p_s, t) is required, and the Mahlangu model (2022) is the documented template for this case [S5].
Limitations, failure modes, and when not to use these models
Three failure modes recur. Failure mode 1, elasticity drift: the constant a and exponent b in D(p) = a·p^(-b) are estimated from a price-demand regression and they shift over the life of a SKU; the optimal p* therefore drifts, and a quarterly re-fit of (a, b) is needed for high-velocity SKUs, otherwise the joint p, T solution becomes a fiction [S3] [S8]. Failure mode 2, concave profit violated: when shortage backorders, rework, or cash discount terms are added, the profit function can lose concavity, and the first-order conditions are no longer sufficient; Tripathi (2021) handles this by partitioning the cycle into four states and checking the concavity of the total revenue in each [S1]. Failure mode 3, poor fit to capital goods: for construction machinery parts and slow-moving capital spares, the price-elastic demand assumption itself is weak, with |b| typically below 0.2, and a joint p, T optimisation gives less than 3 percent improvement over Wilson; in those cases the simpler form is preferred and the extra parameters are not justified [S3] [S7].
Sourcing and standards discipline
For price-dependent EOQ work on dated composites, sourcing decisions, like vendor selection, MOQ negotiation, and lead-time contracting, should be tied to the same (a, b, α, β, γ) tuple used in the model, and the parameters should be re-estimated on at least a quarterly cadence. Inventory managers who already use holding furnace or lamps and light fittings cost models can port the same separable h logic to their consumable SKUs, since the integrand form is dimensionally identical. The verification of the joint optimum is always the same two-step check: ∂(profit)/∂p = 0 yields p*, and ∂(profit)/∂T = 0 yields T*, then a second-derivative or numerical sweep confirms a local maximum rather than a saddle. Items where this discipline pays off most are price-elastic perishables, dated composites, and any multi-item BOM with explicit complements or substitutes, with single-item capital spares the category where the simplest form is usually enough [S3] [S4] [S5] [S8].
Trackable signals for the next 90 days: (1) any update to the Cárdenas-Barrón line of imperfect-quality, distinct-h papers extending the separable h to perishable composites; (2) industry adoption of monthly elasticity re-fits in ERP modules tied to EOQ re-solves; (3) extension of the Mahlangu complement-substitute model to four or more items in a BOM, which would close a known gap in the published literature [S5] [S9].