Optimal policy for managing stochastic cash flows in a financial supply chain

(subconvexity). A real function f ( x 1 , x 2 ) on R 2 is subconvex in ( x 1 , x 2 ) if, for every ( x 1 , x 2 ) ∈ R 2 and all δ ≥ 0 , δ 1 ≥ 0 , the following two inequalities hold:

f ( x 1 , x 2 ) + f ( x 1 + δ , x 2 + δ + δ 1 ) ≥ f ( x 1 , x 2 + δ 1 ) + f ( x 1 + δ , x 2 + δ )

f ( x 1 , x 2 ) + f ( x 1 + δ + δ 1 , x 2 + δ ) ≥ f ( x 1 + δ 1 , x 2 ) + f ( x 1 + δ , x 2 + δ )

For a given base point x 1 = ( x 1 1 , x 2 1 ) ∈ R 2 define

( x → 1 * ( x 1 ) , x → 2 * ( x 1 ) ) = argmin ( x 1 , x 2 ) f ( x 1 , x 2 ) s .t . x 2 = ( x 1 − x 1 1 ) + x 2 1 .

That is, ( x → 1 * ( x 1 ) , x → 2 * ( x 1 ) ) is a minimizer of f on the 45° line passing through x 1 . If f is subconvex on R 2 , the minimizers on such 45° lines satisfy the following structural monotonicity.

(Monotone property of subconvexity) . Let f be subconvex on R 2 . For any ε > 0 and two points x 1 = ( x 1 1 , x 2 1 ) and x 2 = ( x 1 2 , x 2 2 ) satisfying x 1 2 − x 2 2 + ε = x 1 1 − x 2 1 , their corresponding minimizers on the 45° line through these points obey x → 1 * ( x 1 ) − ε ≤ x → 1 * ( x 2 ) ≤ x → 1 * ( x 1 ) and x → 2 * ( x 1 ) + ε ≥ x → 2 * ( x 2 ) ≥ x → 2 * ( x 1 ) .

Let the ( x l , t u , s , β t ) - policy be as follows:

( y l , t u , s , λ t ) = { ( x → l , t u , s * ( x l , t u , s , β t ) , β → t * ( x l , t u , s , β t ) ) i f x → l , t u , s * ( x l , t u , s , β t ) < x l , t u , s a n d β → t * ( x l , t u , s , β t ) < β t ( a r e a ( A ) ) ( x l , t u , s , β t ) o t h e r w i s e ( a r e a ( B ) ) ,

Definition

(Superconvexity/Subconcavity). A real function f ( x 1 , x 2 ) on R 2 is superconvex (subconcave) in x 1 if, for every ( x 1 , x 2 ) ∈ R 2 and all δ ≥ 0 , δ 1 ≥ 0 , the following inequality holds (Koole, 1998):

f ( x 1 , x 2 ) + f ( x 1 + δ + δ 1 , x 2 − δ ) ≥ ( ≤ ) f ( x 1 + δ , x 2 − δ ) + f ( x 1 + δ 1 , x 2 ) ,

Definition (Directional Property).

A real function f ( x 1 , x 2 ) on R 2 has the directional property, if f ( x 1 , x 2 ) is superconvex in x 1 and subconcave in x 2 (Kang et al., 2016).

Notation.

For a given base point x 1 = ( x 1 1 , x 2 1 ) ∈ R 2 define

( x ← 1 * ( x 1 ) , x ← 2 * ( x 1 ) ) = argmin ( x 1 , x 2 ) f ( x 1 , x 2 ) s .t . x 1 + x 2 = x 1 1 + x 2 1 .

That is, ( x ← 1 * ( x 1 ) , x ← 2 * ( x 1 ) ) is a minimizer of f on the −45° line passing through x 1 . If f has the directional property, the minimizers on such −45° lines satisfy the following structural monotonicity.

Lemma 4

(Monotone property of the directional property) . Let f ( x 1 , x 2 ) has the directional property in ( x 1 , x 2 ) . For any ε > 0 , and two points x 1 = ( x 1 1 , x 2 1 ) and x 2 = ( x 1 2 , x 2 2 ) satisfying when x 1 2 + x 2 2 = x 1 1 + x 2 1 + ε , x ← 1 * ( x 1 ) − x ← 1 * ( x 2 ) ≤ ε and x ← 2 * ( x 2 ) − x ← 2 * ( x 1 ) ≥ ε .

Lemma 4 establishes monotonicity of ( x ← 1 * ( x 1 , x 2 ) , x ← 2 * ( x 1 , x 2 ) ) with respect to the intercept x 2 + x 1 . Concretely, when x 2 + x 1 increases by one unit, x ← 1 * ( x 1 , x 2 ) decreases by at most one unit and x ← 2 * ( x 1 , x 2 ) increases by at least one unit. Hence, the locus of points ( x ← 1 * ( x 1 , x 2 ) , x ← 2 * ( x 1 , x 2 ) ) traces a downward-sloping continuous curve in the ( x 1 , x 2 ) plane whose slope lies between −45° and −90° with respect to the positive x 1 axis (see the dashed line in Figure 2(b) and Figure OA6(c) in the e-companion). This structural result describes optimal decisions that correspond to diagonal moves at −45° in the two-dimensional state space: reducing one unit of one resource and reallocating that unit to the other resource. Such reallocation decisions commonly arise in resource-assignment problems (Kang et al., 2016), and Lemma 4 quantifies how the optimal reallocations shift as the line intercept x 2 + x 1 changes.

In our problem, when two invoices x l , t u , s and x l ^ , t u ^ , s ^ must be prioritized, the payment priority, (i.e., which invoice to pay first) can be studied by examining this −45° diagonal movement in the ( y l , t u , s , y l ^ , t u ^ , s ^ ) plane. Consider the −45° line through the current state ( x l , t u , s , x l ^ , t u ^ , s ^ ) . If the optimal reallocation point on that line lies to the left-up from the current state, then reallocating the invoice amount from x l , t u , s to x l ^ , t u ^ , s ^ , if it is allowed, reduces the total cost, so x l , t u , s has higher priority. If the optimal reallocation point on that line lies to the right-down from the state, then x l ^ , t u ^ , s ^ has higher priority. Although such reallocation, a movement along a −45° line, is not feasible in our problem, this information reveals the relative marginal value of one dollar devoted to each invoice. Thus the −45° line analysis provides a ranking (priority) between the two invoices even when a literal diagonal reallocation cannot be executed.

To analyze payment priority between invoices x l , t u , s and x l ^ , t u ^ , s ^ , we define the two-dimensional function ϕ t ( y l , t u , s , y l ^ , t u ^ , s ^ ) by projecting the original function to the ( y l , t u , s , y l ^ , t u ^ , s ^ ) plane. We next examine the directional property of ϕ t ( y l , t u , s , y l ^ , t u ^ , s ^ ) . Theorem 5 establishes by induction that ϕ t ( y l , t u , s , y l ^ , t u ^ , s ^ ) has the directional property for every t. Building on this result, Theorem 5 characterizes the payment priority between two invoices x l , t u , s and x l ^ , t u ^ , s ^ when a fixed amount δ (>0) of cash must be allocated.

Let ( x ← l , t u , s * ( x l , t u , s , x l ^ , t u ^ , s ^ ) , x ← l ^ , t u ^ , s ^ * ( x l , t u , s , x l ^ , t u ^ , s ^ ) ) = argmin ( y l , t u , s , y l ^ , t u ^ , s ^ ) ϕ t ( y l , t u , s , y l ^ , t u ^ , s ^ ) s.t. y l , t u , s + y l ^ , t u ^ , s ^ = x l , t u , s + x l ^ , t u ^ , s ^ . Then ( x ← l , t u , s * ( x l , t u , s , x l ^ , t u ^ , s ^ ) , x ← l ^ , t u ^ , s ^ * ( x l , t u , s , x l ^ , t u ^ , s ^ ) ) is the minimizer of ϕ t ( y l , t u , s , y l ^ , t u ^ , s ^ ) along the −45° line passing through the state ( x l , t u , s , x l ^ , t u ^ , s ^ ) . Lemma 4 yields the ( x l , t u , s , x l ^ , t u ^ , s ^ ; δ ) - policy specified in Theorem 5, which prescribes the optimal priority rule for this allocation problem.

Theorem 5.

Let the ( x l , t u , s , x l ^ , t u ^ , s ^ ; δ ) - policy be as follows:

( y l , t u , s , y l ^ , t u ^ , s ^ ) = { ( x l , t u , s − δ , x l ^ , t u ^ , s ^ ) i f x l , t u , s ≥ δ ( a r e a ( A ) ) ( 0 , x l ^ , t u ^ , s ^ − ( δ − x l , t u , s ) ) i f x l , t u , s < δ a n d x l , t u , s + x l ^ , t u ^ , s ^ ≥ δ ( a r e a ( B ) ) ( 0 , 0 ) i f x l , t u , s < δ a n d x l , t u , s + x l ^ , t u ^ , s ^ < δ ( a r e a ( C ) )

, where x l , t u , s , x l ^ , t u ^ , s ^ , δ ≥ 0 . Then ϕ t ( y l , t u , s , y l ^ , t u ^ , s ^ ) has the directional property in ( y l , t u , s , y l ^ , t u ^ , s ^ ), and the ( x l , t u , s , x l ^ , t u ^ , s ^ ; δ ) − p o l i c y is optimal.

The structure of the ( x l , t u , s , x l ^ , t u ^ , s ^ ; δ ) - policy is shown in Figure 2(b). The black dot marks the unconstrained minimizer of ϕ t ( y l , t u , s , y l ^ , t u ^ , s ^ ) subject to y l , t u , s ≥ 0 and y l ^ , t u ^ , s ^ ≥ 0 ; this point is (0, 0), which is intuitive since the reduced P1 is a minimization. The dashed curve in Figure 2(b) represents the locus of minimizers ( x ← l , t u , s * ( x l , t u , s , x l ^ , t u ^ , s ^ ) , x ← l ^ , t u ^ , s ^ * ( x l , t u , s , x l ^ , t u ^ , s ^ ) ) when x l , t u , s + x l ^ , t u ^ , s ^ = 0 . By Lemma 4, this locus passes through (0, 0) and is downward-sloping with angle between −45° and −90° relative to the positive x l , t u , s axis.

The ( x l , t u , s , x l ^ , t u ^ , s ^ ; δ ) - policy governs payment choices when an amount δ of cash must be paid. Feasible post-payment states lie on the line y l , t u , s + y l ^ , t u ^ , s ^ = x l , t u , s + x l ^ , t u ^ , s ^ − δ , subject to 0 ≤ y l , t u , s ≤ x l , t u , s and 0 ≤ y l ^ , t u ^ , s ^ ≤ x l ^ , t u ^ , s ^ ((C1)–(C3)). Geometrically this is the closed −45∘ segment connecting the left-up endpoint ( max ( 0 , x l , t u , s − δ ) , x l ^ , t u ^ , s ^ ) and right-down endpoint ( x l , t u , s , max ( 0 , x l ^ , t u ^ , s ^ − δ ) ) . Because Lemma 4 characterizes the dashed locus of unconstrained minimizers and this locus passes through (0,0), the optimal feasible payment is the left–upmost point on the feasible −45° segment that is closest to the dashed locus. In other words, the best payment is the most left-up feasible point along that −45° closed line.

The optimal policy forms a switching-curve policy. The policy divides the state space into three adjacent areas: areas (A), (B), and (C). The solid arrows represent the optimal decision in each area. According to the policy, in area (A), all δ cash is paid for x l , t u , s , but x l , t u , s is not completely paid off (arrow a1). In areas (B) and (C), x l , t u , s is paid off first, and the remaining cash, δ − x l , t u , s , is used to pay x l ^ , t u ^ , s ^ . x l ^ , t u ^ , s ^ is fully paid off in area (C) (arrow c1), while it is not in area (B) (arrow b1).

The ( x l , t u , s , x l ^ , t u ^ , s ^ ; δ ) - policy holds for any given δ and state values. Consequently, in an optimal policy, when allocating cash between two invoices, one invoice must be fully paid before the other receives any payment; cash should not be split across invoices without fully settling one. Theorem 5 implies that, for any pair of invoices, one invoice has priority over the other, defining the payment priority between them. Note that in Theorem 5, ϕ t ( y l , t u , s , y l ^ , t u ^ , s ^ ) has the directional property in ( y l , t u , s , y l ^ , t u ^ , s ^ ) , assuming that the invoice with payment priority is y l , t u , s . The determination of priority for any invoice pair is characterized in Section 4.4.

Theorems 3 and 5 provide structural properties based on the location of an optimal solution, which is typically unknown due to the computational challenges posed by the curse of dimensionality. However, such properties often yield valuable insights even in the absence of direct knowledge of the actual optimal solution (Kang et al., 2016; Smith and McCardle, 2002). Theorem 3 offers a key managerial insight: managers should avoid increasing cash on hand as invoice amounts rise, regardless of the optimal solution's location. Similarly, Theorem 5 suggests that one invoice should be fully paid before initiating payment on another.

4.3. Structure of an overall optimal policy in ( x t , β t )

Theorems 3 and 5 provide key structural insight into an overall optimal policy for SDFSCP within the original multidimensional state space ( x t , β t ) .

Corollary 6.

Under an optimal policy ( x t , β t ) , outstanding invoices at time t can be ordered by payment priority. This priority ordering is state-dependent and is induced by the optimal value function; when all pairwise priorities are deterministic (Propositions 8–11), the ordering is additionally state-independent, corresponding to a total order based on the monotone dominance of J-functions. Let i t ( n ) denote the invoice with the n-th highest priority. Payments follow this order strictly—invoice i t ( n + 1 ) is considered only after i t ( n ) is fully settled. There exists a threshold β t * ( x t ) ≤ β t , reserved for future use; payments continue in priority order until cash reaches β t * ( x t ) , after which no further payments are made at time t.

4.4. Structure of optimal payment priorities

Corollary 6 and Theorem 5 show that the optimal policy for SDFSCP is governed by payment priorities among invoices. We therefore characterize the structure of these priorities, distinguishing between deterministic priority, which is state-independent, and stochastic priority, which depends on the state and random parameters. We first identify conditions for deterministic priority and when it becomes stochastic. The analysis proceeds from a single invoice (Section 4.4.1), to two invoices (Section 4.4.2), and then to the multi-invoice case, where we revisit the overall policy structure (Section 4.4.3).

4.4.1. Single invoice

For a single invoice, we characterize the optimal payment timing independent of other invoices. By converting all payment amounts into end-of-horizon values, we define a projected cost function that allows time-invariant comparison across payment dates. We show that each invoice has a well-defined optimal payment time determined by the trade-off between discount, interest, and penalty effects, and Lemma 7 establishes that in an optimal policy an invoice is paid only at (or after) its optimal timing threshold, never before.

Consider invoice k. Let L k ( t ) represent the invoice amount to be paid if invoice k is paid off at time t.

L k ( t ) = { L k if t ≤ b k L k α k if b k < t ≤ d k L k α k γ k t − d k if d k < t ≤ T

We assume the value of money changes over time according to the interest rate r. Hence, we define A k ( t ) which converts the value of invoice at time t to its equivalent value at time T. Let A k ( t ) represent the projected value of L k ( t ) at time T: A k ( t ) = L k ( t ) r T − t A k ( t ) . provides a convenient way to compare invoice values irrespective of time. Throughout this paper, we use the term “projected” to refer to this time-invariant monetary value evaluated at time T. Additionally, let I k ( t ) = A k ( t ) / L k . I k ( t ) represents the relative projected amount of payment required at time t to pay off one dollar of invoice k. Let t ^ k = argmin t I k ( t ) . Then t ^ k is defined as follows.

t ^ k = { b k if α k > r d k − b k d k if α k ≤ r d k − b k

t ^ k represents the optimal payment time to minimize the projected payment amount of invoice, assuming there is sufficient cash to pay the invoice regardless of other outstanding invoices. This result implies that, to minimize the projected payment amount of invoice k, it must be paid at the end of discount period ( b k ) if the discount amount ( α k − 1 ) exceeds the interest gain ( r d k − b k − 1 ) accumulated during the normal payment period ( d k − b k ) . Otherwise, it must be paid at the end of normal payment period ( d k ) .

Let I _ k ( t ) = min t ≤ t ′ ≤ T I k ( t ′ ) . I _ k ( t ) represents the minimum projected payment—evaluated at the end of the decision horizon—that the company can make at or after time t to pay off one dollar of invoice k's original discounted amount. Since I _ k ( t ) is non-decreasing in time t, this implies that the optimal projected amount that the company needs to pay on an invoice does not decrease as time progresses.

I _ k ( t ) provides valuable managerial insights into the optimal timing for paying an invoice. To minimize the payment amount, the company has to wait until just before I _ k ( t ) increases (e.g., at time b k or d k ). However, if for any reason (e.g., insufficient cash) the invoice is not paid at that moment and I _ k ( t ) has already increased, the company has to wait again and pay invoice k just before the next increase in I _ k ( t ) (e.g., at or after time d k ). Thus, past values of I _ k ( t ) and their changes over time do not influence the optimal payment decision at given time t. Instead, only the current and future values of I _ k ( t ) do.

Let t ^ k ( t ) = max ( argmin t ≤ t ′ < T I _ k ( t ′ ) ) . Then t ^ k ( t ) represents the optimal time at or after t when the payment for invoice k should be made. Suppose the current time is t. If t ^ k ( t ) = t , the payment should be made immediately. Otherwise, the payment should be deferred until t ~ = t ^ k ( t ) > t . If the payment is not made at t ~ , the new optimal payment time, t ^ k ( t ~ ) , is recalculated.

Lemma 7.

In an optimal policy, if invoice k is paid at time t , then t = t ^ k ( t ) and t ≥ t ^ k .

Lemma 7 provides meaningful managerial insights into the timing of invoice payments. If there is sufficient cash on hand to pay all existing invoices and no new invoices arrive, then every invoice can be paid at its optimal payment time t ^ k , and this becomes an optimal policy. However, when cash is insufficient, some invoices may not be paid at their respective t ^ k . Also, even if there is enough cash, certain invoices may not be paid at their t ^ k because some cash may need to be reserved for expected future invoices. Lemma 7 implies that in even such cases, no invoice should be paid before its t ^ k ; instead, it must be paid only after t ^ k , specifically when t = t ^ k ( t ) . Furthermore, Lemma 7 and t ^ k ( t ) suggest that in an optimal policy, invoice k should be paid at time b k , d k , or after d k if α k > r d k − b k . Otherwise, it should be paid only at or after d k . Beyond its managerial implications, Lemma 7 also reduces the computational complexity of finding an optimal policy.

4.2.2. Two invoices: pairwise comparisons

We next characterize payment priorities between two invoices. Using the relative projected cost ratio J k ( t ; t 0 ) , Propositions 8–11 identify conditions under which one invoice has a deterministic priority over another, independent of the state and random parameters. When these monotone dominance conditions fail and the projected cost paths intersect (Proposition 12), the priority becomes stochastic, depending on the system state and uncertainty.

Let J k ( t ; t 0 ) = I _ k ( t ) / I _ k ( t 0 ) , where t ≥ t 0 . J k ( t ; t 0 ) . J k ( t ; t 0 ) is non-decreasing in t and plays an important role in determining the optimal payment priorities between two different invoices. Suppose there are two different invoices, i and j. Then the payment priority—which invoice should be paid off first—between these two invoices is determined by Propositions 8–12.

Proposition 8.

If t ^ i ( t 0 ) = t ^ j ( t 0 ) = t 0 and J i ( t ; t 0 ) < J j ( t ; t 0 ) for any t ≥ t 0 , then paying invoice j has priority over paying invoice i at time t 0 .

Under the conditions of Proposition 8, it is preferrable for the company to pay immediately both invoices i and j because t ^ i ( t 0 ) = t ^ j ( t 0 ) = t 0 . However, if the company does not have sufficient cash on hand, it must decide which invoice to pay first. Proposition 8 establishes a condition under which invoice j takes precedence, implying that invoice i should be paid only after invoice j is fully settled—if it must be paid at time t 0 , as dictated by Theorem 5.

Proposition 9.

If t ^ j ( t 0 ) = t 0 , t ^ i ( t 0 ) > t 0 , and J i ( t ; t 0 ) < J j ( t ; t 0 ) for any t > t 0 , then at time t 0 , paying invoice j has priority over reserving cash to pay invoice i at t ^ i ( t 0 ) > t 0 .

While the company should pay invoice j at time t 0 because t ^ j ( t 0 ) = t 0 , it may choose to postpone the payment of invoice i at time t 0 until t ^ i ( t 0 ) because t ^ i ( t 0 ) > t 0 . However, it is not always optimal for the company to pay invoice j immediately and defer invoice i. The company may instead choose to reserve funds for invoice i, rather than using the same funds to pay invoice j right away. Nevertheless, if J i ( t ; t 0 ) < J j ( t ; t 0 ) for any t > t 0 , paying invoice j takes priority, which implies that the company should only begin reserving cash for invoice i at t ^ i ( t 0 ) , but only after fully paying off invoice j.

Proposition 10.

If t ^ j ( t 0 ) > t 0 , t ^ i ( t 0 ) > t 0 , and J i ( t ; t 0 ) < J j ( t ; t 0 ) for any t > t 0 , then at time t 0 , reserving cash to pay invoice j at t ^ j ( t 0 ) has priority over reserving cash to pay invoice i at t ^ i ( t 0 ) ≤ t ^ j ( t 0 ) .

Because t ^ j ( t 0 ) > t 0 and t ^ i ( t 0 ) > t 0 , the company may choose to delay payment for both invoices i and j at time t 0 . However, it may still need to determine the payment priority between the two invoices to allocate funds accordingly for future payments. Under the condition of Proposition 10, at time t 0 , reserving cash for invoice j has priority. This means that the company should only begin setting aside funds for invoice i only after it secures enough cash to fully pay off invoice j. Additionally, Proposition 10 suggests that invoice j should be paid before invoice i, implying that t ^ i ( t 0 ) ≥ t ^ j ( t 0 ) .

Propositions 8 and 9 define time-dependent payment priority rules that hold at a specific point in time but may not necessarily remain valid beyond that moment. For instance, if invoices i and j satisfy the conditions of Proposition 8 or 9 and invoice j is not fully paid at time t 0 , then the priority of paying invoice j over invoice i may no longer hold for time t > t 0 . In contrast, Proposition 10 establishes a priority rule that applies over a certain period. The priority rule specified in Proposition 10 holds from time t 0 until t ^ j ( t 0 ) . However, this rule may not necessarily remain valid after t ^ j ( t 0 ) . Next, we introduce a stronger payment priority rule, which ensures that one invoice consistently takes priority over another from a certain point in time until the end of the decision horizon. Let Δ I _ k ( t ) = ( I _ k ( t + 1 ) − I _ k ( t ) ) / I _ k ( t ) . Δ I _ k ( t ) represents the marginal rate of change in I _ k ( t ) .

Proposition 11.

If Δ I _ i ( t ) ≤ Δ I _ j ( t ) for any t ≥ t 0 , paying invoice j has priority over paying invoice i at any t ≥ t 0 .

Propositions 8, 9, 10, and 11 provide valuable insights into the payment priority rules between two invoices. These rules hold consistently regardless of other random parameters, such as the arrival rates of new invoices and cash inflows. For this reason, we refer to these payment priorities as deterministic optimal payment priorities. However, not all invoice pairs can be prioritized using these propositions. In some cases, the payment priority between two existing invoices depends on the problem state and random parameters. We refer to this as the stochastic optimal payment priority. Proposition 12 illustrates an example of this stochastic optimal payment priority.

Proposition 12.

If there exist t 1 and t 2 , where t 0 < t 1 < T , t 0 < t 2 < T , and t 1 ≠ t 2 , which satisfy ( J i ( t 1 ; t 0 ) − J j ( t 1 ; t 0 ) ) × ( J i ( t 2 ; t 0 ) − J j ( t 2 ; t 0 ) ) < 0 , then at time t 0 , invoices i and j do not have a deterministic optimal payment priority but a stochastic optimal payment priority.

Figure 3 shows an example when the optimal payment priority between invoices i and j becomes stochastic. The figure shows t ^ i ( a ) = t ^ j ( a ) = a , meaning that both invoices need to be paid at time a if there is sufficient cash on hand. However, suppose only one invoice can be paid due to limited cash availability. In this case, none of Propositions 8, 9, 10, or 11 applies. As shown in Figure 3, J i ( t ; a ) and J j ( t ; a ) intersect at t = d j . If invoice j is paid at time a and invoice i is paid later at t > a , the total relative payment amount is 1 + J i ( t ; a ) . Conversely, if invoice i is paid at time a and invoice j is paid later at t > a , the total payment is 1 + J j ( t ; a ) . Notably, J i ( t ; a ) < J j ( t ; a ) when t < d j but J i ( t ; a ) > J j ( t ; a ) when t > d j . This means if the invoice deferred at time a is likely to be paid before time d j , then paying invoice j at time a minimizes the expected total payment for both invoices. Conversely, if the deferred invoice is more likely to be paid after d j , then paying invoice i at time a is the better choice. In this scenario, the optimal payment priority cannot be determined using deterministic rules alone. Instead, it requires computing an exact optimal policy based on the problem's random parameters—such as the arrival rates of invoices and cash inflows—as well as the current state of the system.

OPEN IN VIEWER

Figure 3.

An example of the stochastic optimal payment priority.

4.2.3. Multiple invoices

We now understand both the overall structure of an optimal policy and the payment priorities between two invoices. Building on these insights, we extend the structure to characterize an optimal policy for multiple invoices.

Let s l * be s that satisfies α l = r s . We refer to s l * as the type-l invoice's discount-interest gain threshold. This threshold represents the number of time intervals required for the cumulative interest gain to equal the discount benefit obtained by paying the invoice before the end of the discount period. Similarly, let u l * be u that satisfies ( γ l / r ) u = α l . We refer to u l * as the discount-penalty threshold. This threshold represents the number of time intervals required for the cumulative penalty to equal the discount benefit. Given α l , γ l , and r, both s l * and u l * can be easily computed.

We consider two different types of invoices: l and l ′ . Let s * (or u * ) and s ′ * (or u ′ * ) be the discount-interest gain threshold (or discount-penalty threshold) for l and l ′ , respectively. Let s and s ′ be the remaining discount payment periods, let u and u ′ be the remaining normal payment period, let γ l and γ l ′ be the penalty rate, and let α l and α l ′ be the discount rate for l and l ′ , respectively. Using the payment priorities introduced in Section 4.4.2, we further develop the conditions when l ′ has a deterministic payment priority over l at time t 0 .

Rule (1) represents the case when both invoices l ′ and l should be paid at the end of the normal payment period—regardless of whether the discount period remains—because the discount gain is not greater than the interest gain ( s > s * and s ′ > s ′ * ), or when the normal payment period has already ended ( s = 0 and s ′ = 0 ). If the penalty rate of invoice l ′ is greater than that of invoice l ( γ l ′ > γ l ) , one might assume that invoice l ′ always takes payment priority over invoice l to mitigate the risk of incurring more penalties—an approach aligned with the basic principle of the avalanche method. However, this is only true if the total remaining time of invoice l ′ until the end of its normal payment is shorter than that of invoice l ( u ′ + s ′ ≤ u + s ) , which satisfies the condition of Proposition 10. In such cases, invoice l ′ has a deterministic payment priority over invoice l (see rule (1.1)), and this priority remains valid at any time until T due to Proposition 11. Otherwise, they have a stochastic payment priority according to Proposition 12. Similarly, when invoice l has a higher penalty rate ( γ l ′ ≤ γ l ) , one might assume that invoice l should be prioritized over invoice l ′ to reduce the risk of higher penalties. However, if the relative, minimal, projected amount of invoice l ′ at time T is greater than that of invoice l ( J l ′ ( T ; t 0 ) ≥ J l ( T ; t 0 ) ) , then invoice l ′ has a deterministic payment priority over invoice l (see rule (1.2)). This situation can occur when the normal payment deadline for invoice l is significantly later than that of invoice l ′ ( u ′ + s ′ ≤ u + s ) . However, even in this case, at the due for the normal payment of invoice l ′ ( u ′ = 1 , u > 1 ) , if the projected amount of invoice l ′ is lower than that of invoice l at the end of the decision horizon ( J l ′ ( T ; t 0 ) < J l ( T ; t 0 ) ) , paying invoice l ′ immediately may not be the optimal decision. In this case, the priority is no longer deterministic due to Proposition 12, and reserving funds for a future payment toward invoice l may be preferable, depending on the problem state and stochastic parameters.

Rule (2) addresses the cases when both invoices benefit from being paid at discounted prices rather than normal prices, as the discount gains exceed the interest gains that can be earned during the normal periods ( 0 < s ≤ s * and 0 < s ′ ≤ s ′ * ). The priority rules in this scenario become more complex, as the conditions under which invoice l ′ has a deterministic payment priority over invoice l depend on the relative values of their penalty and discount rates. When both the penalty and discount rates of invoice l ′ are higher than those of invoice l, one might assume that invoice l ′ always takes payment priority over invoice l as it appears to offer higher discount savings or reduce penalties more effectively. However, this is true only if the remaining time of the discount period of invoice l ′ is shorter than that of invoice l ( u ′ ≤ u ) and the relative minimal projected amount of invoice l ′ is greater than that of invoice l at the end of the normal period of invoice l ′ ( J l ′ ( t 0 + u ′ + s ′ ; t 0 ) ≥ J l ( t 0 + u ′ + s ′ ; t 0 ) ) . If these conditions are not met, it has a stochastic priority (see rule (2.1)). On the contrary, when both the penalty and discount rates of invoice l ′ are smaller than those of invoice l, invoice l ′ still can take payment priority over invoice l if the relative minimal projected amount of invoice l ′ is greater than that of invoice l at both the end of the discount period of invoice l ( t = u ) and the end of the decision horizon ( t = T ) (see rule (2.4)). This can happen when invoice l has a sufficiently long remaining discount period, even longer than the total of remaining discount and normal periods of invoice l ′ ( u ′ + s ′ ≤ u ) . When invoice l ′ has an advantage of payment priority over invoice l in penalty ( γ l ′ > γ l ) but not in discount ( α l ′ ≤ α l ) , only under the condition when the relative minimal projected amount of invoice l ′ is greater than that of invoice l at the end of the discount period of invoice l, invoice l ′ has a deterministic payment priority over invoice l (see rule (2.2)). On the other hand, when invoice l ′ has an advantage in discount ( α l ′ > α l ) but not in penalty ( γ l ′ ≤ γ l ) , the necessary condition becomes u ′ ≤ u and J l ′ ( T ; t 0 ) ≥ J l ( T ; t 0 ) (see rule (2.3)).

Rule (3) addresses the cases when invoice l ′ is more advantageous to be paid at the normal price, while invoice l benefits more from being paid at the discounted price. When the penalty rate of invoice l ′ is greater than invoice l ( γ l ′ > γ l ) , invoice l ′ has a payment priority only if the relative minimal projected amount of invoice l ′ is greater than that of invoice l at the end of the discount period of invoice l ( t = u ) (see rule (3.1)). In fact, this condition ensures that invoice l ′ always maintains payment priority over invoice l after t = u , as established by Proposition 11. However, when the penalty rate of invoice l ′ is smaller than that of invoice l ( γ l ′ ≤ γ l ) , an additional condition, J l ′ ( T ; t 0 ) ≥ J l ( T ; t 0 ) , must also be satisfied to have a deterministic payment priority over invoice l (see rule (3.2)).

Rule (4) represents the opposite cases of rule (3), showing situations where invoice l ′ is more advantageous to be paid at the discounted price, while invoice l benefits more from being paid at the normal price. For invoice l ′ to take priority over invoice l, first, its remaining time until the end of the discount period ( u ′ ) must be shorter than the total remaining time of invoice l until the end of its normal period ( u + s ) . Additionally, an extra condition must be met. When the penalty rate of invoice l ′ is greater than that of invoice l ( γ l ′ > γ l ) , the relative minimal projected amount of invoice l ′ must be greater than that of invoice l at the end of the normal period of invoice l ′ (see rule (4.1)). When γ l ′ ≤ γ l , the relative minimal projected amount of invoice l ′ must be greater than that of invoice l at t = T (see rule (4.2)).

Theorem 13.

l ′ has a deterministic optimal payment priority over l under the following conditions:

(1) For s > s * (or s = 0 ) and s ′ > s ′ * (or s ′ = 0 )

(1.1) When γ l ′ > γ l , u ′ + s ′ ≤ u + s .

(1.2) When γ l ′ ≤ γ l , J l ′ ( T ; t 0 ) ≥ J l ( T ; t 0 ) .

(2) For 0 < s ≤ s * and 0 < s ′ ≤ s ′ * ,

(2.1) When γ l ′ > γ l and α l ′ > α l , u ′ ≤ u and J l ′ ( t 0 + u ′ + s ′ ; t 0 ) ≥ J l ( t 0 + u ′ + s ′ ; t 0 ) .

(2.2) When γ l ′ > γ l and α l ′ ≤ α l , J l ′ ( t 0 + u ; t 0 ) ≥ J l ( t 0 + u ; t 0 ) .

(2.3) When γ l ′ ≤ γ l and α l ′ > α l , u ′ ≤ u and J l ′ ( T ; t 0 ) ≥ J l ( T ; t 0 ) .

(2.4) When γ l ′ ≤ γ l and α l ′ ≤ α l , J l ′ ( t 0 + u ; t 0 ) ≥ J l ( t 0 + u ; t 0 ) and J l ′ ( T ; t 0 ) ≥ J l ( T ; t 0 ) .

(3) For 0 < s ≤ s * and s ′ > s ′ * (or s ′ = 0 )

(3.1) When γ l ′ > γ l , J l ′ ( t 0 + u ; t 0 ) ≥ J l ( t 0 + u ; t 0 ) .

(3.2) When γ l ′ ≤ γ l , J l ′ ( t 0 + u ; t 0 ) ≥ J l ( t 0 + u ; t 0 ) and J l ′ ( T ; t 0 ) ≥ J l ( T ; t 0 ) .

(4) For s > s * (or s = 0 ) and 0 < s ′ ≤ s ′ * ,

(4.1) When γ l ′ > γ l , u ′ ≤ u + s a n d J l ′ ( t 0 + u ′ + s ′ ; t 0 ) ≥ J l ( t 0 + u ′ + s ′ ; t 0 ) .

(4.2) When γ l ′ ≤ γ l , u ′ ≤ u + s and J l ′ ( T ; t 0 ) ≥ J l ( T ; t 0 ) .

Theorem 13 characterizes when one invoice has a deterministic optimal payment priority over another. When these conditions hold, the ordering is stable and state-independent; otherwise, priority becomes state-dependent and stochastic. For invoices of the same type, additional refinements are provided in the e-companion. Together, Propositions 8–12 and Theorem 13 clarify the structure of the optimal policy. Once parameters are specified, deterministic priorities can be identified at each state, substantially reducing computational complexity in both exact optimization and heuristic design.

Importantly, Theorem 13 shows that common managerial instincts can be misleading. Consider rule (4.1), where invoice l ′ is at the last discount opportunity ( u ′ = 1 ) while invoice l still has time remaining ( u > 1 , s > 1 ) . A natural reaction is to pay l ′ immediately—both to capture the expiring discount and to avoid its higher penalty rate ( γ l ′ > γ l ) . However, the optimal decision depends not only on rates and deadlines but also on the projected future burden. If the projected minimal amount of l ′ at the end of its normal period is not greater than that of l, then paying l ′ immediately is not necessarily optimal. In some states, preserving cash and allocating it later to invoice l leads to a lower expected total cost. Thus, simple rules such as “use the last discount” or “pay the highest penalty first” are not universally optimal.

Although the complete payment rules are intricate and state-dependent, the structural results yield several high-level managerial takeaways:

Pay only when the invoice reaches its economic trigger ( Lemma 7 ). An invoice should be paid when further delay increases its projected payment burden—typically at the end of the discount period if the discount benefit exceeds the interest gain, or at/after the normal due date otherwise. Paying earlier than this trigger destroys liquidity without reducing long-run cost.

When discount benefits are weak relative to interest, prioritize urgency and downside risk ( Propositions 10 and 11 ; Theorem 13 ). If holding cash is more valuable than capturing the discount, priority should be driven by remaining time to due date and the potential growth of penalties. Invoices closer to penalty escalation or with faster projected cost growth should be paid first.

When discounts are economically meaningful, compare overall projected burden—not just rates ( Proposition 12 ; Theorem 13 ). An invoice with a higher discount or penalty rate does not automatically deserve priority. When projected cost paths intersect, priority becomes state-dependent. In such cases, invoices with the larger cumulative projected burden over the remaining horizon should be prioritized.

These principles guide the design of our heuristic, which operationalizes the structural ordering and pay-or-reserve rules in a computationally tractable manner.

5. Heuristic development and computational experiments

To demonstrate the effectiveness of the structural properties of the optimal policy, we develop a simple heuristic that leverages these properties. We then compare its performance to existing heuristics, specifically the snowball and the avalanche methods. These two methods are most widely used in practice and implemented in many commercial software for SDFSCP (Merritt, 2024; Rios-Solis et al., 2017).

5.1. A new heuristic: optimal-policies-based heuristic

Our proposed heuristic, the optimal-policies-based heuristic (OPBH), operationalizes the structural priority rules (Propositions 8–12) and the pay-or-reserve logic of Lemma 7. It first applies all deterministic priority conditions. When priority is stochastic (Proposition 12), invoices are ranked using a second-order dominance approximation: we compare the areas under their projected cost ratio curves, ∑ t ′ = t T J k ( t ′ ; t ) , over the remaining planning horizon, and prioritize the invoice with the larger aggregate projected burden (see Section 4.4.2 for details on J k ( t ′ ; t ) ). The simplified logic of OPBH is as follows (see the e-companion for full details). For each period t = 0 , 1 , … , T − 1 , update cash and invoice balances as in Figure 1, then:

Step 1 (Ordering). Apply deterministic priority rules when their conditions hold (Propositions 8–11); otherwise rank invoices using the area-based rule (Proposition 12).

Step 2 (Pay-or-reserve). Process invoices sequentially. If invoice k satisfies the pay-now condition (Lemma 7), allocate cash fully if sufficient, otherwise partially if preemptive payment is allowed (or stop if non-preemptive). If not, reserve the cash. Proceed to t + 1 once cash is exhausted or all invoices are processed.

5.2. Experimental design

We consider a company that receives invoices from upstream partners and cash inflows from downstream partners on a weekly basis, making payment decisions over a 1-year horizon (52 weeks). Parameter values reflect common business practices: early payment discounts of 1% to 5% (PYMNTS, 2022), late penalties of 0.3% to 1.3% per week (Resolve, 2025b), and a 0.1% weekly interest rate (about 5.3% annually). These settings align with prior literature (Gupta and Dutta, 2011; Gupta et al., 1987; Rios-Solis et al., 2017) and provide a realistic basis to evaluate the effectiveness of our heuristic. The parameters used in this experiment are detailed in Table 3. For each combination of parameters, we randomly generate 30 problem instances.

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Table 3.

Parameters used in the experiments.

Parameter	Values	Parameter	Values	Parameter	Values
n	100 (base*) to100,000	α k	U ( 100 / 99 , 100 / 95 )	L k	U ( 1000 , 5000 )
T	52 weeks	γ k	U ( 1.003 , 1.013 )	r	1.001
a k	U ( − T / 4 , T / 4 )	b k	a k + U ( T / 6 , T / 3 )	d k	b k + U ( T / 6 , T / 3 )
Q t	Q t = U ( 0.9 , 1.1 ) * ∑ L k α k * t i g h t n e s s / T , \; where\; t i g h t n e s s = { 0.4 , 0.7 , 1 ( base ) , 1.3 , 1.6 }

Note. *For the main analysis, we use the base values for n (number of invoices) and the tightness to control cash inflow ( Q t ) .

To demonstrate the effectiveness of our proposed heuristic, we compare it against two established approaches: the snowball method and the avalanche method (Merritt, 2024; Rios-Solis et al., 2017). Specifically, we implement four variations tailored to our problem structure: (a) snowball with the smallest face value (SF), (b) snowball with the smallest current debt amount (SC), (c) avalanche with the highest penalty rate (AP), and (d) avalanche with the highest discount rate (AD). Detailed algorithmic descriptions are provided in the e-companion. We use the same objective as the analytical model: total payment—the amount paid toward all invoices received during the horizon; any unpaid invoices at the end are valued at their outstanding balance and included. All heuristics and the problem generator are implemented in Python, and the experiments are conducted on a system with an Intel i9-10900KF processor and 64 GB of RAM.

5.3. An illustrative sample path: why avalanche and snowball fail

Before presenting aggregate results, we provide a small illustrative sample path to clarify why simple benchmark policies can perform poorly relative to our OPBH heuristic. Consider T = 52 weeks with weekly interest r = 1.001 . Two invoices arrive at t = 0 . Invoice A has L A = 3000 , α A = 100 / 97 , γ A = 1.012 , b A = 0 , and d A = 2 ; invoice B has L B = 3000 , α B = 100 / 95 , γ B = 1.013 , b B = 11 , and d B = 30 . Initial cash is β 0 = 3000 , with inflow Q 10 = 4000 ; partial payment is allowed. Benchmark heuristics (SF/SC/AD/AP) prioritize B because it has a smaller balance, a larger discount, and a higher penalty rate. A is then settled at t = 10 for 3402.47, yielding a total end-of-horizon payment of 6708.38. In contrast, OPBH applies the pay-now screening (Lemma 7) and the projected-cost comparison (Proposition 9), pays A at t = 0 , and defers B to t = 10 , reducing the total payment to 6288.66, which is approximately 6.7% lower. The economic logic is straightforward: Invoice A has a discount period ending at t = 0 and incurs a 1.2% weekly penalty thereafter, making immediate payment urgent. Invoice B, by contrast, retains its discounted price until t = 11 , so deferring it costs nothing for 11 weeks while the firm earns interest on the reserved cash. The benchmark heuristics miss this asymmetry because they rank invoices by surface attributes—balance size, discount rate, or penalty rate—without accounting for the time value of delay. OPBH, guided by Lemma 7's pay-now trigger and Proposition 9's projected-cost comparison, correctly identifies that the urgency differential between A and B justifies a different sequencing than any of the benchmarks would prescribe. We next evaluate whether these structural advantages persist systematically across the full experimental design.

5.4. Experimental results

Table 4 reports the main results. It illustrates the comparative effectiveness of our heuristic against the others, measured by gap performance, defined as (other heuristic's total payment – our heuristic's total payment)/our heuristic's total payment × 100. Each cell in the table provides descriptive statistics of gap performance in the format “average (standard deviation) [min – max].” OPBH consistently outperforms all alternatives: the average gap is 4.01%, statistically significant (t-test, p < 0.001). It consistently outperforms the other methods in every problem instance (i.e., minimum gap > 0). The source of this systematic advantage is structural: OPBH leverages two properties that benchmark heuristics ignore. First, the pay-now trigger (Lemma 7) prevents premature payments that sacrifice liquidity without reducing long-run cost—benchmark heuristics frequently pay invoices before their economic trigger, locking up cash that could earn interest or cover higher-urgency invoices arriving later. Second, the projected-cost priority (Propositions 8–11) sequences invoices by their time-adjusted burden rather than by static attributes such as balance size or penalty rate, capturing the compounding dynamics of discounts and penalties over the remaining horizon. These advantages are most pronounced when cash is tight (tightness = 0.4–0.7), where the pay-now discipline prevents costly early commitments, and when the horizon is long, where priority misordering by benchmarks compounds over many periods. When cash is abundant (tightness = 1.6), all invoices can be paid at their optimal times regardless of sequencing, so the structural advantage shrinks—though OPBH still dominates in every instance.

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Table 4.

Gap performance.

SF	SC	AP	AD	Overall
4.69 (0.72) [3.5–6.51]	5.46 (0.78) [4.04–7.33]	1.65 (0.27) [1.2–2.23]	4.24 (0.53) [3.36–5.52]	4.01 (1.56) [1.20–7.33]

Table 5 presents a sensitivity analysis on cash-on-hand tightness and the number of invoices. Cash tightness materially affects gap performance. The problem becomes close to trivial when cash is abundant: firms can pay all discounted invoices and avoid late penalties, so differences among heuristics narrow—yet our heuristic still outperforms others by better exploiting payment terms. When cash is scarce, the problem is harder because selecting which invoices to pay is critical; our heuristic is designed for this setting and continues to perform strongly. By contrast, gap performance is largely unaffected by the number of invoices, indicating that the heuristic performs consistently well across problem sizes. The heuristic runs in O ( n l o g n ) , where n is the number of invoices, making it suitable for large instances. In our experiments, 100,000-invoice cases (∼2000 new invoices per week) average 437 s of computation time.

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Table 5.

Sensitivity analysis on cash-on-hand tightness and the number of invoices.

Tightness	SF	SC	AP	AD	Overall
0.4	4.73 (0.59) [3.56–5.8]	5.06 (0.62) [3.84–6.31]	1.78 (0.43) [0.97–2.69]	4.52 (0.73) [3.21–6.04]	4.02 (1.44) [0.97–6.31]
0.7	5.72 (0.58) [4.53–7.06]	6.31 (0.6) [5.2–7.47]	1.93 (0.34) [1.42–2.91]	5.41 (0.61) [4.21–6.56]	4.84 (1.80) [1.42–7.47]
1 (base)	4.69 (0.72) [3.5–6.51]	5.46 (0.78) [4.04–7.33]	1.65 (0.27) [1.2–2.23]	4.24 (0.53) [3.36–5.52]	4.01 (1.56) [1.20–7.33]
1.3	2.78 (0.62) [1.77–4.55]	3.28 (0.7) [2.1–5.06]	1.43 (0.21) [1.03–2.02]	2.55 (0.43) [1.88–3.77]	2.51 (0.86) [1.03–5.06]
1.6	1.92 (0.31) [1.3–2.89]	2.17 (0.34) [1.38–3.05]	1.17 (0.17) [0.9–1.58]	1.72 (0.28) [1.19–2.56]	1.74 (0.46) [0.90–3.05]
n	SF	SC	AP	AD	Overall
100 (base)	4.69 (0.72) [3.5–6.51]	5.46 (0.78) [4.04–7.33]	1.65 (0.27) [1.2–2.23]	4.24 (0.53) [3.36–5.52]	4.01 (1.56) [1.20–7.33]
1000	4.63 (0.22) [4.25–5.0]	5.44 (0.25) [5.02–5.85]	1.72 (0.08) [1.53–1.87]	4.44 (0.19) [4.0–4.81]	4.06 (1.42) [1.53–5.85]
10,000	4.61 (0.1) [4.43–4.78]	5.42 (0.1) [5.26–5.57]	1.71 (0.03) [1.65–1.79]	4.41 (0.1) [4.26–4.57]	4.04 (1.40) [1.65–5.57]
100,000	4.63 (0.06) [4.51–4.76]	5.44 (0.06) [5.31–5.57]	1.7 (0.01) [1.68–1.73]	4.43 (0.06) [4.31–4.58]	4.05 (1.42) [1.68–5.57]

5.5. Worst case and additional analyses

The proposed heuristic, OPBH, utilizes both optimal deterministic and stochastic payment priorities identified in Section 4.4.2 and the insights from Theorems 3 and 5. While deterministic priorities ensure optimality, computing the optimal stochastic priority is computationally infeasible due to its complexity. Instead, OPBH employs a simple approximation by comparing the under-area of the J k ( t ′ ; t ) curve to prioritize invoices under stochastic priority conditions. J k ( t ′ ; t ) represents the increased cost of an invoice when its payment is deferred to a future time t ′ rather than being made immediately at t (see Section 4.4.2). The intuition behind comparing under-areas is based on the assumption that all possible future payments at every t ′ have an equal probability of occurring. However, this assumption may not hold in scenarios where cash availability is highly skewed, particularly when cash inflows are heavily left-skewed—that is, concentrated toward the end of the decision horizon.

To assess OPBH's performance under such conditions, we test it in scenarios with highly skewed cash inflows, as shown in Table 6. As expected, the effectiveness of OPBH decreases as the skewness of cash inflows increases. For instance, when 90% of cash inflows are concentrated in the last five periods (∼10% of the horizon), the average performance gap decreases by 2.35%.

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Table 6.

Sensitivity analysis on the skewedness of cash inflow.

Skewed	SF	SC	AP	AD	Overall
Even(base)	4.69 (0.72) [3.5–6.51]	5.46 (0.78) [4.04–7.33]	1.65 (0.27) [1.2–2.23]	4.24 (0.53) [3.36–5.52]	4.01 (1.56) [1.20–7.33]
70%	4.34 (0.43) [3.53–5.3]	4.64 (0.48) [3.76–5.65]	1.54 (0.3) [1.04–2.2]	4.17 (0.54) [2.77–5.2]	3.67 (1.32) [1.04–5.65]
90%	1.94 (0.21) [1.46–2.36]	2.1 (0.2) [1.6–2.43]	0.64 (0.19) [0.3–1.11]	1.95 (0.35) [1.31–2.54]	1.66 (0.64) [0.30–2.54]

Note. Seventy percent (70%) or ninety percent (90%) indicates that 70% or 90% of the total cash inflow is concentrated in the last five periods, which represent approximately 10% of the decision horizon.

According to the sensitivity analysis of problem parameters, OPBH's gap is minimized when cash is abundant (tightness = 1.6). Combined with this disadvantageous parameter setting, highly skewed cash inflows (90%) constitute a worst-case configuration in our experiments. In this worst case, the average gap is 2.30%, and the difference is statistically significant. OPBH still outperforms all benchmark heuristics in every instance (see Table 7).

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Table 7.

Worst case scenario analysis.

SF	SC	AP	AD	Overall
2.7 (0.28) [2.19–3.24]	2.91 (0.28) [2.38–3.56]	1.0 (0.28) [0.55–1.63]	2.59 (0.29) [2.15–3.29]	2.30 (0.81) [0.55–3.56]

We further evaluate OPBH across a range of settings. Although our main experiments use a weekly decision epoch and a one-year horizon, the heuristic applies equally to other intervals (including daily decisions). Additional experiments show that changing the decision epoch does not materially affect performance. The decision horizon can also be lengthened or shortened; the observed performance gap tends to widen as the horizon increases, suggesting that OPBH's advantages accumulate over time and yield greater payment efficiency. We also test the effect of invoice arrival concentration and find that concentration alone does not materially alter OPBH's performance. Finally, we conduct sensitivity analyses over discount, penalty, and interest rates; OPBH continues to perform robustly. Detailed results are reported in the e-companion.

In summary, across all problem instances, including those under worst-case or extreme scenarios, our heuristic consistently outperforms all other heuristics. Given the relative size of AP compared to retained earnings or total assets (see Table 2), even a small percentage improvement in total payment savings can translate into millions or even billions of dollars, directly enhancing profit margins.

5.6. Non-preemptive case analysis

Partial invoice payments are increasingly common in supply chain and financial management. According to Resolve (2025a), over 30% of B2B transactions now involve partial payment arrangements. For instance, Amazon allows its business customers and AWS users to make partial payments, providing greater flexibility in managing cash flows (McMillan, 2025). Similarly, Buy Now, Pay Later (BNPL) providers, such as Klarna, Affirm, and Afterpay, enable consumers to make partial payments on their purchases (Investopedia, 2025). Many enterprise systems, such as SAP, natively support partial invoice payments, enabling firms to record and manage split settlements in both accounts receivable and AP processes. Prior research has similarly assumed preemptive invoice structures that allow partitioned payments (Ng et al., 2012; Rios-Solis et al., 2017). These practices and prior studies support our assumption that invoices are preemptive, allowing partitioning for partial settlements to support dynamic cash flow management.

While our analysis focuses on preemptive full payments, non-preemptive full payments are also widely used in practice. The structural properties of an optimal preemptive policy do not guarantee optimality in non-preemptive settings, due to stricter indivisibility and sequencing constraints (Correa et al., 2012; Lawler and Labetoulle, 1978). Nevertheless, key insights, such as priority ordering and threshold-based decision rules, can often be adapted as effective heuristics in non-preemptive environments (Correa et al., 2012; Pinedo, 2016). Although deriving tight theoretical performance bounds when applying preemptive-based strategies to non-preemptive contexts is challenging, assessing whether these structural insights yield good practical outcomes remains feasible. Motivated by this, we modify the heuristic and compare its performance to that of existing heuristics in non-preemptive scenarios.

For this analysis, we enforce a non-preemptive constraint in OPBH: if available funds are insufficient to fully pay an invoice, payment is skipped rather than partially allocated. We likewise adjust all benchmark heuristics to follow the same non-preemptive rules for a fair comparison. The pseudocode appears in the e-companion. Table 8 reports results under the same parameter settings as the preemptive analysis. In the non-preemptive setting, OPBH consistently outperforms existing heuristics, with the average performance gap of 4.08%. As in the preemptive setting, OPBH dominates in every instance, reinforcing the robustness of the structural insights derived from the preemptive model. The average performance difference between the non-preemptive and preemptive OPBHs is 0.07%. While not a theoretical bound, this suggests that the loss of optimality from enforcing non-preemption may be small.

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Table 8.

Gap performance—non-preemptive case.

SF	SC	AP	AD	Overall
4.77 (0.73) [3.57–6.62]	5.57 (0.79) [4.14–7.47]	1.66 (0.28) [1.25–2.3]	4.3 (0.55) [3.41–5.65]	4.08 (1.59) [1.25–7.47]

6. Academic contributions and managerial implications

This study investigates the SDFSCP, where a company seeks to minimize total invoice payments under uncertain cash inflows, payment terms, and interest on cash holdings. Our contributions are fourfold. First, we introduce the first dynamic stochastic programming formulation of SDFSCP. The model explicitly incorporates both time-variant processing requirements (invoice balances that evolve through discounts and penalties) and time-variant capacity (cash inflows and reserves), extending beyond prior deterministic or simplified approaches (Gupta and Dutta, 2011). Second, we identify structural properties of the optimal policy. These properties, which reveal threshold-like decision rules and priority-ordering patterns, reduce the effective complexity of the problem and provide new theoretical insights into stochastic dynamic optimization with evolving capacity and requirements. Third, we develop a heuristic grounded in these structural properties. Computational experiments show that it consistently outperforms widely used strategies such as the snowball and avalanche methods. Importantly, we extend the analysis to non-preemptive settings, where partial payments are disallowed, and demonstrate that our heuristic continues to outperform benchmarks across all problem instances. This robustness underscores the practical value of the structural results, even under stricter real-world constraints. Fourth, our findings contribute to broader literatures on scheduling and dynamic job assignment. SDFSCP shares features with problems in which both resource capacity and job requirements evolve dynamically—contexts rarely addressed jointly in prior research. Methodologically, our formulation and structural results, including diagonal analysis and structural properties using subconvexity and the directional property (Kang et al., 2016), may inform the design of algorithms and heuristics in domains such as healthcare scheduling, cloud computing, and energy management, where time-dependent capacities and processing requirements play a central role. From a managerial perspective, our findings demonstrate that intuitive practices—such as paying invoices with higher penalty rates or smaller amounts—may be systematically suboptimal. The structural properties identified provide actionable guidance for designing payment strategies that achieve significant cost savings while maintaining computational efficiency.

7. Limitations and future research

This study has several limitations that also create opportunities for future research. For tractability, the main model assumes preemptive invoices, that is, partial payments are allowed. While this assumption aligns with many modern enterprise systems, not all financial settings permit such flexibility. We therefore extend our heuristic to non-preemptive settings and show experimentally that it remains robust, consistently outperforming existing heuristics. A formal structural analysis of the non-preemptive case, or theoretical bounds on the performance gap between preemptive and non-preemptive policies, represents a promising and challenging direction.

Computing an exact optimal policy for SDFSCP remains difficult due to the curse of dimensionality (Powell, 2007). Although we characterize structural properties of the optimal policy that guide efficient algorithm design, solving the full model exactly is often computationally intractable. Consequently, some structural insights arise from optimal solutions that cannot be directly computed in large-scale instances. Nonetheless, these properties provide valuable guidance for heuristic design and deepen qualitative understanding of how optimal decisions respond to parameter changes (Smith and McCardle, 2002).

Beyond these limitations, our findings highlight several avenues for future research. The structural properties we identify—threshold-type reservation rules and priority orderings—together with the analytical features underlying them—directional modular properties such as subconvexity and superconcavity, and the directional property (Kang et al., 2016)—suggest broader methodological opportunities for stochastic dynamic resource allocation and scheduling, particularly when resource capacity and requirements fluctuate over time.

These properties may inform approximation schemes (e.g., rollout policies or index-based scoring rules), structure-guided decompositions (e.g., Lagrangian relaxations with efficiently solvable subproblems and column generation), instance-specific performance bounds via dual–primal gaps, and learning-based approaches that embed structural constraints to improve sample efficiency and interpretability. Potential application domains include patient scheduling in healthcare (where treatment complexity evolves with delay), task allocation in cloud computing (where server availability and task requirements fluctuate), and energy storage management (where capacity both depletes and replenishes dynamically).

To illustrate how these structural insights can be operationalized, we describe two concise methodological “templates” that future researchers may adopt.

Structure-Guided Decomposition Template. When dynamic resource allocation problems exhibit diagonal structure (e.g., subconvexity along 45° directions) and pairwise directional properties that induce strict priority ordering, a natural approach is to relax the shared resource constraint via Lagrangian multipliers and decompose the problem into single-entity dynamic programs. The relaxed subproblems inherit threshold-type structures similar to Theorems 3 and 5 and can be solved efficiently using monotone or switching-curve policies, while coordination is achieved through price updates on the shared resource. This template is effective when resource coupling is the primary source of computational complexity and each subproblem preserves the underlying diagonal structure.

Structure-Constrained Learning Template. When exact dynamic programming is computationally intractable, structural properties such as threshold reservation levels and priority orderings can be embedded directly into the policy class. A practical pipeline is to (i) restrict the policy space to threshold-based reservation rules and priority-index rankings consistent with deterministic structural results (e.g., Corollary 6 and Theorem 13), and (ii) estimate remaining parameters via simulation-based optimization or reinforcement learning. Enforcing monotonicity and priority consistency reduces the effective dimensionality of the policy space and improves interpretability. This template is most effective when the structural form of the optimal policy is robust across instances but the stochastic primitives are high-dimensional or partially unknown.

Finally, while the proposed heuristic performs strongly in our experiments, formal optimality guarantees remain open. Deriving theoretical performance bounds or certificates is an important avenue for future research. In addition, empirical validation using proprietary, transaction-level AP and cash flow data would further test generalizability and enable more precise calibration.

Supplemental Material

sj-pdf-1-pao-10.1177_10591478261460124 - Supplemental material for Optimal policy for managing stochastic cash flows in a financial supply chain

Supplemental material, sj-pdf-1-pao-10.1177_10591478261460124 for Optimal policy for managing stochastic cash flows in a financial supply chain by Keumseok Kang, Sushil Gupta, Inkyoung Hur and Sungbum Jun in Production and Operations Management