Addressing marketplace logistic tasks in answer set programming

Abstract

Marketplaces bring together products from multiple providers and automatically manage orders that involve several suppliers. We document the use of Answer Set Programming to automatically choose products from various warehouses within a marketplace network to fulfill a specified order. The proposed solution seamlessly adapts to various objective functions utilized at different stages of order management, leading to cost savings for customers and simplifying logistics for both the marketplace and its suppliers.

Keywords

Answer set programming combinatorial search and optimization logistics marketplace

1 Introduction

E-commerce has revolutionized the retail landscape, establishing the Internet as the primary platform for purchasing a wide range of goods [27]. Numerous recent studies address computational challenges associated with e-commerce logistics [11 , 31]. In its early stages, e-commerce primarily centered around business-to-consumer (B2C) transactions, with a primary focus on online stores promoting and selling their own products. Nowadays, marketplaces of varying scales and catering to diverse commercial sectors are emerging, elevating e-commerce to a more lucrative level [25]. Indeed, a marketplace offers a unified perspective on products sold by various suppliers and housed in different warehouses. This capability empowers end-users to fulfill orders that might be impractical through any single supplier alone. The marketplace assumes the responsibility of coordinating the affiliated suppliers to ensure the provision of products requested in a customer’s order. Furthermore, the marketplace platform manages the business-to-consumer (B2C) transaction linked to an order and subsequently handles the business-to-business (B2B) transactions with the suppliers. This simplifies a crucial financial aspect of online sales.

Building a profitable marketplace platform poses various challenges, primarily attributed to the growing complexity of the supply chain. Despite the challenges, marketplaces are gaining widespread popularity, and as of 2022, they represented more than half of the total global e-commerce sales. Considering the substantial financial stakes, optimizing the distribution chain can result in noteworthy cost savings. Primarily, a marketplace must monitor the inventory of multiple suppliers to enable end-users to create valid and eligible orders. Considering that the same product may be offered by different suppliers, an eligible order can be fulfilled by shipping products from various suppliers. Hence, a sensible criterion is needed to determine which suppliers should be engaged in handling an eligible order. Determining the objective function is crucial. One option is to minimize end-user expenses, prioritizing cost-effectiveness. Alternatively, streamlining the logistic processes within the marketplace could be the focus. Additionally, considering the logistic aspects for suppliers is another important consideration. The choice depends on the overarching objectives and priorities of the marketplace.

In this article, which extends our previous work [6] presented at the 26th International Symposium on Practical Aspects of Declarative Languages (PADL24), we consider the setting of the oliveru.comoliveru.com marketplace and some of its logistic problems. In the examined marketplace, suppliers showcase products with prices that have been mutually agreed upon with the platform. Every supplier ensures synchronization of their warehouse inventory with the marketplace platform, allowing end-users to assemble eligible orders. Furthermore, each supplier provides the option to waive shipping fees if the total cost of the products to be shipped meets a specified threshold, a threshold determined by the respective supplier. To enhance the marketplace’s appeal to end-users, the total cost of a composed order is minimized by strategically distributing the order among suppliers with lower shipping fees. This approach aims to capitalize on available opportunities for free shipping whenever possible. The total cost is continuously updated with each addition or removal of a product to ensure a seamless user experience. Consequently, the optimization task needs to be accomplished with minimal computational effort to maintain a pleasant user experience. Prior to finalizing the order, the platform verifies the estimated cost and provides the end-user with information regarding the number of shipments required to distribute the requested products. This information is derived by tackling a slightly more intricate combinatorial problem. It selects, among solutions with the same minimum cost, those that streamline the marketplace logistics to the greatest extent possible. This involves opting for solutions that minimize the number of involved suppliers. After the end-user has paid for the order, the platform engages a broader combinatorial problem. The objective is to further optimize and streamline the logistics of the involved suppliers, specifically by minimizing the number of warehouse–product pairs involved in the process. This way, the marketplace platform provides benefits to both end-users and suppliers, contributing to the overall success of its business.

The combinatorial problems tackled by the examined marketplace are formally defined and addressed through Answer Set Programming (ASP) specifications (see [21, 24]. ASP has demonstrated success in numerous planning and scheduling applications, as evidenced by studies such as [13, 17]. Its effectiveness is attributed to a rich set of linguistic constructs that facilitate the representation of intricate knowledge. Additionally, the availability of efficient engines for automated reasoning, as highlighted in [5 , 16], contributes to its utility. Actually, a small fragment of ASP is sufficient to empower the proposed specifications. The problem prototyping was accomplished using ASP CHEF [4], a straightforward and user-friendly web application for analyzing answer sets. It is specifically designed to facilitate the creation of pipelines of operations over sequences of interpretations. An experiment, conducted to evaluate the quality of the developed specifications, compares solutions obtained using ASP with those obtained through a greedy algorithm. ASP not only yields significantly superior performance but also offers the flexibility to uniformly address all three combinatorial tasks of interest. The experiment is further elaborated by incorporating an alternative solution implemented using MINIZINC [29]. MINIZINC is a solver-agnostic modeling language designed for defining and solving combinatorial satisfaction and optimization problems.

The remainder of this article is structured as follows. Section 2 introduces the required background on ASP. Section 3 formalizes the analyzed problem (Section 3.1), its complexity (Section 3.2), and how to address it in ASP (Section 3.3). After that, Section 4 reports on the implementation of the proposed solution and its empirical assessment: the ASP implementation is described in Section 4.1; the greedy algorithm used as baseline is reported in Section 4.2; the empirical assessment is detailed in Section 4.3; an additional comparison with an alternative implementation in MINIZINC is given in Section 4.4. Finally, Section 5 discusses recent literature regarding e-commerce marketplaces, and Section 6 concludes the article.

2 Background

We adopt a restricted syntax in which aggregates solely appear in (weak) constraints, and all other rules take the form of choice rules aimed at selecting exactly one atom. The language fragment is associated with decision tasks in the complexity class $Δ_{2}^{p}$ [9], and can easily encode optimization problems like Partial Weighted MaxSat [12]. This constrained syntax proves to be adequate for our encodings, streamlining the definition of the language’s semantics. For a comprehensive description of a broader fragment of ASP, we refer to the ASP-Core-2 format outlined in [10].

A constant is any natural number in $ℕ$ . Strings starting by an uppercase letter denote (object) variables. Interpreted functions are prefixed by @: an interpreted function maps a sequence of constants to a single constant. Terms are inductively defined as follows: constants and object variables are terms; if @ f is an interpreted function and $\bar{t}$ is a sequence of terms, then $@ f (\bar{t})$ is a term. For a sequence $\bar{t}$ , let $| \bar{t} |$ denote its length and ${\bar{t}}_{i}$ denote its i-th element. An atom is of the form $p (\bar{t})$ , where p is a predicate name and $\bar{t}$ is a sequence of n ≥ 0 terms. An interval has the form t = l . . u where t, l, u are terms; t is referred to as the left-hand-side of the interval. A (sum) aggregate has the form

$# sum {\bar{t} : conj} ⊙ b$ (1) where $\bar{t}$ is a sequence of terms, conj is a sequence of atoms and intervals (a conjunction), ⊙ ∈ {< , ≤ , ≥ , > , = , ≠}, and b is a term (the bound).

A (rule) body is a conjunction of atoms, intervals and aggregates. An (exactly-one) choice rule has the form

${α : conj} = 1 : - body .$ (2) where α is an atom, conj is a (possibly empty) sequence of atoms and intervals, and body is an aggregate-free body. The left-hand-side of a choice rule is called the head of the rule. A fact is a choice rule of the form ${p (\bar{t})} = 1 : -$ , and usually simply denoted as $p (\bar{t})$ . A constraint has the form

$: - body .$ (3) where body is a body. A weak constraint has the form

$: \tilde{} body . [w @ l, \bar{t}]$ (4) where body is a body, w and l are terms, and $\bar{t}$ is a sequence of terms. Choice rules, constraints and weak constraints are jointly called rules. A program Π is a set of rules.

Example 1. Let us consider an instance of Partial Weighted MaxSat [12] comprising clauses x₁ ∨ x₂ and ¬x₁ ∨ ¬ x₂, and soft clauses ¬x₁ ∨ x₂ (with weight 1) and x₁ ∨ ¬ x₂ (with weight 2). The instance has two satisfying assignments, namely (i) x₁ = 1 and x₂ = 0, with cost 1; (ii) x₁ = 0 and x₂ = 1, with cost 2. The assignment (i) is optimal.∥Let Π be the following program: $\begin{matrix} var (x 1) . var (x 2) . \\ clause (c 1) . pos (c 1, x 1) . pos (c 1, x 2) . \\ clause (c 2) . neg (c 2, x 1) . neg (c 2, x 2) . \\ soft_clause (s 1, 1) . neg (s 1, x 1) . pos (s 1, x 2) . \\ soft_clause (s 2, 2) . pos (s 2, x 1) . neg (s 2, x 2) . \\ {assign (X, V) : V = 0 . . 1} = 1 : - var (X) . \\ : - clause (C), \\ # sum {1, X : pos (C, X), assign (X, 1)} = 0, \\ # sum {1, X : neg (C, X), assign (X, 0)} = 0 . \\ : \tilde{} soft_clause (C, W), \\ # sum {1, X : pos (C, X), assign (X, 1)} = 0, \\ # sum {1, X : neg (C, X), assign (X, 0)} = 0 . \\ [W @ 1, C] \end{matrix}$ Intuitively, the first rules (facts) encode the Partial Weighted MaxSat instance. The choice rule guesses an assignment for each variable. Satisfaction of all clauses is enforced by the constraint: there must be at least one positive literal assigned 1 or one negative literal assigned 0. The weak constraint encodes the objective function: unsatisfied soft clauses provides penalties in the measure of their weight. ■

Variables of different rules are distinct (even if they share the same string representation). Global variables of a rule are those occurring in at least one atom of its body and those on the left-hand-side of an interval of its body. Global variables are (virtually) expanded by substituting them with constants, in all possible ways. Any remaining (local) variable inside an aggregate of the form (1) or in a head of a choice rule of the form (2) is safe if it occurs in at least one atom or left-hand-side of an interval in conj. Any remaining variable in weak constraints (in squared brackets) is unsafe. Only programs with safe variables are considered here. An expression (term, atom, interval, and so on) is ground if all its terms are constants. Let expand(E) be the set obtained by expanding an expression E.

Example 2. [Continuing Example 1]In the choice rule, variable X is global, and variable V is local. The expansion of the global variables in the choice rule comprises the following rules (among other rules of no interest, which are not materialized by ASP systems):∥ $\begin{matrix} {assign (x 1, V) : V = 0 . . 1} = 1 : - var (x 1) . \\ {assign (x 2, V) : V = 0 . . 1} = 1 : - var (x 2) . \end{matrix}$ Note that the local variable V is still present. ■

An interpretation I is a set of ground atoms. Variable-free terms are interpreted independently by the chosen interpretation as follows: I (c) = c if $c \in ℕ$ ; $I (@ f (\bar{t})) = f (I (t_{1}), \dots, I (t_{| \bar{t} |}))$ , where f is the mapping associated with @ f. An interpretation I is a model of a program Π if I ⊨ Π, where relation ⊨ is defined as follows: $I ⊨ p (\bar{t})$ if $p (I (\bar{t})) \in I$ ; I ⊨ t = l . . u if I (l) ≤ I (t) ≤ I (u); I ⊨ A, where A is an aggregate of the form (1), if

$(\sum_{{\bar{t^{'}} ∣ (\bar{t^{'}} : {conj}^{'}) \in expand (\bar{t} : conj)}} {\bar{t^{'}}}_{1}) ⊙ I (b);$ (5)I ⊨ body if I ⊨ body_i for all i = 1 . . |body|; I ⊨ r, where r is a choice rule of the form (2), if I ⊭ body or

$\begin{matrix} | {α^{'} ∣ (α^{'} : {conj}^{'}) \in expand (α : conj), \\ I ⊨ {conj}^{'}} | = 1; \end{matrix}$ (6)I ⊨ r, where r is a constraint of the form (3), if I ⊭ body; I ⊨ r, where r is a weak constraint of the form (4), if I ⊭ body; I ⊨ Π if I ⊨ r for all choice rules and constraints r ∈ expand (Π). Weak constraints of Π associate a cost function $c o s t_{Π} : I \times ℕ \to ℕ$ to I as follows:

${cost}_{Π} (I, l) = \sum_{\begin{matrix} (w, \bar{t}) : (: \sim body . [w @ l, \bar{t}]) \\ \in expand (Π), I ⊭ body \end{matrix}} w .$ (7) The answer sets of Π are its subset-minimal models. An answer set I of Π is optimal if there is no answer set J of Π such that there exists $l \in ℕ$ with $c o s t_{Π} (J, l^{'}) = c o s t_{Π} (I, l^{'})$ for all l′ > l, and $c o s t_{Π} (J, l) < c o s t_{Π} (I, l) .$ .

Example 3. [Continuing Examples 1–2] Let I comprises all facts of Π, and the atoms assign(x1,1) and assign(x2,0). The two choice rules in Example 2 are satisfied by I according to (6). Indeed, the expansion of assign(x1,V): V = 0..1 comprises assign(x1,0): 0 = 0..1and assign(x1,1): 1 = 0..1; I satisfies both 0 = 0 . .1 and 1 = 0 . .1; I ⊭ assign (x1, 0), and I ⊨ assign (x1, 1). It can be checked that I is actually the optimal model of Π. ■∥Development in ASP is often accomplished with the help of productivity tools. Among such tools, we mention ASP CHEF [4], which introduced the notion of recipe in ASP: a chain of ingredients combining computational tasks typical of ASP with other operations of data manipulation and visualization. More formally, a recipe is a tuple of the form (encode, Ingredients, decode), where Ingredients is a (finite) sequence of ingredients, and encode and decode are Boolean values. An ingredient is an instantiation of a parameterized operation with side output; operations are essentially functions manipulating sets of answer sets. Among the operations supported by ASP CHEF there are the computation of optimal answer sets, the parsing of comma-separated values (CSV), and the visualization of graphs. The input of a recipe is processed by the pipeline of ingredients, possibly producing some side output along the way. If encode is true, the input is Base64-encoded (for example, to be processed as CSV). Similarly, the output is Base64-decoded if decode is true.∥Example 4. [Continuing Examples 1–3] An ASP CHEF recipe for the considered instance of Partial Weighted MaxSat could start with the following input: $\begin{matrix} \begin{matrix} 1 & 2 \\ - 1 & - 2 \\ 1 & | & - 1 & 2 \\ 2 & | & 1 & - 2 \end{matrix} \end{matrix}$ Since the input is not in the form of facts, the encode flag must be set to true. After that, the above CSV can be processed by a Parse CSV operation to obtain a relational representation in terms of facts of the form __cell__(row,col,value). Such facts can be mapped to predicates var/1, clause/1, soft_clause/2, pos/2, neg/2 by using a Search Models operation (combining the facts with an ASP program). After that, an optimal answer set can be computed by using an Optimize operation (taking the choice rule, constraint and weak constraint in Example 1). Finally, the output can be sorted and shown in the form of a table with the help of other operations. The recipe can be tested in the browser at https://asp-chef.alviano.net/s/example@IA2024https://asp-chef.alviano.net/s/example@IA2024. ■

3 Problem statement and solution

This section starts by formalizing the problem introduced in Section 1 (see Section 3.1) and shows its OptP-hardness (see Section 3.2). Subsequently, it introduces a greedy algorithm that does not ensure the computation of optimal solutions (see Section 4.2). Finally, it provides a declarative ASP encoding that guarantees the computation of optimal solutions (see Section 3.3).

3.1 Problem formalization

The input includes the following elements:

a set P of products, i.e., the integer interval [1 . . |P|];

a set W of warehouses, i.e., the integer interval [1 . . |W|];

a function $p r o d u c t_p r i c e : P \to ℕ^{+}$ assigning a price to each product;

a function $p r o d u c t_r e q u e s t : P \to ℕ^{+}$ representing the requested amount of each product;

a function $s h i p p i n g_f e e : W \to ℕ^{+}$ assigning a shipping fee to each warehouse;

a function $s h i p p i n g_f r e e_t h r e s h o l d : W \to ℕ^{+}$ representing the minimum expense to waive the shipping fee of each warehouse;

a function $p r o d u c t s_i n_w a r e h o u s e : P \times W \to ℕ$ providing the available amount of each product in each warehouse.

The goal is to select the requested amount of products from warehouses (i.e., computing a function

s e l e c t : P \times W \to ℕ

) minimizing shipping costs, possibly breaking ties by minimizing the number of warehouses involved, and possibly breaking further ties by minimizing the number of pairs warehouse–product involved.

Example 5. Let us consider an instance with P = [1 . .2], W = [1 . .3], $p r o d u c t_r e q u e s t = {1 \mapsto 8, 2 \mapsto 8},$ $p r o d u c t_p r i c e = {1 \mapsto 4, 2 \mapsto 2},$ $p r o d u c t s_i n_w a r e h o u s e = {(1, 1) \mapsto 6, (1, 2) \mapsto 1, (1, 3) \mapsto 9, (2, 1) \mapsto 4, (2, 2) \mapsto 8, (2, 3) \mapsto 0},$ $s h i p p i n g_f e e = {1 \mapsto 1, 2 \mapsto 1, 3 \mapsto 2},$ $s h i p p i n g_f r e e_t h r e s h o l d = {1 \mapsto 8, 2 \mapsto 5, 3 \mapsto 20}$ . The instance parameters are summarized in Fig. 6a. Below are four admissible solutions: ${\begin{matrix} (1, 1) \mapsto 6, (1, 2) \mapsto 1, (1, 3) \mapsto 1, \\ (2, 1) \mapsto 4, (2, 2) \mapsto 4, (2, 3) \mapsto 0 \end{matrix}}$ (8) ${\begin{matrix} (1, 1) \mapsto 2, (1, 2) \mapsto 0, (1, 3) \mapsto 6, \\ (2, 1) \mapsto 4, (2, 2) \mapsto 4, (2, 3) \mapsto 0 \end{matrix}}$ (9) ${\begin{matrix} (1, 1) \mapsto 0, (1, 2) \mapsto 1, (1, 3) \mapsto 7, \\ (2, 1) \mapsto 0, (2, 2) \mapsto 8, (2, 3) \mapsto 0 \end{matrix}}$ (10) ${\begin{matrix} (1, 1) \mapsto 0, (1, 2) \mapsto 0, (1, 3) \mapsto 8, \\ (2, 1) \mapsto 0, (2, 2) \mapsto 8, (2, 3) \mapsto 0 \end{matrix}}$ (11)

Note that (8) involves warehouse 3 without reaching the threshold for waiving its shipping fee, while (9)–(11) benefit from completely free shipping. Additionally, (10)–(11) also minimize the number of warehouses involved. Finally, (11) also minimizes the number of warehouse–product pairs involved. ■

3.2 Complexity lower bound

We provide a reduction from Partial Weighted MaxSat [12], hence establishing OptP-hardness of the problem [18]. The gadgets used in the reduction are shown in Fig. 1. Without loss of generality, we associate weights with Boolean variables instead of clauses [3].

Fig. 1

Gadgets used in the reduction from PARTIAL WEIGHTED MAXSAT: (a) product P_{x
_j} (of cost n + 2) must be selected from W_{x
_j} or W_{¬x_j} (leading to free shipping), and W_{x
_j} is preferable as it also has product P_{+x_j} (which otherwise must be selected from W_{+x_j}); (b) product P_{γ_i} must be selected from one warehouse among W_ℓ₁, …, W_{ℓ_k}, preferably from one already selected (unless the initial theory Γ is unsatisfiable).

Theorem 1. Given a set of products and warehouses, functions product price, product request, shipping fee, shipping free threshold, and product in warehouse, selecting the requested amount of products from warehouses minimizing shipping costs is OptP-hard even without breaking ties, and even if functions product request and shipping free threshold are constant.

Proof. Let us fix integers m, n ≥ 0. In the following, we will use i = 1 . . n and j = 1 . . m. Let Γ = {γ₁, …, γ_n} be a set of clauses over the Boolean variables $v a r s (Γ) = {x_{1}, \dots, x_{m}}$ , that is, each γ_i is a disjunction of literals (where a literal is either a variable x_j or its negation ¬x_j). Let $w : vars (Γ) \to ℕ^{+}$ be a weight function. The task is to find a Boolean assignment I : vars (Γ) → {0, 1} that satisfies Γ and maximizes ∑_jw (x_j) · I (x_j), or equivalently that minimizes ∑_jw (x_j) · (1 - I (x_j)).

Let ∞ be the ∑_jw (x_j) + 1 (an integer larger than the sum of all weights). We fix functionproduct_request to the constant 1, and function shipping free threshold to the constant n + 2. We associate each variable x_j with products P_{x
_j}, P_{+x_j}, and each clause γ_i with one product P_{γ_i}. Variable x_j is also associated with three warehouses, namely W_{x
_j}, W_{¬x_j} and W_{+x_j}. Let $p r o d u c t_p r i c e (P_{x_{j}}) = n + 2$ and $p r o d u c t_p r i c e (P_{+ x_{j}}) = p r o d u c t_p r i c e (P_{γ_{i}}) = 1$ . Let $s h i p p i n g_f e e (W_{x_{j}}) = s h i p p i n g_f e e (W_{\neg x_{j}}) = \infty$ and $s h i p p i n g_f e e (W_{+ x_{j}}) = w (x_{j}) .$ . Regarding product_in_warehouse, we set one unit of P_{x
_j} in W_{x
_j} and W_{¬x_j}, one unit of P_{+x_j} in W_{x
_j} and W_{+x_j}, and one unit of P_{γ_i} in each W_ℓ such that ℓ ∈ γ_i.

There is a one-to-many mapping from assignments satisfying Γ and selections with shipping cost smaller than ∞. Let I : vars (Γ) → {0, 1} satisfy Γ. We select P_{x
_j} from W_{x
_j} if I (x_j) =1, and from W_{¬x_j} otherwise, obtaining in any case free shipping from the used warehouse. Since I satisfies Γ, for each γ_i there is (at least one) ℓ ∈ γ_i such that I (ℓ) =1; we select P_{γ_i} from W_ℓ (for one of such ℓ). Finally, we select P_{+x_j} from W_{x
_j} if I (x_j) =1 (as W_{x
_j} is already used), or from W_{+x_j} otherwise (paying the shipping fee w (x_j). We have that the total shipping cost is ∑_jw (x_j) · (1 - I (x_j)).

To complete the proof, we observe that any assignment not satisfying Γ maps to a selection using both W_{x
_j} and W_{¬x_j} for at least one j: Let γ_i be unsatisfied by I. Product P_{γ_i} must be selected from some W_ℓ such that ℓ ∈ γ_i, but since I (ℓ) =0 we also use $W_{\bar{ℓ}}$ .

■

3.3 ASP encoding

Given the formalization of the problem presented in Section 3.1, it is natural to develop an ASP encoding for the computation of optimal solutions for instances of the problem. Products in P and warehouses in W are identified by constants. The input is encoded by the following facts:

product(p), for each p ∈ P;

warehouse(w), for each w ∈ W;

product_price(p,p′), for each p ∈ P, where product_price(p) = p’;

product_request(p,r), for each p ∈ P, where product_request(p) = r;

shipping_fee(w,f), for each w ∈ W, where shipping_fee(w) = f;

shipping_free_threshold(w,t), for each w ∈ W, where shipping_free_threshold(w) = t;

product_in_warehouse(p,w,q), for each p ∈ P and each w ∈ W, where product_in_warehouse(p,w) = q.

The encoding of the instance from Example 5 is shown in Fig. 2. The output is encoded by facts select(p,w,q), for each p ∈ P and w ∈ W, where q is the quantity of product p selected from warehouse w.

Fig. 2

ASP representation of the instance from Example 5

The code snippet presented in Fig. 3 encapsulates the necessary conditions. Specifically, it relies on the @-term @min implemented in Python (lines 1–5) to restrict the selection of products from each warehouse (lines 9–10) to cover all requested products (line 12). The objective functions are encoded by the weak constraints in lines 13–22. To minimize the shipment cost, it is sufficient to include the weak constraint in lines 13–18 alone (resulting in the encoding Π_$). If ties need to be broken by minimizing the number of involved warehouses, then the weak constraint in lines 19–20 is also included (resulting in the encoding Π_$W). If further ties need to be broken by minimizing the number of warehouse–product pairs involved in the solution, then all weak constraints are included in the encoding (resulting in the encoding Π_$WP).

Fig. 3

ASP encoding to select products from warehouses

Example 6. [Continuing Example 5] Program Π_$ accepts all solutions (9)–(11) as answer sets. On the other hand, Program Π_$W accepts solutions (10)–(11) as answer sets. Finally, Π_$WP only accepts solution (11) as answer set. ■

4 Implementation and experiment

This section provides a few details on the implementation of a prototype addressing the computational tasks formalized in Section 3, and its empirical assessment. Specifically, the main development steps are summarized in Section 4.1, which also describes some technical details of the implementation and its prototyping in ASP CHEF [4]. After that, Section 4.3 delves into the experimental design, outlining the objective and metrics guiding our evaluation. The conducted experiments aim to assess the performance and efficacy of the proposed method using a greedy algorithm as baseline (introduced in Section 4.2). Finally, Section 4.4 presents an alternative implementation of our formalization in MINIZINC [23], and extends the empirical analysis accordingly.

4.1 ASP implementation

The ASP encodings presented in Section 3.3 were fast prototyped creating a recipe with ASP CHEF [4]. Figure 4 shows the interface of ASP CHEF and the first ingredients of a recipe addressing the marketplace logic tasks analyzed in this article. (Recipe and Example 5 available on https://asp-chef.alviano.net/s/marketplace@IA2024https://asp-chef.alviano.net/s/marketplace@IA2024.) The window is split in two columns, where input and output are placed on the right panel and the recipe on the left panel. Below is a brief description of the recipe shown in Fig. 4.

Fig. 4

First eight ingredients of an ASP CHEF recipe stacking solutions for the three marketplace logistic tasks analyzed in this article. The instance in input is the one introduced in Example 5.

#1. ParseCSV and #2. Search Models are used to accommodate the input in a format suitable for ASP. First of all, the comma-separated values (CSV) in input are mapped to a sequence of facts __cell__(row, col, value) by the ParseCSV ingredient. After that, the Search Models ingredient combines these facts with an ASP encoding producing the facts described in Section 3.3. For example,

product(1..V) :- __cell__(1,1,V).

warehouse(1..V) :- __cell__(1,2,V).

are used to produce facts representing the products and warehouses in the given instance using the values provided in the first row of the CSV.

#3. Set Optimization Strategy is used to configure CLINGO-WASM (https://github.com/domoritz/clingo-wasmhttps://github.com/domoritz/clingo-wasm; accessed on 18 February 2024) for using un-satisfiable core analysis algorithm K [3] with reitera-ted geometric search unsatisfiable core shrinking [2].

#4. Lua @-terms is used to encode the @-term @min(a,b) (essentially, the Lua equivalent of lines 3–4 in Fig. 3). The encoded content is stored in an instance of predicate __program__/1.

#5. Encode is used to store in an instance of predicate __program__/1 the common rules of the three ASP encodings introduced in Section 3.3, that is, program Π_$ (lines 6–18 from Fig. 3).

#6. Recipe: Show as Table is a recipe ingredient, that is, a recipe run inside the main recipe. In this case, the inner recipe is available online 1 ¹ and comprises the ingredients Search Models, Graph and Select Predicates. The Search Models ingredient produces instances of predicate __graph__ that are then interpreted by the Graph ingredient to obtain a side output visualization of the input instance and of the solutions computed by CLINGO-WASM, as shown in Fig. 6a. Without going into too much detail, the ASP program used by the Search Models ingredients includes the rules shown in Fig. 5. The first two lines above define the constants radius and size, which are then used by the first rule to set some defaults properties of the graph, among them the shape and size of nodes. The last three rules define how to show warehouses in the header of the table: first of all, the label and position of the node are given; after that, the node is colored red if selected, and black otherwise. Finally, the Select Predicates ingredient preserves only the facts representing the input instance, as well as instances of predicate __program__/1.

Fig. 5

A snippet of code to produce a table representation of input instances

Fig. 6

Stacked representation provided by ASP CHEF of the input (a) and solutions (b)–(d) for the instance from Example 5.

#7. Optimize searches for an optimal answer set of the program stored in instances of predicate __program__/1. Instances of __program__/1 are left in the output of the ingredient so that subsequent ingredients can also use them. In this case, the program is actually Π_$, and the computed solution is shown by #8. Recipe: Show as Table (see Fig. 6b).

#9. Encode adds the weak constraint shown in lines 19–20 of Fig. 3 as an instance of predicate __program__/1, so that #10. Optimize searches for an optimal answer set of program Π_$W, which is visualized by #11. Recipe: Show as Table (see Fig. 6c).

#12. Encode adds the weak constraint shown in lines 21–22 of Fig. 3 as an instance of predicate __program__/1, so that #13. Optimize searches for an optimal answer set of program Π_$WP, which is visualized by #15. Recipe: Show as Table (see Fig. 6d).

Before the last visualization, #14. Generate CSV stores the answer set as CSV in an instance of predicate __base64__/1, the only predicate selected by #16. Select Predicates. The CSV is finally shown in the output panel.

4.2 Greedy algorithm

Sub-optimal solutions of the problem formalized in Section 3.1 can be obtained by Algorithm 1. It is a greedy algorithm that iteratively selects a warehouse maximizing the number of requested products available within its supplies. This process continues until all requested products are covered by the selection. In more detail, the algorithm constructs a select function with the signature $s e l e c t : W \times P \to ℕ$ , representing the quantity of products taken from each warehouse to fulfill the requested amount of products. To achieve this, the algorithm iteratively makes greedy choices until there are products to cover (line 1):

Each warehouse is assigned a utility value, which is the number of requested products available within its supplies (lines 2–3).

A warehouse w with the maximum utility is selected (line 4), and the maximum quantity of requested products is taken from w (lines 6–10).

If no new products are covered by w, the algorithm terminates, reporting the inconsistency of the input instance (line 5).

Note that each warehouse is selected at most once, given that its product supply is updated by line 9, rendering its utility value 0 for the next iteration.

Example 7. [Continuing Example 5] On the first iteration of the main loop, we have utility₁ = 10 and utility₂ = utility₃ = 9. Hence, warehouse 1 is selected: it will provide 6 units of product 1 and 4 units of product 2. On the second iteration we have to cover 1 unit of product 1 and 4 units of product 2, so that utility₁ = 0, utility₂ = 5 and utility₃ = 1. Hence, warehouse 2 is selected: it will provide 1 unit of product 1 and 4 units of product 2. On the third iteration there is only 1 unit of product 1 to cover, so that utility₁ = utility₂ = 0 and utility₃ = 1. Hence, warehouse 3 is selected and it will provide 1 unit of product 1. On the fourth iteration there are not units of product to cover, so that utility₁ = utility₂ = utility₃ = 0 and the algorithm terminates returning the solution (8). ■

Algorithm 1 finds an admissible solution if one exists, but cannot guarantee its optimality, as shown by the above example. For another example, consider a trivial instance with only one unit of product 1, whose cost is 1. Product 1 is available in warehouses 1 and 2, whose thresholds are respectively 1 and 2. The algorithm cannot distinguish between 1 and 2, so it possibly selects warehouse 2, failing to obtain free shipping (which is instead possible if warehouse 1 is selected). For yet another example, consider an instance with one unit of product 1 and one unit of product 2, whose costs is 1. Product 1 is available in warehouse 1, and product 2 is available in warehouse 2; both warehouses have threshold 1. There is a third warehouse with both products, whose threshold is 3 (unreachable), and the algorithm eventually selects this third warehouse, failing to obtain free shipping (which is instead possible if warehouses 1 and 2 are selected). Moreover, the algorithm cannot guarantee the inclusion of a minimal number of warehouses or pairs of warehouse–product in the computed solution. Consequently, the more sophisticated objective functions defined in Section 3.1 are beyond its capabilities. Nonetheless, it can serve as a baseline for more sophisticated algorithms.

4.3 Empirical assessment

The most common scenario for the oliveru.comoliveru.com marketplace involves end-users purchasing no more than 10 different products, typically with a single unit per product. However, for the experiment, much larger instances were generated. These instances included a random number of products ranging from 1 to 20, each with a random multiplicity in the range of 1 to 20. The number of warehouses was randomly chosen between 1 and 20, and the available products within each warehouse were randomly selected from the range of 1 to 20. A total of 10000 instances were generated for the experiment. The generated instances were processed using CLINGO (version 5.6.2) with the activation of the unsatisfiable core analysis algorithm K [3] along with reiterated geometric search unsatisfiable core shrinking [2] (command line: –opt-strategy=usc,k,0,5 –opt-usc-shrink=rgs). The tests were conducted on an Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz with 12GB of RAM, utilizing a timeout of 3 seconds. The measured values include the execution time, the cost of the computed solution, the number of involved warehouses, and the number of warehouse–product pairs involved.

The experimental results are presented in Fig. 7. Firstly, it is observed that the greedy algorithm completes in less than 0.01 seconds for all tested instances, while running the ASP encodings Π_$, Π_$W, Π_$WP takes approximately 0.04s, 0.05s, and 0.07s on average, respectively. Furthermore, 804 out of the 10,000 generated instances do not have feasible solutions, and their statistics are depicted on the rightmost positions in the plots. It is noteworthy that CLINGO identifies the inconsistency with negligible effort. The other plots in Fig. 7 quantify the quality of the computed solutions. Concerning the shipping cost, all ASP encodings yield the same result, and thus, we focus on Π_$. Interestingly, the shipping cost is frequently 0, indicating that shipping fees are correctly waived when feasible. Regarding the number of warehouses involved, Π_$W and Π_$WP yield the same result. Hence, we focus on Π_$ and Π_$W, along with the greedy algorithm. It is evident that the greedy algorithm prioritizes minimizing the number of warehouses involved rather than minimizing the resulting shipping cost. Nevertheless, activating the weak constraint of level 2 significantly reduces the number of warehouses involved, aligning them with the small number obtained by the greedy algorithm while maintaining the minimum shipping cost. A similar observation is applicable to the number of warehouse–product pairs involved in the process. The greedy algorithm performs well, and ASP slightly improves upon this baseline by activating the weak constraint of level 1 (i.e., program Π_$WP).

Fig. 7

Experimental results comparing the greedy algorithm and CLINGO running algorithm K (bar plots with instance number of the x-axis).

We ran CLINGO a second time without activating the unsatisfiable core analysis, that is, using the default configuration of the solver [15]. In this case, CLINGO implements a linear search sat-unsat that iteratively improves the latest computed solution by imposing a constraint on the admissible value of the objective function, until the unsatisfiability of the processed instance is proved (witnessing that the latest computed solution is optimal). The results are shown in Fig. 8. It can be observed that algorithm K is usually faster than the default configuration on the tested instances. The gap between the two settings is more evident for program Π_$WP (Fig. 8b), for which the default configuration leads to 2373 timeouts. Even if we have not registered any timeout for programs Π_$W and Π_$ (Fig. 8a), we note that the performance of CLINGO is improved in around 48% of the tested instances, and worsened in around 36% of the cases; algorithm K provides a more sensible particular gain on program Π_$ (5808 times faster than the default configuration, and only 1175 times providing some negligible overhead). All in all, we conclude that running CLINGO with its default configuration is not aligned with the requirements of the marketplace.

Fig. 8

Execution time of CLINGO using its default configuration or algorithm K; time bounded to 3 seconds in (b). For Π_$ and Π_$W, the result is mixed but in favor of algorithm K: in (a) the majority of points is above the dashed line, and specifically the number of wins is 5808 vs 1175 (Π_$) and 3789 vs 2474 (Π_$W) in favor of K. The result is more uniform and in favor of K for Π_$WP, as shown in (b), even if there are some exceptions: the number of wins is 8226 vs 534 in favor of K.

4.4 Comparison with an alternative implementation in MINIZINC

Constraint Satisfaction is a Knowledge Representation and Reasoning (KRR) formalism at the core of many applications in Artificial Intelligence [30]. A (finite-domain) Constraint Satisfaction Problem (CSP) can be expressed in the following form: Given a set of variables, together with a finite set of possible values that can be assigned to each variable, and a list of constraints, find values of the variables that satisfy every constraint [8]. Many combinatorial problems, such as scheduling, timetabling and resource allocation, can be formulated as CSPs. Some of these problems involve an objective function to minimize or maximize, leading to the definition of Constraint Satisfaction and Optimization Problem (CSOP). MINIZINC is a solver-agnostic modeling language for defining and solving CSPs and CSOPs. MINIZINC provides a solver-independent modeling language that is supported by several constraint-programming solvers, mixed integer programming solvers, SAT and SAT modulo theories solvers, and hybrid solvers [29]. In MINIZINC, the declarative specification for a problem is encoded in a file that takes the name of model file (or simply model). A typical model file includes the following elements: directives to define the format of the instances of the addressed problem, that is, the parameters of the problem; statements to define the search space, that is, variables and their domains; one solving instruction, possibly paired with an objective function. Instances of a problem are then written in files that take the name of data files. A data file contains assignments for all parameters of the associated problem.

The MINIZINC model $M_{$}$ reported in Fig. 9 encodes the first problem addressed in this article, that is, selecting products from warehouses to fulfill the given order with the objective of minimizing the shipping cost. Parameters of the problem are declared according to the formalization given in Section 3.1 (lines 1–8). After that, the search space is defined (lines 9–13) as the function select assigning a natural number to each warehouse–product pair; here the range of the function must be finite, and a common upper bound to all warehouse–product pairs is obtained by taking the largest number of requested units among the product in the given order (line 12). The search space is then restricted by enforcing that all selected units in each warehouse are actually available in that warehouse (lines 14–17) and that all and only the requested products in the given order are selected (lines 18–21). Finally, the solving instruction in lines 26–36 imposes to minimize the shipping cost as objective function: Each selected warehouse applies its shipping fee unless the order includes its products for a total cost reaching the warehouse threshold for free shipping; the actual shipping cost is obtained via the utility function shipping_cost defined in lines 22–25. Figure 10 shows the data file for the model $M_{$}$ and the instance from Example 5: two products (line 1) with their requested quantities (line 5) and prices (line 6), three warehouses (line 2) with their shipping fees (line 8) and thresholds to waive shipping fees (line 7), and the available units of products in each warehouse (lines 3–4).

Fig. 9

A MINIZINC model (referred to as $M_{$}$ ) to select products from warehouses minimizing the shipping cost.

Fig. 10

MINIZINC data file for the instance from Example 5 and model $M_{$}$ (shown in Fig. 9).

To the best of our knowledge, MINIZINC does not support any syntactic construct to specify multi-level optimization. A simple and common strategy to address multi-level optimization in MINIZINC is by first obtaining the optimum value of the top level with one model, and then use such an optimum value in a second model targeting solutions that optimize the next level. The process is reiterated until there are no more levels to optimize. Following such a strategy, we defined the MINIZINC model $M_{$ W}$ shown in Fig. 11. We observe that $M_{$ W}$ starts with the first 25 lines of $M_{$}$ , and expands the format of data files with the additional parameter shipping_cost (lines 37–39), which must be obtained by computing a solution of $M_{$}$ . The given optimum shipping cost is then enforced by the constraint in lines 40–50, which is essentially the objective function of $M_{$}$ (lines 26–36). Finally, the objective function is given in lines 51–54: the number of selected warehouses is minimized. The same strategy was applied to define a third model $M_{$ WP}$ to address our three-level optimization problem, shown in Fig. 12. It starts with the first 50 lines of $M_{$ W}$ , and expands the format of data files with the additional parameter selected_warehouses (lines 55–57), which must be obtained by computing a solution of $M_{$ W}$ . The given optimum number of warehouses is then enforced by the constraint in lines 58–62, which is essentially the objective function of $M_{$ W}$ (lines 51–54). Finally, the objective function is given in lines 63–66: the number of involved warehouse–product pairs is minimized.

Fig. 11

Lines 1–25 of model $M_{$}$ (shown in Fig. 9) are extended to obtain a second MINIZINC model (referred to as $M_{$ W}$ ) to minimize warehouses given the optimum shipping cost.

Fig. 12

Lines 1–50 of model $M_{$ W}$ (shown in Figure 11) are extended to obtain a third MINIZINC model (referred to as $M_{$ WP}$ ) to minimize warehouse–product pairs given the optimum shipping cost and the optimum number of warehouses.

We ran the three MINIZINC models on the instances used in Section 4.3, on the same machine we ran the other experiments, using OR-TOOLS as backend (https://github.com/google/or-toolshttps://github.com/google/or-tools). All instances are solved by MINIZINC, which obtained the same optimal values of CLINGO. Our analysis is then focused on the execution time of the two solvers, which is summarized in Fig. 13. In the plots, the execution time of model $M_{$ W}$ includes also the execution time of $M_{$}$ (the value of shipping_cost must be computed using $M_{$}$ in order to use $M_{$ W}$ ), and the execution time of model $M_{$ WP}$ includes also the execution time of $M_{$ W}$ (the value of selected_warehouses must be computed using $M_{$ W}$ in order to use $M_{$ WP}$ ). In the bar plot in Fig. 13a, it can be observed that CLINGO (with algorithm K) is faster than MINIZINC on the tested instances, as already the execution time of $M_{$ W}$ tends to be higher than the execution time of Π_$WP. The gap is more evident for the instances without feasible solutions (the rightmost instances in the bar plot), but such a delay can be eliminated by first running the heuristic algorithm to decide satisfiability of the instance, and then running MINIZINC only on satisfiable instances. The instance-by-instance comparison on the execution time of the two solvers given in Fig. 13b highlights the different performance of the ASP encodings Π_$, Π_$W and Π_$WP with respect to $M_{$}$ , $M_{$ W}$ and $M_{$ WP}$ , respectively. The scatter plot confirms that the performance of CLINGO is superior on the tested instances, and also highlights that the gap between the two approaches increases with the number of levels in the optimization problem.

Fig. 13

Execution time of CLINGO and MINIZINC; time bounded to 3 seconds in (a); time bounded to 1 second in (b). CLINGO is faster than MINIZINC with a few exceptions (16 wins and 9 draws) for the task of minimizing the shipping cost; in (b) the majority of points are above the dashed line.

A different view on the experimental results is given by the plots in Fig. 14. In the plots, instances are sorted by increasing the number of warehouses, products, and requested products. The whisker plots report data on the execution time of CLINGO (running K) and MINIZINC (minimum, maximum, median, lower and upper quartiles). As a general comment, execution time tends to increase with such parameters, as expected. The benefits of CLINGO are consistently spread across various groups of instances, despite MINIZINC displaying a more gradual increase in execution time for the basic objective function in our benchmark. Notably, for this specific objective function, the quartiles of MINIZINC are also more closely clustered compared to those of CLINGO. On the other hand, the fastest execution time obtained by MINIZINC is always above the upper quartile of CLINGO (in all plots).

Fig. 14

Execution time of CLINGO and MINIZINC with instances sorted according to different criteria (whisker plots reporting minimum, lower quartile, median, upper quartile and maximum values).

5 Related work

The paper relates to literature which already studied the E-commerce Marketplace optimization of retail chain. Dutta et al. [14] study the need for sustainable practices in Indian E-commerce by proposing a multi-objective logistics model. It optimizes cost, environmental impact, and social responsibility in the reverse logistics network, considering various facilities and technologies. The efficiency of the E-commerce marketplace is a relevant topic from an economic and environmental point of view. Zhang and Ma [31] investigate the possibility of the supplier and the platform reaching a consensus on the implementation of the e-commerce marketplace channel through the integration of a logistics service strategy. The study identifies the significant impact of logistics service levels on consumers’ channel preferences. While suppliers are generally inclined to embrace the marketplace channel and its associated logistics responsibilities, the platform’s benefit depends on the effectiveness of logistics service or commission rates. However, the authors do not consider any specific optimization in the delivery and logistic of the solution. Qin et al. [26] analyze the dynamics of logistics service sharing within e-commerce platforms, where the platform extends marketplace services to sellers in addition to its retailing function. The study conducts a strategic and economic analysis of this sharing arrangement. Results indicate that the outcome of logistics service sharing hinges on factors such as the service level of third-party logistics service providers (TPLPs) and the market potential. Kumar and Mahapatra [19] leverage the Rain Optimization Algorithm (ROA) designed by Moazzeni and Kamehchi [22] and addresses the challenge of item deterioration in inventory management, focusing on non-instantaneous items. Specifically, this research incorporates a multi-warehouse setup, with one owned warehouse and others rented, each with limited storage capacity. The primary objective is to determine retailers’ replenishment policies that maximize total profit for each item while minimizing overall costs.

The order splitting problem (also known as split delivery) means that multi-item orders are split into several suborders fulfilled by multiple warehouses separately. The suborders are packed in different packages and delivered with multiple shipments. Order splitting is becoming particularly prominent for large-scale online retailers with the multi-item order feature. The problem of split orders in retail is of particular relevance in the E-commerce Marketplace due to the nature of the business model itself. It has become a great challenge to online retailers for fulfilling multi-item orders in a multi-warehouse storage network. A split order entails additional cost for the retailed that certainly are reflected on the cost of the goods. Zhang et al. [32] presents a multi-warehouse package consolidation approach aimed at consolidating multiple suborders’ stock-keeping units (SKUs) through transshipments among warehouses. They propose a combined multi-commodity network flow model and an enhanced logic-based Bender’s decomposition algorithm to effectively generate schemes for the multi-warehouse package consolidation problem. Lei, Jasin and Sinha [20] consider a multi-period joint pricing and fulfillment (JPF) problem where an e-commerce sells multiple products to customers coming from multiple demand locations and demands are fulfilled in real-time through multiple fulfillment centers. They propose a tractable deterministic approximation for the problem to maximize the total expected profits. Catalán and Fisher [11] introduce the problem of assigning stock-keeping units to distribution centers to minimize split orders. Specifically, the research investigates the problem of shipping products from different suppliers and demonstrates the NP-hardness of the problem. The authors then focus on developing several heuristics mainly taking into account the average number of shipments per order and the workload distribution across suppliers, hence deviating from the objective functions of the marketplace considered in this paper. Nonetheless, extensions of the ASP specifications presented here to accommodate workload balancing and fair distribution of sales represent interesting aspects to explore.

6 Conclusion

ASP demonstrated effectiveness in resolving the combinatorial problems associated with the analyzed platform, particularly for instances of anticipated size corresponding to the volume of business managed by the marketplace. In addition to computational efficiency, the ASP specification presented in Section 3.3 offers a consistent solution to the three combinatorial problems under consideration. This solution is contingent on the activation of specific weak constraints. Interestingly, the standard algorithm used by CLINGO to address combinatorial optimization is not sufficient to cope with the more sophisticated objective function that also minimizes the logistic effort for suppliers. This is due to the uncontrolled improvements provided by linear search sat-unsat: the algorithm may proceed with small improvements even if currently far from optimality. Moreover, the constraint enforced by linear search sat-unsat involves all weak constraints, leading to an expensive computation that makes particularly challenging to prove optimality of the last computed solution. The observed inefficiency was quickly fixed by switching to the unsatisfiable core analysis algorithm K with reiterated geometric search unsatisfiable core shrinking. Within this algorithm, the search directly targets optimality, actually building a proof of optimality during the whole search rather than as a final step. (We refer [1, 3] for details on the algorithm.) The assignment of products generated for delivery from each warehouse can be further optimized in the realm of in-warehouse logistics. For instance, in an automated warehouse scenario, there would be a necessity to compute a collision-free schedule of robot movements and actions to efficiently deliver all the required products (we refer [28] for details).

Acknowledgment

This work was partially supported by Italian Ministry of University and Research (MUR) under PRIN project PRODE “Probabilistic declarative process mining”, CUP H53D23003420006 under PNRR project FAIR “Future AI Research”, CUP H23C22000860006, under PNRR project Tech4You “Technologies for climate change adaptation and quality of life improvement”, CUP H23C22000370006, and under PNRR project SERICS “SEcurity and RIghts in the CyberSpace”, CUP H73C22000880001; by Italian Ministry of Health (MSAL) under POS projects CAL.HUB.RIA (CUP H53C22000800006) and RADIOAMICA (CUP H53C22000650006); by Italian Ministry of Enterprises and Made in Italy under project STROKE 5.0 (CUP B29J23000430005); by the LAIA lab (part of the SILA labs) and by GNCS-INdAM.

Disclaimer

The contents of this document represents the views of the authors only and are their sole responsibility; it cannot be considered to reflect the views of the European Commission and/or the Joint Research Centre (JRC), or any other body of the European Union. The European Commission and JRC do not accept any responsibility for use that may be made of the information it contains.

Footnotes

https://asp-chef.alviano.net/s/marketplace-show-astable@AI2024

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