Improving delivery planning via an optimization technique for a case study of a construction material store

Abstract

The construction material store (CMS) industry in Thailand is highly competitive, boasting 16,000 stores and experiencing rapid growth (GSB Research). Despite this, back-office management, such as delivery and appointment scheduling, struggles to keep up with the increasing customer demand. The present study aims to develop an operational model for efficient delivery planning in a CMS. The model accurately calculates the minimum delivery costs while adhering to certain constraints. This research employs a mixed-integer linear programming (MILP) mathematical model specifically designed for planning routes and organizing products for delivery. In this study, the CMS delivers products to customers through a third-party trucking company, and a practical delivery plan is developed based on existing conditions and limitations. The delivery plan ensures that products are supplied to customers using appropriate vehicles for the product type, while staying within truckload capacities. The CMS operates by outsourcing vehicles of various sizes, and different products are selected for delivery. All products are available for shipment to customers at any given time. The improved delivery planning considers routing, product selection for trucks, capacity limitations, customer demand, and distance management. Certain limitations of the model, such as routing policy, truck capacity, and product weight, are addressed. The results demonstrate that implementing this model can decrease planning time by up to 80%, reduce total costs by up to 20%, lower delivery distances by up to 19%, and enhance customer satisfaction.

Keywords:

• delivery planning
• construction material
• single depot
• split delivery
• heterogeneous fleet
• an optimization model for delivery management

1. Introduction

The construction material industry is enhancing its last-mile delivery capabilities to satisfy rapidly growing customer demands. Customers expect work to be completed more quickly, accurately, and with higher quality. Consequently, construction businesses are constantly seeking strategies to streamline processes and improve operational efficiency. Delivery planning encompasses supply chain activities that support a business advantage (Sabri & Beamon, 2000). Delivery service is one of many activities that entrepreneurs practice to create customer satisfaction. As noted by Al-Werikat (2017), Biswas and Sarker (2020), Vukić et al. (2021), Gupta et al. (2021), Haq et al. (2021), Donmuen and Pitiruek (2022), Richey et al. (2022), Li et al. (2023), and Van Nguyen et al. (2023), delivery planning is essential for enabling cost-cutting and profit-boosting opportunities in supply chains.

Most construction stores in Thailand are classified as single depots, where all materials are stored and distributed from the same location. This case study is also categorized as a single depot. Additionally, stores often have numerous products to be ordered and distributed, resulting in challenges when categorizing products according to truck capacities and types. Tavakkoli et al. (2015) proposed a multi-product vehicle routing problem with heterogeneous fully loaded vehicles with different capacities. Ayyildiz et al. (2022) developed a multi-depot, multi-product split delivery vehicle routing method with time windows for money distribution to an Automated Teller Machine (ATM) of a public bank in Turkey. This method led to effective solutions for their cash-in-transit operations with different currencies.

Due to transportation limitations, it is sometimes impossible to deliver all materials for one customer at once. Hence, split delivery can be applied. According to Ray et al. (2014), Ozbaygin et al. (2018), and Munari and Savelsbergh (2022), in this type of problem, a vehicle can deliver any quantity less than or equal to the customer’s demand when visiting a customer. This implies that customers can be visited by more than one vehicle.

Given that shipping plans vary daily, effective plans must be created accordingly. Capacitated vehicle routing problem (CVRP) is adopted to address these problems. Basuki et al. (2019) developed a CVRP for a printing business, where products were shipped to customers at various distances. The developed model was successfully used to reduce total costs. Dellaert et al. (2021) developed model formulations and a solution approach for a multi-commodity, two-echelon CVRP with time windows. Alesiani et al. (2022) solved a CVRP with NP-hard conditions by applying constrained clustering capacitated vehicle routing, which included several constraints.

However, these previously reported models are insufficient for more complex situations. In this case study, the examined construction material store (CMS) company has many different products, and some products can only be loaded on specific vehicles. The details of the products are required before planning the shipment. Therefore, we developed a technique that can respond to all these limitations. Hence, in this study, a single depot split delivery with multi-product heterogeneous capacitated vehicle routing (SDMHCVRP) is proposed.

2. Literature review

The planning and operation of supply chains in manufacturing and distribution centers are crucial to their smooth operation. Optimal decision-making is necessary to lower costs in supply chains (Biswas & Sarker, 2020). Many researchers such as Abdulkader et al. (2018) have developed mathematical models to designate delivery routes for customers with the same truck type. Ibrahim et al. (2019) proposed a capacitated vehicle routing problem (CVRP) to determine the shortest route for delivery, with the condition that the truck should begin and end its route at the same location. Setiawan et al. (2019) developed a mathematical model to determine the transportation routes for delivering various goods with different vehicle types for a drug store in Indonesia. The proposed model provided for a 48.78% cost savings. Yuliza et al. (2020) defined a route with a constraint on the truck capacity for a liquid propane gas delivery business. They used the Clarke and Wright algorithm to decrease the total delivery distance. Weerakkody et al. (2021) developed a vehicle routing problem (VRP) with multiple goods and validated their model with 3PL, one of the most prominent transportation businesses in Sri Lanka. Their result was similar to that of Utama et al. (2020) and Naclerio and De Giovanni (2022). Basuki et al. (2019) developed a CVRP for a printing business. The products of this printing business were shipped to many customers located at various distances, and the developed model was successfully used to reduce total costs. Alesiani et al. (2022) solved a CVRP with NP-Hard conditions by applying constrained clustering capacitated vehicle routing. The constraints included: 1) a vehicle serves more than one group, 2) a cluster must have at least one customer, 3) customers are located within one cluster only, and 4) customers in the same cluster use only one vehicle.

Numerous VRP studies have been performed to solve problems in transportation businesses. Ozbaygin et al. (2018) reported an exact method for a split delivery VRP (SDVRP) that allowed customers to receive more than one shipment. This method was different from CVRP, but it also reduced costs. Bortfeldt and Yi (2020) and Munari and Savelsbergh (2022) also proposed split delivery routing problems. Aganis and Sirawadee (2019) presented a shipping management model for a beverage company with different vehicles of various sizes to address the problem of customer demands that were larger than a single vehicle’s capacity. Their proposed method resulted in total cost savings of 10.66% to 11.01%. Siriruk and Tangmo (2017) determined the delivery routes of a construction material business by initially grouping the customers before planning. Nine customer groups were set. Consequently, different vehicles were also planned for delivering the products, resulting in cost savings. Chaiwuttisak et al. (2018) proposed a model that employed vehicle limitations on product placement.

Other researchers, such as Siriruk and Tangmo (2017), have studied the differential vehicle routing problem of a building material retailer (HVRP) with five types of vehicles. However, their results only indicated the aggregate demand, cost, distance, and delivery route of each vehicle, while ignoring limitations such as the size, shape, and weight of products. Other methods are unable to separate and transfer products. Therefore, when the VRP algorithm is applied, the results only provide the distance and cost of the transport. Additionally, Aganis and Sirawadee (2019) investigated the transport routing problem of vehicles with different load capacities (VRPSDHF) by developing a mathematical model. Their results showed the vehicle type, number of pallets to be loaded and shipped to customers, and delivery route. For Aganis et al and Siriruk et al, a mathematical model was developed to plan for multiple shipments from a single warehouse to multiple customers using multiple trucks. The resulting delivery plan also selected the correct type of vehicle to deliver each product with the lowest shipping cost.

On the other hand, the past relevant research is as follows: Wang et al. (2020) used MILP to develop a product delivery plan that allowed a specific vehicle type to correctly deliver each category of product and arrange each product class correctly and completely. Their delivery plan achieved the lowest delivery costs, according to van der Vorst (2000), Mula et al. (2010), Munyimi and Guo (2019), and Ardliana et al. (2022).

As a result of literature research, it is seen that problem handled in this study is delivery planning. This study deals with single depot, split delivery, multi-product, heterogeneous fleet, and capacitated vehicle routing. This study also has a real-life application for delivery planning of a construction material in a case study.

3. Methodology

In this research, we proposed a mathematical model for delivery planning in a case study of a construction material store. This research aims to achieve a delivery plan with the lowest delivery costs. The objective function and its constraints are expressed in Equations (1)(1) $min\mathit{z}={\sum }_{\begin{array}{c}i\ne j\\ i=1\end{array}}^N{\sum }_{j=1}^N{\sum }_{k=1}^K\mathit{F}{C}_k{x}_{\mathit{ijk}}+{\sum }_{i\ne j}^{\mathrm{N}}{\sum }_{j=1}^N\mathit{Dis}{t}_{\mathit{ij}}{\sum }_{k=1}^{\mathrm{N}}\mathit{V}{C}_k\cdot x_{\mathit{ijk}}+{\sum }_{\begin{array}{c}i\ne j\\ i\ne 1\\ j\ne 1\end{array}}^{\mathrm{N}}{\sum }_{j\ne 1}^N{\sum }_{k=1}^{\mathrm{N}}\mathit{D}{C}_k\cdot x_{\mathit{ijk}}\mathit{\hspace{0.17em}}\left(1\right)\mathit{\hspace{0.17em}}$(1) to (18(18) $S_{\mathit{ijkl}}\in \left\{\mathit{I}\right\};\mathrm{\forall i},\mathit{j},\mathit{k},\mathit{l}$(18) ).

3.1. Assumptions

To develop the proposed mathematical model, a few important assumptions were made, as follows:

1. There is only one distribution center, and the vehicles must start and end their journeys at the same location.

2. The store operates by outsourcing vehicles of three different sizes and the model uses five different products.

3. All vehicles and drivers are ready to work at any time. No vehicle can travel more than 480 km per trip.

4. Some product types can only be shipped by a 6-wheel crane truck.

5. The total weight of a vehicle’s load must not exceed its load capacity.

6. All products are ready for delivery.

7. All customers are ready to receive products at any time.

8. The placement of the products as well as their loading and unloading times are negligible.

3.2. Notation

The parameters, indices, and decision variables are defined as follows (See Table 1):

Table 1. The solution method of each study in the literature

Reference	Problem characteristics					Solution method
	Single depot	Split delivery	Multi-product	Heterogeneous fleet	Capacitated vehicle routing
Aganis and Sirawadee (2019)	✓	✓		✓		Branch-and-reduce
Ibrahim et al. (2019)	✓				✓	LP
Ozbaygin et al. (2018)		✓				Exact algorithm
Munari and Savelsbergh (2022)		✓				Branch-and-cut
Tavakkoli et al. (2015)			✓	✓		MILP
Ayyildiz et al. (2022)		✓	✓			MIP
Siriruk and Tangmo (2017)				✓		Cluster-First Route-Second
Setiawan et al. (2019)			✓	✓		MILP
Basuki et al. (2019)			✓	✓		Saving Matrix
Dellaert et al. (2021)					✓	Exact algorithm
This study	✓	✓	✓	✓	✓	MILP

3.3. Indices

$\mathit{i},\mathit{j}$ the customer/delivery point/order. $\mathit{i},\mathit{j}$; $\mathit{i},\mathit{j}$ = 1., $\mathit{N}$

$\mathit{p}$ the delivery point. $\mathit{p}$; $\mathit{p}$ = 1., $\mathit{N}$

$\mathit{k}$a vehicle. $\mathit{k}$; $\mathit{k}$ = 1., $\mathit{K}$

$l_{\mathit{\hspace{0.17em}}}$ the product. $l_{\mathit{\hspace{0.17em}}}$; $l_{\mathit{\hspace{0.17em}}}$ = 1., $\mathit{L}$

$\mathit{N}$ the total number of customers/delivery points/orders.

$\mathit{K}$ the total number of vehicles.

$\mathit{L}$ the total number of products.

3.4. Parameters

$\mathit{F}{C}_k$ the fixed cost of vehicle $\mathit{k}$(THB per trip)

$\mathit{V}{C}_k$ the variable cost of vehicle $\mathit{k}$ (THB per km)

$\mathit{D}{C}_k$ the delivery cost of vehicle $\mathit{k}$ (THB per delivery point)

$W_l$the weight of product $\mathit{l}$ (kg)

$\mathit{Dis}{t}_{\mathit{ij}}$ the distance from delivery point $\mathit{i}$ to delivery point $\mathit{j}$ (km)

$\mathit{Mdist}$the maximum distance traveled by a vehicle per day (km)

$\mathit{Ca}{p}_k$ the maximum capacity of vehicle $\mathit{k}$ (kg)

$\mathit{rul}{e}_{\mathit{lk}}$ 1 if vehicle $\mathit{k}$ can transport product l

and 0 otherwise.

$\mathit{D}{M}_{\mathit{il}}$ the demand for product $\mathit{l}$ of customer/delivery point/order $\mathit{i}$ (units)

3.5. Decision variables

$X_{\mathit{ijk}}$ 1 if vehicle $\mathit{k}$travels from the delivery point $\mathit{i}$ to delivery point $\mathit{j}$

and 0 otherwise.

$D_{\mathit{ikl}}$ the product $\mathit{l}$ delivered in order $\mathit{i}$ by vehicle $\mathrm{k}$ (units).

$S_{\mathit{ijkl}}$ the product $\mathit{l}$ delivered by vehicle $\mathit{k}$ from delivery point $\mathit{i}$ to delivery point $\mathit{j}$ (units).

3.6. Objective function

Equation (1)(1) $min\mathit{z}={\sum }_{\begin{array}{c}i\ne j\\ i=1\end{array}}^N{\sum }_{j=1}^N{\sum }_{k=1}^K\mathit{F}{C}_k{x}_{\mathit{ijk}}+{\sum }_{i\ne j}^{\mathrm{N}}{\sum }_{j=1}^N\mathit{Dis}{t}_{\mathit{ij}}{\sum }_{k=1}^{\mathrm{N}}\mathit{V}{C}_k\cdot x_{\mathit{ijk}}+{\sum }_{\begin{array}{c}i\ne j\\ i\ne 1\\ j\ne 1\end{array}}^{\mathrm{N}}{\sum }_{j\ne 1}^N{\sum }_{k=1}^{\mathrm{N}}\mathit{D}{C}_k\cdot x_{\mathit{ijk}}\mathit{\hspace{0.17em}}\left(1\right)\mathit{\hspace{0.17em}}$(1) : The objective equation developed to minimize the total cost of delivery.

(1) $min\mathit{z}={\sum }_{\begin{array}{c}i\ne j\\ i=1\end{array}}^N{\sum }_{j=1}^N{\sum }_{k=1}^K\mathit{F}{C}_k{x}_{\mathit{ijk}}+{\sum }_{i\ne j}^{\mathrm{N}}{\sum }_{j=1}^N\mathit{Dis}{t}_{\mathit{ij}}{\sum }_{k=1}^{\mathrm{N}}\mathit{V}{C}_k\cdot x_{\mathit{ijk}}+{\sum }_{\begin{array}{c}i\ne j\\ i\ne 1\\ j\ne 1\end{array}}^{\mathrm{N}}{\sum }_{j\ne 1}^N{\sum }_{k=1}^{\mathrm{N}}\mathit{D}{C}_k\cdot x_{\mathit{ijk}}\mathit{\hspace{0.17em}}\left(1\right)\mathit{\hspace{0.17em}}$(1)

This equation represents the sum of the fixed cost of the chosen vehicle type, the variable costs of the vehicle as a function of the total distance of the delivery routes, and the cost of the transport vehicles, which varies according to the number of delivery points that each vehicle visits.

3.7. Constraint functions

Equations (2)(2) ${\sum }_{k=1}^{\mathrm{K}}{\sum }_{i\ne j}^{\mathrm{N}}{x}_{\mathit{ijk}}\ge 1;\mathrm{\forall j}$(2) to (7(7) ${\sum }_{i\ne j}^{\mathrm{N}}{\sum }_{j=1}^N\mathit{Dis}{t}_{\mathit{ij}}{\sum }_{k=1}^{\mathrm{N}}{x}_{\mathit{ijk}}\le \mathit{Mdist};\mathrm{\forall k}$(7) ) are the equations for the routing of each vehicle, representing the delivery order from $X_{\mathit{ijk}}$ when $X_{\mathit{ijk}}$ is 1. Equations (8)(8) ${\sum }_{k=1}^{\mathrm{K}}{D}_{\mathit{ikl}}=\mathit{D}{M}_{\mathit{il}};\mathrm{\forall i},\mathit{l}$(8) to (10(10) ${\sum }_{i=1}^N{D}_{\mathit{ilk}}=0;\mathrm{\forall l},\mathit{k}\in \mathit{L},\mathit{K}|\mathit{rul}{e}_{\mathit{lk}}=1\mathrm{\forall l}<3,\mathit{k}=3$(10) ) represent the product loading on each vehicle, where $D_{\mathit{ikl}}$ represents how much product each vehicle carries for chosen orders. Equations (11)(11) ${\sum }_{j=1}^{\mathrm{N}}(S_{\mathit{jikl}}-S_{\mathit{ijkl}})=D_{\mathit{ikl}};\mathrm{\forall i}>1,\mathit{k},\mathit{l}$(11) and (12(12) ${\sum }_{j=1}^{\mathrm{N}}{\sum }_{l=1}^{\mathrm{L}}(S_{\mathit{jikl}}-S_{\mathit{ijkl}})=0;\mathrm{\forall i}>1,\mathit{k}$(12) ) are subtour elimination equations. Equations (13)(13) ${\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{1\mathit{jkl}}\le \mathit{Ca}{p}_k;\mathrm{\forall j}>1,\mathit{k}$(13) and (14(14) ${\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{\mathit{ijkl}}\le x_{\mathit{ijk}}\cdot \mathit{Ca}{p}_k;\mathrm{\forall i}\ne \mathit{j},\mathit{j},\mathit{k}$(14) ) are capacity limitations and Equation (15)(15) ${\sum }_{i=1}^{\mathrm{N}}{\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{\mathit{i}1\mathit{kl}}=0;\mathrm{\forall k}$(15) ensures that no products remain when the trucks return to the depot. Equation (16)(16) $x_{\mathit{ijk}}\in \left\{0,1\right\};\mathrm{\forall i},\mathit{j},\mathit{k}$(16) is a binary constraint. Equations (17)(17) $D_{\mathit{ikl}}\in \left\{\mathit{I}\right\};\mathrm{\forall i},\mathit{k},\mathit{l}$(17) and (18(18) $S_{\mathit{ijkl}}\in \left\{\mathit{I}\right\};\mathrm{\forall i},\mathit{j},\mathit{k},\mathit{l}$(18) ) are integer constraints.

(2) ${\sum }_{k=1}^{\mathrm{K}}{\sum }_{i\ne j}^{\mathrm{N}}{x}_{\mathit{ijk}}\ge 1;\mathrm{\forall j}$(2)

According to Equation (2)(2) ${\sum }_{k=1}^{\mathrm{K}}{\sum }_{i\ne j}^{\mathrm{N}}{x}_{\mathit{ijk}}\ge 1;\mathrm{\forall j}$(2) , every order can be split more than one trip.

(3) ${\sum }_{i=1}^N{x}_{\mathit{ipk}}-{\sum }_{j=1}^{\mathrm{N}}{x}_{\mathit{pjk}}=0;\mathrm{\forall k},\mathit{p}$(3)

Equation (3)(3) ${\sum }_{i=1}^N{x}_{\mathit{ipk}}-{\sum }_{j=1}^{\mathrm{N}}{x}_{\mathit{pjk}}=0;\mathrm{\forall k},\mathit{p}$(3) requires that the number of vehicles entering a delivery point must be equal to the number of vehicles leaving a delivery point.

(4) ${\sum }_{j=1}^{\mathrm{N}}{x}_{\mathit{ijk}}\le 1;\mathrm{\forall i}>1,\mathit{k}$(4)

Equation (4)(4) ${\sum }_{j=1}^{\mathrm{N}}{x}_{\mathit{ijk}}\le 1;\mathrm{\forall i}>1,\mathit{k}$(4) requires that each vehicle can only deliver each order once.

(5) ${\sum }_{j=1}^{\mathrm{N}}{x}_{\mathit{jik}}\le 1;\mathrm{\forall i}>1,\mathit{k}$(5)

Equation (5)(5) ${\sum }_{j=1}^{\mathrm{N}}{x}_{\mathit{jik}}\le 1;\mathrm{\forall i}>1,\mathit{k}$(5) requires that every vehicle can only travel from the construction material store once.

(6) ${\sum }_{j=2}^N{x}_{1\mathit{jk}}\le 1;\mathrm{\forall k}$(6)

Equation (6)(6) ${\sum }_{j=2}^N{x}_{1\mathit{jk}}\le 1;\mathrm{\forall k}$(6) requires that each vehicle can only travel to the construction material store once.

(7) ${\sum }_{i\ne j}^{\mathrm{N}}{\sum }_{j=1}^N\mathit{Dis}{t}_{\mathit{ij}}{\sum }_{k=1}^{\mathrm{N}}{x}_{\mathit{ijk}}\le \mathit{Mdist};\mathrm{\forall k}$(7)

Equation (7)(7) ${\sum }_{i\ne j}^{\mathrm{N}}{\sum }_{j=1}^N\mathit{Dis}{t}_{\mathit{ij}}{\sum }_{k=1}^{\mathrm{N}}{x}_{\mathit{ijk}}\le \mathit{Mdist};\mathrm{\forall k}$(7) states that the delivery route for each vehicle cannot exceed the maximum distance traveled by the vehicle per day.

(8) ${\sum }_{k=1}^{\mathrm{K}}{D}_{\mathit{ikl}}=\mathit{D}{M}_{\mathit{il}};\mathrm{\forall i},\mathit{l}$(8)

Equation (8)(8) ${\sum }_{k=1}^{\mathrm{K}}{D}_{\mathit{ikl}}=\mathit{D}{M}_{\mathit{il}};\mathrm{\forall i},\mathit{l}$(8) requires that the total number of products that all vehicles deliver to the customers is equal to the product demand for each customer order.

(9) ${\sum }_{i=1}^{\mathrm{N}}{\sum }_{l=1}^{\mathrm{L}}{D}_{\mathit{ikl}}{W}_l\le \mathit{Ca}{p}_k;\mathrm{\forall k}$(9)

Equation (9)(9) ${\sum }_{i=1}^{\mathrm{N}}{\sum }_{l=1}^{\mathrm{L}}{D}_{\mathit{ikl}}{W}_l\le \mathit{Ca}{p}_k;\mathrm{\forall k}$(9) requires that the product weight of the orders on each vehicle does not exceed the maximum capacity of the vehicle.

(10) ${\sum }_{i=1}^N{D}_{\mathit{ilk}}=0;\mathrm{\forall l},\mathit{k}\in \mathit{L},\mathit{K}|\mathit{rul}{e}_{\mathit{lk}}=1\mathrm{\forall l}<3,\mathit{k}=3$(10)

Equation (10)(10) ${\sum }_{i=1}^N{D}_{\mathit{ilk}}=0;\mathrm{\forall l},\mathit{k}\in \mathit{L},\mathit{K}|\mathit{rul}{e}_{\mathit{lk}}=1\mathrm{\forall l}<3,\mathit{k}=3$(10) assures that only product l that satisfy the condition rule, $\mathit{rul}{e}_{\mathit{lk}}=$1, can be delivered by vehicle type $\mathit{k}$.

(11) ${\sum }_{j=1}^{\mathrm{N}}(S_{\mathit{jikl}}-S_{\mathit{ijkl}})=D_{\mathit{ikl}};\mathrm{\forall i}>1,\mathit{k},\mathit{l}$(11)

Equation (11)(11) ${\sum }_{j=1}^{\mathrm{N}}(S_{\mathit{jikl}}-S_{\mathit{ijkl}})=D_{\mathit{ikl}};\mathrm{\forall i}>1,\mathit{k},\mathit{l}$(11) ensures that the number of all products of all orders delivered on the vehicles subtracted from the number of products remaining on each vehicle must be equal to the number of demands, eliminating subtour.

(12) ${\sum }_{j=1}^{\mathrm{N}}{\sum }_{l=1}^{\mathrm{L}}(S_{\mathit{jikl}}-S_{\mathit{ijkl}})=0;\mathrm{\forall i}>1,\mathit{k}$(12)

Equation (12)(12) ${\sum }_{j=1}^{\mathrm{N}}{\sum }_{l=1}^{\mathrm{L}}(S_{\mathit{jikl}}-S_{\mathit{ijkl}})=0;\mathrm{\forall i}>1,\mathit{k}$(12) states that the previous orders must be greater than the remaining orders on each vehicle.

(13) ${\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{1\mathit{jkl}}\le \mathit{Ca}{p}_k;\mathrm{\forall j}>1,\mathit{k}$(13)

Equation (13)(13) ${\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{1\mathit{jkl}}\le \mathit{Ca}{p}_k;\mathrm{\forall j}>1,\mathit{k}$(13) requires that the weight of all products of all types of orders on each vehicle leaving the construction material store must not exceed the loading capacity of the vehicle.

(14) ${\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{\mathit{ijkl}}\le x_{\mathit{ijk}}\cdot \mathit{Ca}{p}_k;\mathrm{\forall i}\ne \mathit{j},\mathit{j},\mathit{k}$(14)

Equation (14)(14) ${\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{\mathit{ijkl}}\le x_{\mathit{ijk}}\cdot \mathit{Ca}{p}_k;\mathrm{\forall i}\ne \mathit{j},\mathit{j},\mathit{k}$(14) constrains the total product weight on each vehicle so that it does not exceed the load capacity of the vehicle.

(15) ${\sum }_{i=1}^{\mathrm{N}}{\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{\mathit{i}1\mathit{kl}}=0;\mathrm{\forall k}$(15)

Equation (15)(15) ${\sum }_{i=1}^{\mathrm{N}}{\sum }_{l=1}^{\mathrm{L}}{W}_l\cdot S_{\mathit{i}1\mathit{kl}}=0;\mathrm{\forall k}$(15) stipulates that when returning to the construction material store, every vehicle must have no product.

(16) $x_{\mathit{ijk}}\in \left\{0,1\right\};\mathrm{\forall i},\mathit{j},\mathit{k}$(16)

Equation (16)(16) $x_{\mathit{ijk}}\in \left\{0,1\right\};\mathrm{\forall i},\mathit{j},\mathit{k}$(16) sets the solution value format of the $X_{\mathit{ijk}}$ parameter as a binary.

(17) $D_{\mathit{ikl}}\in \left\{\mathit{I}\right\};\mathrm{\forall i},\mathit{k},\mathit{l}$(17)

Equation (17)(17) $D_{\mathit{ikl}}\in \left\{\mathit{I}\right\};\mathrm{\forall i},\mathit{k},\mathit{l}$(17) states that the number of product deliveries is an integer.

(18) $S_{\mathit{ijkl}}\in \left\{\mathit{I}\right\};\mathrm{\forall i},\mathit{j},\mathit{k},\mathit{l}$(18)

Equation (18)(18) $S_{\mathit{ijkl}}\in \left\{\mathit{I}\right\};\mathrm{\forall i},\mathit{j},\mathit{k},\mathit{l}$(18) states that the number of delivery points is an integer.

3.7.1. Data used for testing

Table 2 shows the distances between construction material store and the delivery point are unequal distances. Let $\mathit{i}$ = 1 and $\mathit{j}$ = 1 be the construction material store:

Table 2. Distance matrix (km)

$\mathit{Dis}{\mathit{t}}_{\mathit{ij}}$	1	2	3	4	5	6	7	8	9	10	11	12
1	0	10	53	18	13	11	5	70	59	62	26	61
2	10	0	35	29	14	18	10	71	56	47	30	11
3	53	34	0	16	36	29	33	61	50	37	53	40
4	18	24	16	0	29	17	16	60	41	35	46	33
5	12	10	34	27	0	23	14	69	61	45	25	17
6	14	19	29	19	28	0	11	79	32	55	44	27
7	2	10	35	21	17	10	0	70	47	46	34	18
8	69	71	57	60	73	79	70	0	100	56	72	77
9	63	56	50	41	65	32	48	91	0	49	81	64
10	65	46	32	36	48	54	45	55	49	0	65	53
11	26	30	53	47	24	43	34	72	81	65	0	20
12	60	11	41	35	20	31	22	76	69	53	20	0

Table 3 shows the maximum payload of a pickup truck, box truck, and 6 wheel (6W) crane truck. The costs of each type of vehicle are fixed (THB per trip), variable costs of the vehicle are based on distance traveled (THB per km), and the delivery cost of the vehicle are based on the number of deliveries (THB per delivery point).

Table 3. Load weight and cost of each type of transport vehicle

${\mathit{k}}_{\mathit{\hspace{0.17em}}}$	Truck Type	$\mathit{Ca}{\mathit{p}}_{\mathit{k}}$	$\mathit{F}{\mathit{C}}_{\mathit{k}}$	$\mathit{V}{\mathit{C}}_{\mathit{k}}$	$\mathit{D}{\mathit{C}}_{\mathit{k}}$
1	Pickup	1,000	275	13	30
2	Box	1,500	300	14	30
3	6W Crane	5,000	3,600	29	50

Table 4 shows the weight of the products and the limitations of each type of goods. X indicates unloading and Y is loading. From the table, it is found that steel and wood planks can only be transported by a 6-wheel crane truck, while cement bags, tiles, and lightweight bricks can be delivered by every vehicle type.

Table 4. Product weight (kg) and limitations of each product

$\mathit{l}$	Product Type	${\mathit{W}}_{\mathit{l}}$	Pickup	Box	6W Crane
1	Steel	3.95	X	X	Y
2	Wood Planks	7	X	X	Y
3	Cement Bags	50	Y	Y	Y
4	Tiles	25	Y	Y	Y
5	Lightweight Brick	6.5	Y	Y	Y

Table 5 shows the level of demand for each type of product for each order (units). For example, $\mathit{D}{M}_{2,1}$ means that an order has 200 units of demand for Product Type 1. For instance, $\mathit{D}{M}_{2,2}\mathit{\hspace{0.17em}}$ indicates that an order has demand for 200 units of Product Type 2, and etc.

Table 5. Demand for each type of each order (units)

$\mathit{D}{\mathit{M}}_{\mathit{il}}$	1	2	3	4	5
1	0	0	0	0	0
2	200	0	0	0	0
3	0	50	0	10	0
4	0	0	20	0	0
5	0	0	0	40	100
6	0	0	0	15	0
7	200	50	5	5	20
8	0	0	0	5	200

3.7.2. Mathematical model test

The data in Tables 2 to 5 were input into a mathematical model developed using the LINGO program and the system used in this research is LINGO/WIN64 19.0.40.

The details of the solution values are shown in Equation (1)(1) $min\mathit{z}={\sum }_{\begin{array}{c}i\ne j\\ i=1\end{array}}^N{\sum }_{j=1}^N{\sum }_{k=1}^K\mathit{F}{C}_k{x}_{\mathit{ijk}}+{\sum }_{i\ne j}^{\mathrm{N}}{\sum }_{j=1}^N\mathit{Dis}{t}_{\mathit{ij}}{\sum }_{k=1}^{\mathrm{N}}\mathit{V}{C}_k\cdot x_{\mathit{ijk}}+{\sum }_{\begin{array}{c}i\ne j\\ i\ne 1\\ j\ne 1\end{array}}^{\mathrm{N}}{\sum }_{j\ne 1}^N{\sum }_{k=1}^{\mathrm{N}}\mathit{D}{C}_k\cdot x_{\mathit{ijk}}\mathit{\hspace{0.17em}}\left(1\right)\mathit{\hspace{0.17em}}$(1) . The sum of the delivery costs of all transport vehicles is 9,703 THB, which is equal to the objective value shown in Table 6:

Table 6. All objective function value and delivery planning results

${\mathit{x}}_{\mathit{ijk}}$	Distance (km)	$\mathit{F}{\mathit{C}}_{\mathit{k}}$	$\mathit{V}{\mathit{C}}_{\mathit{k}}$	$\mathit{D}{\mathit{C}}_{\mathit{k}}$	Total cost (THB)
1,4,1	18	275	234		509
4,1,1	18		234		234
1,8,2	70	300	980		1,280
8,7,2	70		980		980
7,1,2	2		28	30	58
1, 6, 3	11	3,600	319		3,919
6, 3, 3	29		841		841
3, 5, 3	36		1,044	50	1,094
5, 2, 3	10		290	50	340
2, 7, 3	10		290	50	340
7, 1, 3	2		58	50	108

According to the results, the delivery routes for each vehicle were developed and the results in Table 6 were plotted as shown in Figure 1. For a pickup truck, the delivery route was 1-4-1. The delivery route for a box truck was 1-8-7-1, and the route for a 6-wheel crane truck was 1-6-3-5-2-7-1, respectively.

Figure 1. Delivery routes of the different vehicles, where node 1 represents the construction material store and nodes 2–8 are customer locations.

The details of the number of products for each vehicle loading and loading weight as the results from Tables 2 to 6 are shown in Table 7.

Table 7. The number of products for each vehicle loading

${\mathit{k}}_{\mathit{\hspace{0.17em}}}$	${\mathit{D}}_{\mathit{ikl}}$	${\mathit{D}}_{\mathit{ikl}}\mathit{\hspace{0.17em}}\left(\mathit{units}\right)$	${\mathit{W}}_{\mathit{l}}$	${\mathit{D}}_{\mathit{ikl}}{\mathit{W}}_{\mathit{l}}$	${\displaystyle \sum _{\mathit{i}=1}^8\mathit{ }}{\displaystyle \sum _{\mathit{l}=1}^5{D}_{\mathit{ikl}}}{W}_l$	$\mathit{Ca}{\mathit{p}}_{\mathit{k}}$	$\mathbf{\%}\mathit{Capacityutilizatio}{\mathit{n}}_{\mathit{\hspace{0.17em}}}$
1	4,1,3	20	50.00	1,000.00	1,000.00	1,000	100.00
2	7,2,4	2	25.00	50.00	1,488.00	1,500	99.20
	8,2,4	5	25.00	125.00
	7,2,5	2	6.50	13.00
	8,2,5	200	6.50	1,300.00
3	2,3,1	200	3.95	790.00	4,997.00	5,000	99.94
	7,3,1	200	3.95	790.00
	3,3,2	50	7.00	350.00
	7,3,2	50	7.00	350.00
	7,3,3	5	50.00	250.00
	3,3,4	10	25.00	250.00
	5,3,4	40	25.00	1,000.00
	6,3,4	15	25.00	375.00
	7,3,4	3	25.00	75.00
	5,3,5	100	6.50	650.00
	7,3,5	18	6.50	117.00

Table 7 shows that the warehouse operator must arrange 20 units of cement bags for a pickup truck, 7 units of tiles and 202 units of lightweight bricks for a box truck, and 400 units of steel, 100 units of wood plank, 5 units of cement bags, 68 units of tile, and 118 units of lightweight bricks for a 6-wheel crane truck. These results from Table 7 can also be vividly illustrated in Figure 2.

Figure 2. Vehicle loading for different vehicle types.

In this research, 20 scenarios, ranging from small to medium deliveries, were used to test the proposed model. Table 8 shows the results of these scenarios.

Table 8. The results of 20 scenarios

Scenario	# Customer	# Vehicle	# Product	Runtime (hh:mm:ss)
1	5	3	10	0:00:04
2	6	3	10	0:00:07
3	7	3	5	0:01:22
4	8	3	10	0:00:51
5	9	7	10	0:07:46
6	10	3	10	0:04:42
7	11	3	10	0:27:15
8	12	3	10	0:43:22
9	13	3	5	0:01:28
10	14	6	5	0:00:05
11	16	1	1	0:00:02
12	20	1	1	0:00:04
13	25	1	1	0:00:02
14	31	1	1	3:45:40
15	32	1	1	3:38:26
16	33	1	1	3:35:06
17	34	1	1	2:02:10
18	35	1	1	8:09:02
19	9	20	10	8:49:43
20	9	10	20	2:48:27

The proposed model is capable of solving for as large as 34 customers with the reasonable computational times. For the complete data sets of the 20 scenarios used to input to the model can been seen at the extended data (https://drive.google.com/drive/folders/1kthUDAepT48a2wGE3mX2zuc8NoDM07dU?usp=sharing).

4. Result and discussion

The results displayed vehicle type, number of products to be loaded and shipped to customers, and the delivery route. In this study, a mathematical model was developed to plan multiple shipments from a single warehouse to multiple customers using various trucks. Ultimately, the resulting delivery plan selected the appropriate type of transport for each product, achieving the lowest shipping cost. This method was compared to the conventional approach in which skilled and experienced staff manually defined daily shipping plans.

The advantage of the model in this study is that it realizes the lowest total delivery cost for delivery planning. The delivery plan outlines the number of each product to load on each vehicle and delivery routing to each customer for each vehicle. Additionally, the proposed model outperforms the existing method, as computational time decreased by 80% and the total cost decreased by 20%.

The operation models can also be used to accurately plan and deliver orders to customers according to the applied constraints and limitations, which include:

All vehicles must begin and end their journeys at the construction material store; the store operates with three different sizes of outsourced vehicles; up to five different products are used in this model; some product types can only be delivered by a 6-wheel crane truck; the total weight of each load must not exceed the vehicle’s load capacity; all products are ready for delivery, and all customers are prepared to receive products at any time; all vehicles and drivers are available to work at any time; vehicles can travel no more than 480 km per trip, as regulated by Thailand’s transportation law; and product placement and loading/unloading times are negligible.

Moreover, one of the many advantages of this proposed model is its adaptability. The model can include more constraints and parameters to handle larger problems or cater to similar business units. Additionally, the proposed model is simple to extend and user-friendly for new users, while still providing accurate results and relatively short calculation times when compared to existing case methods.

5. Conclusions

In this study, a mathematical model was developed to plan the delivery of goods according to specific conditions and constraints. The resulting delivery plan selected the appropriate type of transport for each product type. This led to more efficient delivery methods, which resulted in an average reduction in total delivery distance of 19% and an average reduction in shipping costs of 20% when compared to the current method. Therefore, by arranging the appropriate shipping order and choosing the right truck for each order, total shipping costs were reduced by determining which trucks to use and their delivery routes. This model is beneficial to operators involved in various departments with delivery activities, such as planning, warehousing, transportation, and accounting, among others.

However, the proposed model can be valid only for this case or other similar small-scale cases. If a larger scale problem is encountered, the developed model might not be sufficient. Consequently, heuristic methods and different algorithms should be further examined to address large-scale problems while maintaining reasonable computational times (Abdulkader et al., 2018; Wang et al., 2020).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by Research and Graduate Studies, Khon Kaen University and the Supply Chain and Logistics System Research Unit, Khon Kaen University.

Notes on contributors

Siranya Prachyathawornkul

Siranya Prachyathawornkul is currently a M.Eng. student in the Department of Industrial Engineering, Faculty of Engineering, Khon Kaen University, Thailand. She received her B.E. in industrial engineering from Khon Kaen University in 2015. Her research of interest includes supply chain & logistics optimization.

Arthit Apichottanakul

Arthit Apichottanakul has been working as an assistant professor in the Department of Production System Technology and Industrial Management, Faculty of Technology, Khon Kaen University, Thailand. He completed B.Sc in Statistics from Mahasarakam Univeristy and M.E. and Ph.D. in Industrial engineering both from Khon Kaen University. His current research interest includes intelligent applications, optimization and data science in logistics and supply chain management.

Komkrit Pitiruek

Komkrit Pitiruek has been working as an associate professor in the Department of Industrial Engineering, Faculty of Engineering, Khon Kaen University, Thailand. He completed his undergrad degree from King Mongkut University of Technology North Bangkok and Ph.D. in industrial and systems engineering from Auburn University, USA. His research interest includes the applications of engineering economics, engineering statistics, and operations research.