An optimization model for the operations of a pallet pool with both radio-frequency identification

Abstract

In a pallet pool, pallets would be delivered through a supply chain. The operation procedure that consists of at least five operation processes as distribution, reposition, recycling, purchase (or rent), and maintenance is quite complex. These pallets are likely to be damaged, lost, destroyed, and so on. So, it is necessary to monitor the pallets using radio-frequency identification technology. However, there is no literature on the management of a pallet pool with both radio-frequency identification–tagged pallets and non-tagged pallets being put into consideration. In our research, an optimization model is presented to manage such a pallet pool. The objective of the optimization model is to minimize the total operation cost of a pallet pool including distribution cost, reposition cost, recycling cost, purchase or rent cost, loss cost, maintenance cost, loading and unloading cost, storage cost, and punishment cost. A particle swarm optimization algorithm is developed in Microsoft Visual Basic. Our numerical example shows that the optimization model and particle swarm optimization algorithm are effective. It is proved that the model and algorithm also can be used to measure whether the investment of a radio-frequency identification system is valuable or not. We proposed some suggestions for the pallet pools management.

Keywords

Pallet radio-frequency identification closed-loop system particle swarm optimization logistics

Introduction

Pallets are the most popular equipments in the logistics industry. There are about 2 billion pallets in circulation in the United States, 280 million pallets in the European Union, and 1.2 billion pallets in China.^1–3 The most common pallet materials are wood and plastic.^4–6 About 80% of pallets are wooden pallets. Flat pallets are the most common structure of pallets. The dimensions of flat pallets are various. For example, 1016 mm × 1219 mm (famous as 40 in × 48 in) is the most commonly used in the North America, 1000 mm × 1200 mm in Europe, and 1100 mm × 1100 mm in Asia. As important assets, pallets always are managed very carefully. There are three industry strategies for managing pallets as follows: open-loop system with no salvage value (single-use expendable pallet system), open-loop system with salvage value (buy and sell system), and closed-loop rental system (pallet pool system).^2,7 The closed-loop system is the most popular.^8,9

There is one issue which ever plagues the pallet industry—proprietary pallets that have leaked out of a closed-loop system.¹⁰ Ilic et al.¹¹ reported that the annual loss rate of pallets was about 10%. Moreover, LeBlanc¹² and Brindley¹³ found that a lot of pallets in CPC (Canadian Pallet Association) and PECO (PECO Pallet Pooling Co., Ltd) were damaged. In order to reduce the loss rate and damage rate, radio-frequency identification (RFID) technology is always used.¹⁴ Intelligent Global Pooling Systems (iGPS, a leading provider of pallet and pooling services) is the first pallet pool operator in the world to build an advanced tracking and management system that take the advantage of RFID. After about 2 months, in October 2006, Commonwealth Handling Equipment Pool (CHEP, an iGPS competitor) announced that it would also deploy an RFID solution which allowed CHEP to offer track-and-trace asset visibility solutions to its customers. Now, iGPS is the operator of the world’s first 100% RFID-tagged all-plastic pallet pool, while CHEP is the leading pool providing both RFID-tagged plastic and wooden pallets. Moreover, almost all of the pallet pools could provide RFID-tagged pallets.

In fact, RFID could not only reduce the loss and damage rate but also help the managers to make scientific decisions. For example, using RFID, the managers of a pallet pool could get more information about their pallets so as to make a reasonable and scientific pallet allocation planning against the uncertainty of some parameters (such as the position and condition of the pallets).

RFID have been deployed in many fields. In the manufacturing management field, Barenji et al.¹⁵ discussed a multi-agent architecture devised to deploy RFID-enabled distributed control and monitoring system for a manufacturing shop in 2013. Then, a multi-agent-based architecture for scheduling and control in the manufacturing industry was designed and developed based on RFID.¹⁶ This scheduling and control system was tested, and the results highlighted the potential of employing both the multi-agent system and RFID technology as new paradigms for retrofitting the current manufacturing system.¹⁷ A simulation test platform also had been developed, and it was proved that the RFID-enabled multi-agent scheduling and control system could increase the uptime productivity and production rate of a flexible assembly line.^18,19 Recently, an RFID-enabled multi-agent-based manufacturing control system integrated with indirect coordination mechanism had been presented, and it performed better in comparison with the RFID-enabled multi-agent-based manufacturing control system integrated with direct coordination mechanism.²⁰ In the warehouse management field, Choy et al.²¹ proposed an RFID-based storage assignment system which could enhance the efficiency of order picking in a warehouse. Zhou et al.²² presented a novel RFID localization approach for pallets checking-in in a warehouse. In the supply chain management field, although Veronneau and Roy²³ stated that RFID could not achieve direct gains significant enough on a pallet-level-tagging deployment to justify the expenditure, a lot of researchers proved that the implementation of RFID practices significantly positively affect the supply chain performance in the areas as supplier, inventory, distribution, and so on.^24,25 It not only enables the supply chain partners to improve their utilities but also promotes the efficiency of supply chain management as a whole.²⁶

There are several literatures on the operations of a pallet pool. Ren and Zhang^27,28 proposed an integer programming model and a stochastic programming model for the pallet recycling over a pallet pool. Then, they studied on the pallet allocation model with the method of stochastic chance constrained programming.²⁹ In this model, they considered distribution, reposition, and recycling. Ren et al.⁸ pointed out a multi-scenario model for multi-type pallet allocation over a pallet pool considering the uncertain future without adequate historical data. In this model, the processes of pallet allocation include purchase (or rent), distribution, reposition, and recycling. A pallet allocation model over a railway pallet pool was presented by Zhou et al.³⁰ They also established a pallet allocation model for a normal pallet pool in which allocation time was considered.³¹ Wen and Qing³² proposed a reverse logistics network model for rental pallet recycling considering transport capacity. Based on the lifecycle analysis of pallets in a pallet pool, Ni et al.³³ modeled the pallet allocation considering production, distribution, reposition, recycling, and maintenance. Doungpattra et al.³⁴ studied on how to minimize the cost of pallet allocation over a pallet pool in the pet food industry. However, there is no literature on the management of a pallet pool with both RFID-tagged pallets and non-tagged pallets. This article will focus on the topic.

In section “Problem description,” the problem addressed is described. In section “Model development,” an optimization model for the problem is proposed. A case study is introduced in section “Numerical example and analysis.” Some suggestions for the managers are given in section “Assessing the value of an RFID system.”

Problem description

In a pallet pool, pallets would be moved through a supply chain. As shown in Figure 1, the operation processes are quite complex. There are at least five procedures including distribution, reposition, recycling, purchase (or rent), and maintenance.^29,35,36 The pallets may be damaged, lost, destroyed, and so on:

A customer requires the number of $d$ pallets from a pallet rental service supplier (supplier). The supplier moves the right number of pallets from the right service station to the customer (distribution).

The customer loads cargos on these pallets and moves these loaded pallets to the destination. After unloading cargos from these pallets, the customer requires the supplier to take these pallets away.

The supplier (or a recycler who is the partner of the supplier) checks the quantity and quality of these pallets. The number of $(1 - α) d$ pallets is lost, where $1 - α$ is the loss rate. The number of $α β d$ pallets is in good condition, so $1 - β$ is the damage rate. The supplier (or a recycler) delivers the $α β d$ pallets to the customer who requires pallets (reposition).

The supplier moves the number of $α (1 - β) d$ pallets to a maintenance station. The number of $α (1 - β) (1 - γ) d$ pallets has to be disposed because of unable to be recovered. $γ$ is the rate of damaged pallets could be recovered. The number of $α (1 - β) γ d$ pallets is moved to a service station for reusing (recycling).

If there are not enough pallets to meet the demand $(α (1 - β) γ d + S + α β d \leq d)$ , the supplier will have to purchase (or rent) the number of $(1 - α β - α (1 - β) γ) d - S$ pallets from a manufacturer or pallet rental company (purchase or rent). $S$ is the number of pallets stock at the supplier.

The customer moves the loaded pallets with cargos to the destination. Then, the customer requires the supplier to take these pallets away after cargos unloaded.

The following steps are as steps 3–5.

Figure 1.

The operation processes of a pallet pool.

As mentioned above, the loss rate and damage rate of pallets in a pallet pool are quite high. In fact, the loss rate is about 15%–20%, according to Deloitte. And, the damage rate of wooden pallets is more than 22%, according to a Chinese report. Consequently, it is hard to estimate how many pallets will be returned and whether they are in good condition. The application of RFID technology can significantly improve the visibility of the supply chain. Specifically, it is helpful to improve the accuracy of the value of $α$ and $β$ . This is one of the reasons that more and more pallet service suppliers prefer to use RFID technology.

Model development

Objective function

The objective of the optimization model is to minimize the total operation cost of a pallet pool. The managers of 10 pallet pools in China were visited. The cost list suggested by them is as follows:

The transportation cost of pallets from supplier’s service stations to demand customers (this kind of customers require pallets for freights carriage). It also could be named as distribution cost. $X_{i j^{0}}^{t' t}$ is the number of non-tagged pallets moved from $i$ (a supplier’s service station) at time $t'$ to $j^{0}$ (a demand customer) at time $t$ ${XR}_{i j^{0}}^{t' t}$ is the number of RFID-tagged pallets moved from $i$ at time $t'$ to $j^{0}$ at time $t$ . The unitary transportation cost from $i$ to $j^{0}$ of the two kinds of pallets is the same, and it is represented as $C_{i j^{0}}$ .

The transportation cost of pallets from supply customers (this kind of customers have many pallets which need to be taken away; in Figure 1, the customer at destination is a supply customer) to demand customers. It also could be named as reposition cost. $X_{j^{1} j^{0}}^{t' t}$ indicates the number of non-tagged pallets moved from $j^{1}$ (a supply customer) at time $t'$ to $j^{0}$ at time $t$ . ${XR}_{j^{1} j^{0}}^{t' t}$ indicates the number of RFID-tagged pallets moved from $j^{1}$ at time $t'$ to $j^{0}$ at time $t$ . The unitary transportation cost from $j^{1}$ to $j^{0}$ of the two kinds of pallets is the same, and it is represented as $C_{j^{1} j^{0}}$ .

The transportation cost of pallets from supply customers to supplier’s service stations. This kind of cost also could be named as recycling cost. $X_{j^{1} i}^{t' t}$ means the number of non-tagged pallets moved from $j^{1}$ at time $t'$ to $i$ at time $t$ . ${XR}_{j^{1} i}^{t' t}$ means the number of RFID-tagged pallets moved from $j^{1}$ at time $t'$ to $i$ at time $t$ . The unitary transportation cost from $j^{1}$ to $i$ of the two kinds of pallets is the same, and it is represented as $C_{j^{1} i}$ .

The total purchase or rent cost. $H_{i}^{t}$ implies the number of non-tagged pallets purchased from a manufacturer (or rent from a rental company) by a supplier’s service station $i$ at time $t$ . $C_{h}$ represents the unitary purchase (or rent) cost of non-tagged pallets. ${HR}_{i}^{t}$ implies the number of RFID-tagged pallets purchased from a manufacturer (or rent from a rental company) by a supplier’s service station $i$ at time $t$ . $C R_{h}$ represents the unitary purchase (or rent) cost of RFID-tagged pallets.

The total loss cost. $(1 - L_{j^{1}})$ is the loss rate of non-tagged pallets at $j^{1}$ . $(1 - L_{j^{1}}) S_{j^{1}}^{t}$ indicates the number of lost non-tagged pallets at $j^{1}$ at time $t$ . $C_{l}$ is the loss cost of a non-tagged pallet. $(1 - L R_{j^{1}})$ is the loss rate of RFID-tagged pallets at $j^{1}$ . $(1 - L R_{j^{1}}) {SR}_{j^{1}}^{t}$ indicates the number of lost RFID-tagged pallets at $j^{1}$ at time $t$ . $C R_{l}$ is the loss cost of an RFID-tagged pallet.

The total maintenance cost. $(1 - M)$ means the damage rate of non-tagged pallets. $P$ shows the recovery percentage of damaged non-tagged pallets. $P (1 - M) X_{j^{1} i}^{t' t}$ indicates how many damaged non-tagged pallets which are moved from $j^{1}$ at time $t'$ to $i$ at time $t$ are recovered. $C_{m}$ and $C_{mp}$ represents the maintenance cost of a recovered and un-recovered non-tagged pallet, respectively. $(1 - MR)$ means the damage rate of RFID-tagged pallets. $PR$ shows the recovery percentage of damaged RFID-tagged pallets. $PR (1 - M R_{j^{1}}) {XR}_{j^{1} i}^{t' t}$ indicates how many damaged RFID-tagged pallets which are moved from $j^{1}$ at time $t'$ to $i$ at time $t$ are recovered. $C R_{m}$ and $C R_{mp}$ represents the maintenance cost of a recovered and un-recovered RFID-tagged pallet, respectively.

The loading and unloading cost at all supplier’s service stations. The unitary loading and unloading cost at $i$ is represented by $C_{li}$ , and the loading and unloading amount is calculated by $\sum_{j^{0} = 1}^{J^{0}} X_{i j^{0}}^{t' t} + \sum_{j^{0} = 1}^{J^{0}} {XR}_{i j^{0}}^{t' t} + \sum_{j^{1} = 1}^{J^{1}} X_{j^{1} i}^{t' t} + \sum_{j^{1} = 1}^{J^{1}} {XR}_{j^{1} i}^{t' t}$ .

The loading and unloading cost at all supply customers. The unitary loading and unloading cost at $j^{1}$ is represented by $C_{l j^{1}}$ , and the loading and unloading amount is calculated by $\sum_{j^{0} = 1}^{J^{0}} X_{j^{1} j^{0}}^{t' t} + \sum_{j^{0} = 1}^{J^{0}} {XR}_{j^{1} j^{0}}^{t' t} + \sum_{i = 1}^{I} X_{j^{1} i}^{t' t} + \sum_{i = 1}^{I} {XR}_{j^{1} i}^{t' t}$ .

The loading and unloading cost at all demand customers. The unitary loading and unloading cost at $j^{0}$ is represented by $C_{l j^{0}}$ , and the loading and unloading amount is calculated by $\sum_{i = 1}^{I} X_{i j^{0}}^{t' t} + \sum_{i = 1}^{I} {XR}_{i j^{0}}^{t' t} + \sum_{j^{1} = 1}^{J^{1}} X_{j^{1} j^{0}}^{t' t} + \sum_{j^{1} = 1}^{J^{1}} {XR}_{j^{1} j^{0}}^{t' t}$ .

The total storage cost at all supplier’s service stations. The number of non-tagged and RFID-tagged pallets stored at $i$ at time $t$ is $K_{i}^{t}$ and ${KR}_{i}^{t}$ , respectively. The unitary storage cost of the two kinds of pallets at $i$ is the same, and it is represented as $C_{Ki}$ .

The total punishment cost of failing to take away all pallets from all supply customers. We assumed the supplier could fail to take away all pallets if the requests are uncertain. The number of non-tagged and RFID-tagged pallets stored at $j^{1}$ at time $t$ is $K_{j^{1}}^{t}$ and ${KR}_{j^{1}}^{t}$ , respectively. The unitary storage cost of the two kinds of pallets at $j^{1}$ is the same, and it is represented as $C_{K j^{1}}$ .

The total punishment cost of exceeding the demands for all pallets of all demand customers. We assumed the supplier could exceed the demands if the demands were uncertain. The number of non-tagged and RFID-tagged pallets stored at $j^{0}$ at time $t$ is $K_{j^{0}}^{t}$ and ${KR}_{j^{0}}^{t}$ , respectively. The unitary storage cost of the two kinds of pallets at $j^{0}$ is the same, and it is represented as $C_{K j^{0}}$ .

Therefore, the objective function could be described as equation (1)

\begin{array}{l} \min f = \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{i = 1}^{I} \sum_{j^{0} = 1}^{J^{0}} C_{i j^{0}} X_{i j^{0}}^{t' t} + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{j^{1} = 1}^{J^{1}} \sum_{j^{0} = 1}^{J^{0}} C_{j^{1} j^{0}} X_{j^{1} j^{0}}^{t' t} + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{j^{1} = 1}^{J^{1}} \sum_{i = 1}^{I} C_{j^{1} i} X_{j^{1} i}^{t' t} \\ + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{i = 1}^{I} \sum_{j^{0} = 1}^{J^{0}} C_{i j^{0}} X R_{i j^{0}}^{t' t} + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{j^{1} = 1}^{J^{1}} \sum_{j^{0} = 1}^{J^{0}} C_{j^{1} j^{0}} X R_{j^{1} j^{0}}^{t' t} + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{j^{1} = 1}^{J^{1}} \sum_{i = 1}^{I} C_{j^{1} i} X R_{j^{1} i}^{t' t} \\ + \sum_{t = t 1}^{T} \sum_{i = 1}^{I} C_{h} H_{i}^{t} + \sum_{t = t 1}^{T} \sum_{i = 1}^{I} C R_{h} H R_{i}^{t} \\ + \sum_{t = t 1}^{T} \sum_{j^{1} = 1}^{J^{1}} C_{l} (1 - L_{j^{1}}) S_{j^{1}}^{t} + \sum_{t = t 1}^{T} \sum_{j^{1} = 1}^{J^{1}} C R_{l} (1 - L R_{j^{1}}) S R_{j^{1}}^{t} \\ + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{i = 1}^{I} \sum_{j^{1} = 1}^{J^{1}} C_{m} P (1 - M) X_{j^{1} i}^{t' t} + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{i = 1}^{I} \sum_{j^{1} = 1}^{J^{1}} C R_{m} P R (1 - M R) X R_{j^{1} i}^{t' t} \\ + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{i = 1}^{I} \sum_{j^{1} = 1}^{J^{1}} C_{m p} (1 - P) (1 - M) X_{j^{1} i}^{t' t} + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{i = 1}^{I} \sum_{j^{1} = 1}^{J^{1}} C R_{m p} (1 - P R) (1 - M R) X R_{j^{1} i}^{t' t} \\ + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{i = 1}^{I} C_{l i} (\sum_{j^{0} = 1}^{J^{0}} X_{i j^{0}}^{t' t} + \sum_{j^{0} = 1}^{J^{0}} X R_{i j^{0}}^{t' t} + \sum_{j^{1} = 1}^{J^{1}} X_{j^{1} i}^{t' t} + \sum_{j^{1} = 1}^{J^{1}} X R_{j^{1} i}^{t' t}) \\ + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{j^{1} = 1}^{J^{1}} C_{l j^{1}} (\sum_{j^{0} = 1}^{J^{0}} X_{j^{1} j^{0}}^{t' t} + \sum_{i = 1}^{I} X_{j^{1} i}^{t' t} + \sum_{j^{0} = 1}^{J^{0}} X R_{j^{1} j^{0}}^{t' t} + \sum_{i = 1}^{I} X R_{j^{1} i}^{t' t}) \\ + \sum_{t' = t 1}^{t} \sum_{t = t'}^{T} \sum_{j^{0} = 1}^{J^{0}} C_{l j^{0}} (\sum_{i = 1}^{I} X_{i j^{0}}^{t' t} + \sum_{j^{1} = 1}^{J^{1}} X_{j^{1} j^{0}}^{t' t} \sum_{i = 1}^{I} X R_{i j^{0}}^{t' t} + \sum_{j^{1} = 1}^{J^{1}} X R_{j^{1} j^{0}}^{t' t}) \\ + \sum_{t = t 1}^{T} \sum_{i = 1}^{I} C_{K i} (K_{i}^{t} + K R_{i}^{t}) + \sum_{t = t 1}^{T} \sum_{j^{1} = 1}^{J^{1}} C_{K j^{1}} (K_{j^{1}}^{t} + K R_{j^{1}}^{t}) + \sum_{t = t 1}^{T} \sum_{j^{0} = 1}^{J^{0}} C_{K j^{0}} (K_{j^{0}}^{t} + K R_{j^{0}}^{t}) \end{array}

(1)

Constraints

As the survey results shown, the constraints should include distribution, reposition, demand, recycling, storage, transportation capacity, loading and unloading capacity, and so on.

Distribution

The number of non-tagged (or RFID-tagged) pallets moved from a pallet service supplier station to all demand customers cannot exceed the non-tagged (or RFID-tagged) pallets supply capacity. The supply capacity is the sum of the number of storage pallets, purchased (or rent) pallets, recycled pallets, and recovered pallets. The constraint sets equations (2) and (3) show the distribution constraints of non-tagged and RFID-tagged, respectively

\begin{array}{l} \sum_{t^{″} = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X_{i j^{0}}^{t t^{″}} \leq K_{i}^{t - 1} + \sum_{t^{'} = t 1}^{t} H_{i}^{t^{'} t} + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} M X_{j^{1} i}^{t^{'} t} \\ + {(\sum_{t^{‴} = t 1}^{t^{'}} \sum_{t^{'} = t^{‴}}^{t} \sum_{j^{1} = 1}^{J^{1}} P (1 - M) X_{j^{1} i}^{t^{‴} t^{'}})}^{t} \end{array}

(2)

\begin{array}{l} \sum_{t^{″} = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X R_{i j^{0}}^{t t^{″}} \leq K R_{i}^{t - 1} + \sum_{t^{'} = t 1}^{t} H R_{i}^{t^{'} t} + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} M X R_{j^{1} i}^{t^{'} t} \\ + {(\sum_{t^{‴} = t 1}^{t^{'}} \sum_{t^{'} = t^{‴}}^{t} \sum_{j^{1} = 1}^{J^{1}} P R (1 - M R) X R_{j^{1} i}^{t^{‴} t^{'}})}^{t} \end{array}

(3)

Reposition

The constraint set equation (4) guarantees that the number of non-tagged pallets moved from a supply customer to all demand customers does not exceed the available non-tagged pallets that are in good condition at this supply customer. The constraint set equation (5) guarantees that the number of RFID-tagged pallets moved from a supply customer to all demand customers does not exceed the available RFID-tagged pallets that are in good condition at this supply customer

\sum_{t ″ = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X_{j^{1} j^{0}}^{tt ″} \leq L_{j^{1}} (M) S_{j^{1}}^{t}

(4)

\sum_{t ″ = t}^{T} \sum_{j^{0} = 1}^{J^{0}} {XR}_{j^{1} j^{0}}^{tt ″} \leq L R_{j^{1}} (MR) {SR}_{j^{1}}^{t}

(5)

Demand

The constraint set equation (6) ensures that the requests of a demand customer have to be met using of non-tagged and RFID-tagged pallets

\begin{array}{l} \sum_{t^{'} = t 1}^{t} \sum_{i = 1}^{I} X_{i j^{0}}^{t^{'} t} + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} X_{j^{1} j^{0}}^{t^{'} t} + \sum_{t^{'} = t 1}^{t} \sum_{i = 1}^{I} X R_{i j^{0}}^{t^{'} t} \\ + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} X R_{j^{1} j^{0}}^{t^{'} t} \geq D_{j^{0}}^{t} \end{array}

(6)

Recycling

The number of pallets moved from a supply customer could not exceed the available pallets at this supply customer. The constraint sets equations (7) and (8) show the recycling constraints of non-tagged and RFID-tagged pallets, respectively

\sum_{t ″ = t}^{T} \sum_{i = 1}^{I} X_{j^{1} i}^{tt ″} \leq L_{j^{1}} S_{j^{1}}^{t} - \sum_{t ″ = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X_{j^{1} j^{0}}^{tt ″}

(7)

\sum_{t ″ = t}^{T} \sum_{i = 1}^{I} {XR}_{j^{1} i}^{tt ″} \leq L R_{j^{1}} {SR}_{j^{1}}^{t} - \sum_{t ″ = t}^{T} \sum_{j^{0} = 1}^{J^{0}} {XR}_{j^{1} j^{0}}^{tt ″}

(8)

Storage

The constraint sets equations (9) and (10) show the number of storage of non-tagged and RFID-tagged pallets at a supplier’s service station, respectively

\begin{matrix} K_{i}^{t} = K_{i}^{t - 1} + \sum_{t' = t 1}^{t} H_{i}^{t' t} + \sum_{t' = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} {MX}_{j^{1} i}^{t' t} \\ + {(\sum_{t ‴ = t 1}^{t'} \sum_{t' = t ‴}^{t} \sum_{j^{1} = 1}^{J^{1}} P (1 - M) X_{j^{1} i}^{t ‴ t'})}^{t} - \sum_{t ″ = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X_{i j^{0}}^{tt ″} \end{matrix}

(9)

\begin{matrix} {KR}_{i}^{t} = {KR}_{i}^{t - 1} + \sum_{t' = t 1}^{t} {HR}_{i}^{t' t} + \sum_{t' = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} {MXR}_{j^{1} i}^{t' t} \\ + {(\sum_{t ‴ = t 1}^{t'} \sum_{t' = t ‴}^{t} \sum_{j^{1} = 1}^{J^{1}} P R (1 - MR) {XR}_{j^{1} i}^{t ‴ t'})}^{t} \\ - \sum_{t ″ = t}^{T} \sum_{j^{0} = 1}^{J^{0}} {XR}_{i j^{0}}^{tt ″} \end{matrix}

(10)

According to constraint equation (11), pallets (including non-tagged and RFID-tagged pallets) stored at a supplier’s service station are not supposed to beyond its storage capacity

K_{i}^{t} + {KR}_{i}^{t} \leq K_{0 i}^{t}

(11)

The constraint sets equations (12) and (13) show the number of storage of non-tagged and RFID-tagged pallets at a supply customer, respectively

\begin{array}{l} K_{j^{1}}^{t} = K_{j^{1}}^{t - 1} + S_{j^{1}}^{t} - (1 - L_{j^{1}}) (S_{j^{1}}^{t}) \\ - \sum_{t^{″} = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X_{j^{1} j^{0}}^{t t^{″}} - \sum_{t^{″} = t}^{T} \sum_{i = 1}^{I} X_{j^{1} i}^{t t^{″}} \end{array}

(12)

\begin{array}{l} K R_{j^{1}}^{t} = K R_{j^{1}}^{t - 1} + S R_{j^{1}}^{t} - (1 - L R_{j^{1}}) (S R_{j^{1}}^{t}) \\ - \sum_{t^{″} = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X R_{j^{1} j^{0}}^{t t^{″}} - \sum_{t^{″} = t}^{T} \sum_{i = 1}^{I} X R_{j^{1} i}^{t t^{″}} \end{array}

(13)

The constraint sets equations (14) and (15) ensure the number of non-tagged and RFID-tagged pallets which have not been taken away is non-negative

K_{j^{1}}^{t} \geq 0

(14)

{KR}_{j^{1}}^{t} \geq 0

(15)

The constraint set equation (16) shows the number of pallets exceeding the requests of a demand customer

\begin{array}{l} K_{j^{0}}^{t} = \sum_{t^{'} = t 1}^{t} \sum_{i = 1}^{I} X_{i j^{0}}^{t^{'} t} + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} X_{j^{1} j^{0}}^{t^{'} t} \\ + \sum_{t^{'} = t 1}^{t} \sum_{i = 1}^{I} X R_{i j^{0}}^{t^{'} t} + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} X R_{j^{1} j^{0}}^{t^{'} t} - D_{j^{0}}^{t} \end{array}

(16)

Transportation capacity

The constraint sets equations (17)–(19) impose an upper transportation capacity on the number of the pallets that can be moved between two locations

X_{i j^{0}}^{t' t} + {XR}_{i j^{0}}^{t' t} \leq {MA}_{i j^{0}}^{t' t}

(17)

X_{j^{1} j^{0}}^{t' t} + {XR}_{j^{1} j^{0}}^{t' t} \leq {MA}_{j^{1} j^{0}}^{t' t}

(18)

X_{j^{1} i}^{t' t} + {XR}_{j^{1} i}^{t' t} \leq {MA}_{j^{1} i}^{t' t}

(19)

Loading and unloading capacity

The constraint sets equations (20)–(22) impose an upper loading and unloading capacity on the number of pallets that can be loaded or unloaded at a supplier’s service station, a supply customer, and a demand customer, respectively

\begin{array}{l} \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} X_{j^{1} i}^{t^{'} t} + \sum_{t^{″} = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X_{i j^{0}}^{t t^{″}} \\ + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} X R_{j^{1} i}^{t^{'} t} + \sum_{t^{″} = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X R_{i j^{0}}^{t t^{″}} \leq L_{i}^{t} \end{array}

(20)

\begin{array}{l} \sum_{t^{″} = t}^{T} \sum_{i = 1}^{I} X_{j^{1} i}^{t t^{″}} + \sum_{t^{″} = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X_{j^{1} j^{0}}^{t t^{″}} \\ + \sum_{t^{″} = t}^{T} \sum_{i = 1}^{I} X R_{j^{1} i}^{t t^{″}} + \sum_{t^{″} = t}^{T} \sum_{j^{0} = 1}^{J^{0}} X R_{j^{1} j^{0}}^{t t^{″}} \leq L_{j^{1}}^{t} \end{array}

(21)

\begin{array}{l} \sum_{t^{'} = t 1}^{t} \sum_{i = 1}^{I} X_{i j^{0}}^{t^{'} t} + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} X_{j^{1} j^{0}}^{t^{'} t} \\ + \sum_{t^{'} = t 1}^{t} \sum_{i = 1}^{I} X R_{i j^{0}}^{t^{'} t} + \sum_{t^{'} = t 1}^{t} \sum_{j^{1} = 1}^{J^{1}} X R_{j^{1} j^{0}}^{t^{'} t} \leq L_{j^{0}}^{t} \end{array}

(22)

The value of decision variables

The constraint set equation (23) indicates that all decision variables are non-negative integer values

X_{i j^{0}}^{t' t}, X_{j^{1} j^{0}}^{t' t}, X_{j^{1} i}^{t' t}, H_{i}^{t}, {XR}_{i j^{0}}^{t' t}, {XR}_{j^{1} j^{0}}^{t' t}, {XR}_{j^{1} i}^{t' t}, {HR}_{i}^{t} \geq 0, and int

(23)

Numerical example and analysis

Numerical examples

There are two supplier’s service stations $(i = a, b)$ , two supply customers $(j^{1} = c, d)$ , and two demand customers $(j^{0} = e, f)$ in a pallet pool. The dimension of all pallets in this pool is 1000 mm × 1200 mm. All pallets are made of wood. Some pallets are RFID-tagged, while the others are non-tagged. If there are not enough pallets, the supplier have to purchase pallets from a manufacturer and theses pallets will be delivered to the supplier’s service stations immediately. The purchase cost of a non-tagged pallet is 100, while the RFID-tagged pallet is 102. The loss cost is the same as purchase cost. The maintenance cost of a recovered pallet is 1, while an un-recovered pallet is 110. The damage rate of non-tagged pallets and RFID-tagged pallets is 0.1 and 0.01, respectively. The recovery percentage of damaged non-tagged pallets and RFID-tagged pallets is 0.9 and 0.99, respectively. The loss rate of non-tagged pallets at $c$ is random number between 0 and 0.3. The loss rate of RFID-tagged pallets at $c$ is random number between 0 and 0.1. Both the loss rate of non-tagged pallets and RFID-tagged pallets at $d$ is 0. There are none pallets that will arrive at the supplier’s service stations at the beginning of this period. The other parameters are shown in Tables 1 –4.

Table 1.

Transportation capacity.

	a	b	c	d	e	f
a	−	−	1000	1000	400	500
b	−	−	0	700	300	500
c	1000	0	−	−	400	700
d	1000	700	−	−	250	0
e	400	300	400	250	−	−
f	500	500	700	0	−	−

Table 2.

Unitary transportation cost.

	a	b	c	d	e	f
a	−	−	3	4	5	6
b	−	−	$\infty$	2	4	5
c	3	$\infty$	−	−	7	8
d	4	2	−	−	2	$\infty$
e	5	4	7	2	−	−
f	6	5	8	$\infty$	−	−

Table 3.

Demand and supply.

	The number of pallets stored at a supplier’s service station at the beginning of the period (non-tagged/RFID-tagged)	The number of pallets that have to be taken away from a supply customer (non-tagged/RFID-tagged)	The number of pallets requested by a demand customer
a	95/105	−	−
b	150/200	−	−
c	−	100/120	−
d	−	390/420	−
e	−	−	700
f	−	−	800

RFID: radio-frequency identification.

Table 4.

Other parameters.

	Storage capacity	Unitary holding cost (unitary punishment cost)	Loading and unloading capacity	Unitary loading and unloading cost
a	2000	1	8000	1
b	1800	2	7000	1
c	−	10	5000	1
d	−	10	4000	1
e	−	6	2000	1
f	−	6	2000	1

Particle swarm optimization algorithm and experimental results

Particle swarm optimization (PSO) is a novel swarm intelligence algorithm presented by Kennedy and colleagues.^37,38 It has been applied in a wide range of problems: vehicle routing, berth allocation, machine scheduling, order allocation, and empty container allocation.^39–42 PSO was inspired on the behavior of birds in flocks where solutions to a given optimization problem, called particles, “fly” (like birds) through a multidimensional search space.^43,44

The pallet pool operation problem is a non-deterministic polynomial-time (NP) problem. Due to the computational complexity, researchers have developed heuristics to solve the pallet pool operation model in the literature. Zhou et al.⁹ developed an immune clone algorithm (ICA) for a pallet allocation model, and it was proved more effective than the traditional genetic algorithm (GA). However, compared with some other evolutionary algorithms (EAs) as ICA and GA, PSO is easier to implement and fewer control parameters are needed.⁴⁵ To our best knowledge, none of research we found has used PSO to solve the pallet pool optimization problem. In that case, we developed a PSO algorithm for pallet pool operation. The procedures of the PSO algorithm are as follows.

Step 1. Randomly generate a population of particles with velocity. The population size is 40.⁴⁶ The initial position of each particle is given with $p_{ij}^{o} = int (1000 {rnd}_{ij}^{p})$ , while the initial velocity is calculated with ${v_{i}}_{j}^{o} = 8 ({rnd}_{ij}^{v} - 0.5)$ . ${rnd}_{ij}^{p}$ and ${rnd}_{ij}^{v}$ are random numbers between 0 and 1. In order to get the optimization scheme faster, the position of the 40th particle is instead of a feasible solution of the problem (as shown in Table 5).

Step 2. Calculate constraints and fitness function. Calculate constraint sets equations (2)–(23) as showed in section “Constraints.” If one of the constraints is not met, then set the fitness function with $f (i) = M$ . Where $M$ is a data with a very large value (we set M = 888,888). Otherwise, compute the fitness function which is the objective function as equation (1). ${\begin{matrix} if f (i) < f (pbes t_{ij}) then pbes t_{ij} = p_{ij} \\ if f (i) < f (gbes t_{i}) then gbes t_{i} = p_{ij} \end{matrix}$ , where $pbes t_{ij}$ is the particle’s own previous best searching experience and $gbes t_{i}$ is the previous best searching experience by all particles.

Step 3. Update velocity and positions. The velocity is updated with $v_{ij}^{n} = w * v_{ij}^{o} + c 1 * rn d_{1} * (pbes t_{ij} - p_{ij}^{o}) + c 2 * rn d_{2} * (gbes t_{i} - p_{ij}^{o})$ , while the particle position is updated with $p_{ij}^{n} = p_{ij}^{o} + v_{ij}^{n}$ . ${v_{i}}_{j}^{n}$ is the new particle velocity and $v_{ij}^{o}$ is the current particle velocity. $w = w_{min} + (w_{max} - w_{min}) (k / k_{max})$ is the inertia weight, where $k$ is the iterations, $k_{max} = 1000$ is the maximum number of iterations, $w_{min} = 0.4$ , and $w_{max} = 0.9$ . $c 1 = 2.5 - k / 1000$ is self-confidence, while $c 2 = 2.5 - k / 1000$ is swarm confidence. $rn d_{1}$ and $rn d_{2}$ are random numbers between 0 and 1 which represent the stochastic element of the PSO. $p_{ij}^{n}$ is the new particle position and $p_{ij}^{o}$ is the current particle position. The velocity should be between $v_{max}$ and $v_{min}$ . $v_{max} = 10$ is the maximum velocity, and $v_{min} = - 10$ is the minimization velocity. If $v_{ij}^{n} > v_{max}$ , then $v_{ij}^{n} = v_{max}$ . If $v_{ij}^{n} < v_{min}$ , then $v_{ij}^{n} = v_{min}$ .

Step 4. Stop the algorithm when the maximum number of iterations $(k_{max} = 1000)$ is reached; otherwise, go to step 2.

Table 5.

Initial position of the 40th particle.

$p_{40, 1}^{0}$	$p_{40, 2}^{0}$	$p_{40, 3}^{0}$	$p_{40, 4}^{0}$	$p_{40, 5}^{0}$	$p_{40, 6}^{0}$	$p_{40, 7}^{0}$
a-e,N	a-e,R	a-f,N	a-f,R	b-e,N	b-e,R	b-f,N
200	200	300	0	150	150	450
$p_{40, 8}^{0}$	$p_{40, 9}^{0}$	$p_{40, 10}^{0}$	$p_{40, 11}^{0}$	$p_{40, 12}^{0}$	$p_{40, 13}^{0}$	$p_{40, 14}^{0}$
b-f,R	c-e,N	c-e,R	c-f,N	c-f,R	d-e,N	d-e,R
50	0	0	0	0	0	0
$p_{40, 15}^{0}$	$p_{40, 16}^{0}$	$p_{40, 17}^{0}$	$p_{40, 18}^{0}$	$p_{40, 19}^{0}$	$p_{40, 20}^{0}$	$p_{40, 21}^{0}$
d-f,N	d-f,R	c-a,N	c-a,R	c-b,N	c-b,R	d-a,N
0	0	100	120	0	0	390
${p_{40,}}_{22}^{0}$	$p_{40, 23}^{0}$	$p_{40, 24}^{0}$	$p_{40, 25}^{0}$	$p_{40, 26}^{0}$	$p_{40, 27}^{0}$	$p_{40, 28}^{0}$
d-a,R	d-b,N	d-b,R	a,P,N	a,P,R	b,P,N	b,P,R
420	0	0	405	95	450	0

N: Non-tagged pallet; R: RFID-tagged pallet; P: purchase.

For example, $p_{40, 1}^{0}$ indicates move non-tagged pallets from a to e, and the number of pallets is 200 because its value is 200, and $p_{40, 28}^{0}$ indicates b purchase RFID-tagged pallets, and the number of pallets is 0 because its value is 0.

The PSO algorithm was developed in Microsoft Visual Basic 6.0 and ran on a PC with a 2.50-GHz Intel CPU and 4.00 GB RAM, under the Windows 7 operating system. The result shows the minimization operation cost is 66,577. And, the optimization scheme is shown in Table 6.

Table 6.

Optimization scheme of the example.

	a	b	e	f	Purchase	Loss	Recycling fulfillment
a	−	−	79/96	231/9	215/0	−
b	−	−	61/131	390/69	301/0	−
c	8/2	0/0	46/37	26/75	−	20/6	100%
d	0/0	140/420	250/0	0/0	−	0/0	100%
Demand fulfillment	−	−	100%	100%	−	−	−

As shown in Table 6, both the recycling fulfillment and demand fulfillment are 100%. So, the model and PSO algorithm are useful to manage a pallet pool with both RFID-tagged pallets and non-tagged pallets.

Assessing the value of an RFID system

The model and PSO algorithm could also be used to measure whether the investment of an RFID system is valuable or not. There are four steps to do so. The first step is to use the model and calculate the operation cost of a pallet pool without RFID-tagged pallets. The second step is to use the model and calculate the operation cost of a pallet pool with RFID-tagged pallets. The third step is to calculate how much operation cost could be reduced using RFID-tagged pallets. The fourth step is to calculate whether the investment of an RFID system could be covered with the operation cost saving. Certainly, the decision period should be reasonable and the data should be accurate. For example:

Step 1. Assumed there are only non-tagged pallets in a pallet pool. The demand and supply are shown in Table 7. We set the purchase cost of an RFID-tagged pallet as 1020, because it is hoped that supplier’s service stations will never purchase an RFID-tagged pallet. The other parameters values are the same as the section “Numerical example.” As shown in Table 8, the result shows the minimization operation cost is 74,007.

Step 2. Assumed there are only RFID-tagged pallets in a pallet pool. The demand and supply are shown in Table 9. We set the purchase cost of a non-tagged pallet as 1000, because it is hoped that supplier’s service stations will never purchase a non-tagged pallet. The other parameters values are the same as the section “Numerical example.” As shown in Table 10, the result shows the minimization operation cost is 68,587.

Step 3. Obviously, the operation cost could be reduced 5420 using RFID-tagged pallets in a period.

Step 4. If the cost of an RFID system is 50,000 and the maintenance cost of the system is 50 per year, the cost of the RFID system could be covered with the operation cost saving as long as the decision period is longer than 18 periods (with the method of net present value (NPV)). The relationship between the cost of an RFID system and save of operation cost (income) is shown in Figure 2. Figure 3 shows the accumulated revenue (total income minus total maintenance cost). Figure 4 shows the NPV.

Table 7.

Demand and supply where there are only non-tagged pallets.

	The number of pallets stored at a supplier’s service station at the beginning of the period (non-tagged)	The number of pallets that have to be taken away from a supply customer (non-tagged)	The number of pallets requested by a demand customer
a	200	−	−
b	350	−	−
c	−	220	−
d	−	810	−
e	−	−	700
f	−	−	800

Table 8.

Optimization scheme where there are only non-tagged pallets.

	a	b	e	f	Purchase	Loss	Recycling fulfillment
a	−	−	139	500	439	−
b	−	−	173	300	123	−
c	16	−	138	0	−	66	100%
d	−	560	250	0	−	0	100%
Demand fulfillment	−	−	100%	100%	−	−	−

Table 9.

Demand and supply where there are only RFID-tagged pallets.

	The number of pallets stored at a supplier’s service station at the beginning of the period (RFID-tagged)	The number of pallets that have to be taken away from a supply customer (RFID-tagged)	The number of pallets requested by a demand customer
a	200	−	−
b	350	−	−
c	−	220	−
d	−	810	−
e	−	−	700
f	−	−	800

RFID: radio-frequency identification.

Table 10.

Optimization scheme where there are only RFID-tagged pallets.

	a	b	e	f	Purchase	Loss	Recycling fulfillment
a	−	−	251	319	370	−
b	−	−	187	297	154	−
c	2	−	12	184	−	2	100%
d	−	560	250	0	−	0	100%
Demand fulfillment	−	−	100%	100%	−	−	−

Figure 2.

Income (save of operation cost) and cost (maintenance cost) of each period.

Figure 3.

Accumulated revenue.

Figure 4.

Net present value.

Conclusion

There is one issue which ever plagues the pallet industry—proprietary pallets that have leaked out of a closed-loop system. Moreover, a lot of pallets in a closed-loop system are damaged that contributes to manage pallets difficultly. The implementation of RFID technology plays an important role in not only reducing the loss rate and damage rate but also offering support for the managers to make scientific decisions.

An optimization model was proposed to manage a pallet pool with both RFID-tagged pallets and non-tagged pallets. The objective of the model is to minimize the total operation cost including the transportation cost, purchase or rent cost, loss cost, maintenance cost, loading and unloading, storage cost, and punishment cost. Several constraints are considered in the model, such as distribution, reposition, demand, recycling, storage, transportation capacity, and loading and unloading capacity. This is the first article to study the five processes (including distribution, reposition, recycling, purchase (or rent), and maintenance) of pallet pool operation in an optimization model. A PSO algorithm was proposed to solve the optimization model. Compared with other EAs as ICA and GA, PSO is easier to implement and there are fewer control parameters to adjust. A numerical example was used to prove the useful of the model and PSO algorithm. To the best of our knowledge, this is the first study to provide an optimization model and PSO algorithm for the operations of a pallet pool with both RFID-tagged pallets and non-tagged pallets.

Although the investment of an RFID system is expensive, it will help us to reduce operation cost. The managers of a pallet pool can measure whether it is worthwhile to invest in an RFID system based on the methodology proposed and combined empirical experience. The four steps presented can be used to do the job. Obviously, with promising foreground of the pallet pool industry, the managers who apply RFID will take advantages in the competitive market.

The objective of the optimization model is to minimize the total operation cost of a pallet pool, but the average operation cost per unit time may be better than that. Future work will involve research on a more scientific optimization model and empirical study.

Footnotes

Acknowledgements

The authors thank Dr Bo Liu at University of Science and Technology Beijing for his help on the research. The authors are grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this article.

Handling Editor: Chenguang Yang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (71502087), China Postdoctoral Science Foundation (2017M611204), the Research foundation of China Society of Logistics (2016CSLKT3-015), and the Postdoctoral Research Foundation of Inner Mongolia University.

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	The number of pallets stored at a supplier’s service station at the beginning of the period (non-tagged)	The number of pallets that have to be taken away from a supply customer (non-tagged)	The number of pallets requested by a demand customer
a	200	−	−
b	350	−	−
c	−	220	−
d	−	810	−
e	−	−	700
f	−	−	800

	The number of pallets stored at a supplier’s service station at the beginning of the period (RFID-tagged)	The number of pallets that have to be taken away from a supply customer (RFID-tagged)	The number of pallets requested by a demand customer
a	200	−	−
b	350	−	−
c	−	220	−
d	−	810	−
e	−	−	700
f	−	−	800

	The number of pallets stored at a supplier’s service station at the beginning of the period (non-tagged)	The number of pallets that have to be taken away from a supply customer (non-tagged)	The number of pallets requested by a demand customer
a	200	−	−
b	350	−	−
c	−	220	−
d	−	810	−
e	−	−	700
f	−	−	800

	The number of pallets stored at a supplier’s service station at the beginning of the period (RFID-tagged)	The number of pallets that have to be taken away from a supply customer (RFID-tagged)	The number of pallets requested by a demand customer
a	200	−	−
b	350	−	−
c	−	220	−
d	−	810	−
e	−	−	700
f	−	−	800

An optimization model for the operations of a pallet pool with both radio-frequency identification–tagged pallets and non-tagged pallets

Abstract

Keywords

Introduction

Problem description

Model development

Objective function

Constraints

Distribution

Reposition

Demand

Recycling

Storage

Transportation capacity

Loading and unloading capacity

The value of decision variables

Numerical example and analysis

Numerical examples

Particle swarm optimization algorithm and experimental results

Assessing the value of an RFID system

Conclusion

Footnotes

Acknowledgements

Declaration of conflicting interests

Funding

References

	The number of pallets stored at a supplier’s service station at the beginning of the period (non-tagged)	The number of pallets that have to be taken away from a supply customer (non-tagged)	The number of pallets requested by a demand customer
a	200	−	−
b	350	−	−
c	−	220	−
d	−	810	−
e	−	−	700
f	−	−	800

	The number of pallets stored at a supplier’s service station at the beginning of the period (RFID-tagged)	The number of pallets that have to be taken away from a supply customer (RFID-tagged)	The number of pallets requested by a demand customer
a	200	−	−
b	350	−	−
c	−	220	−
d	−	810	−
e	−	−	700
f	−	−	800