An Architecture for Distributed Energies Trading in Byzantine-Based Blockchain

Here, the aforementioned DES is implemented by the combined heat and power (CHP) system. The CHP system consumes natural gas to generate electricity and heat that serve its community or sell to the aggregators of its corresponding city, shown in Fig. 2. The gas is fed into the gas turbine (GT) which will generate electricity $E_{g}$ and emit high-temperature waste heat $Q_{w}$. The heat $Q_{w}$ can be recovered by heat recovery system that can generate heat $Q_{r}$. Here, $E_{use}$ (resp. $Q_{use}$) is used to supply electricity (resp. heat) to community, and $E_{exc}$ (resp. $Q_{exc}$) is sold to EA (resp. HA).

Shown as Fig. 2, the total electricity generated by GT is $E_{g}=E_{use}+E_{exc}$. Measured in days, the units of quantities denoted by $E$ and $Q$ are $({\rm J}/{\rm day})$. The gas consumption per day $F$ $({\rm m}^{3}/{\rm day})$ can be defined as

where $q$ $({\rm J}/{\rm m}^{3})$ is the calorific value of natural gas, thereby the total energy generated by $F$ is $q\cdot F$ definitely. The $\eta_{g}$ is the electric conversion of GT, percentage energy that transferred to electricity. Given a specific GT, its electric conversion can be considered as a constant. Besides, let $\eta_{r}$ be the thermal efficiency of heat recovery system, and $\eta_{h}=1$ be the thermal efficiency of heat component. We have $Q_{r}=Q_{w}\cdot\eta_{r}=Q_{pre}+Q_{exc}$ and $Q_{use}=Q_{pre}$ respectively.

As mentioned above, for a given CHP system, the electricity (resp. heat) generated by it can be divided into two parts: one is used to serve local residents, another is sold to EA (resp. HA). Thus, we define two dispatching factor $\alpha,\beta\in[0,1]$ for this CHP, where $\alpha=E_{use}/E_{g}$ is the electric dispatching factor and $\beta=Q_{pre}/Q_{r}$ is the heat dispatching factor. This DES needs to buy natural gas from the gas company. The company is for profit, thus it is valid to assume the gas company always supply enough gas that is able to meet the DES’s requirement. Given a smart city ${\rm S}_{i}$ and a smart community ${\rm C}_{ij}\in{\rm S}_{i}$, the energy relationships in the CHP system of ${\rm C}_{ij}$ has been obtained, that is

In this model, we assume all the CHP equipments in this ecosystem $\mathbb{S}$ has the same efficiency parameters $\eta_{g}$ and $\eta_{r}$. Each ${\rm CHP}_{ij}$ can determine the amount of electricity (resp. heat) that can be sold to ${\rm EA}_{i}$ (resp. ${\rm HA}_{i}$) by adjusting its dispatching factor $\alpha^{ij}$ (resp. $\beta^{ij}$) automonously. For example, when $\alpha^{ij}$ is one, it means that ${\rm CHP}_{ij}$ will not sell any electricity to ${\rm EA}_{i}$ for making revenue.

Consider a smart city ${\rm S}_{i}$, the ${\rm EA}_{i}$ (resp. ${\rm HA}_{i}$) offers a unit price $p_{e}^{i}$ (resp. $p_{h}^{i}$) to collect surplus electricity (resp. heat) generated by ${\rm DES}_{ij}\in{\rm S}_{i}$, where the units of $p_{e}^{i}$ and $p_{h}^{i}$ are ${\rm coin}/{\rm J}$. For each ${\rm DES}_{ij}\in{\rm S}_{i}$, it is a risk-averse agent in the energy market. If ${\rm DES}_{ij}$ chooses dispatching factor $\alpha^{ij}$, $\beta^{ij}$, and consume natural gas $F^{ij}$, that is

where $W_{e}^{ij}$ (resp. $W_{h}^{ij}$) is the satisfaction function of community ${\rm C}_{ij}$ that provides electricity (resp. heat) to satisfy the usage of local residents in this community, and $E_{use}^{ij}$, $E_{exc}^{ij}$, $Q_{use}^{ij}$, and $Q_{exc}^{ij}$ are defined from (2) to (5). Here, $c_{f}$ $({\rm coin}/{\rm m}^{3})$ is the unit cost of natural gas.

From our simplified CHP model, we denote the cost of electricity (resp. heat) produced from ${\rm DES}_{ij}$ by $c_{e}$ (resp. $c_{h}$). Then, the cost $({\rm coin}/{\rm J})$ of electricity and heat can be quantified, that is $c_{e}=c_{f}/q$ and $c_{h}=c_{f}/(q\cdot\eta_{r})$. Thus, we have $c_{f}\cdot F^{ij}=\eta_{g}\cdot c_{e}\cdot(qF^{ij})+(1-\eta_{g})\cdot c_{h}\cdot\eta_{r}\cdot(qF^{ij})$. If the price $p_{e}^{i}$ (resp. $p_{h}^{i}$) offered by ${\rm EA}_{i}$ (resp. ${\rm HA}_{i}$) is less than the cost $c_{e}$ (resp. $c_{h}$), this ${\rm DES}_{ij}$ will not sell any electricity (resp. heat) to them. It will reduce the gas intake $F^{ij}$ such that only meet its local requirement. Like this, there is no energies trading between aggregators and DESs, and obviously, it is not what we want to see. Thus, it is reasonable to consider the prices offered by aggregators satisfy $p_{e}^{i}\geq c_{e}$ and $p_{h}^{i}\geq c_{h}$. At this time, for each ${\rm DES}_{ij}\in{\rm S}_{i}$, it will produce electricity and heat as much as possible, because of the fact that it is always profitable to sell them to the aggregators. For maximizing its utility, each CHP system will run at full capacity. Here, for each ${\rm CHP}_{ij}$, we define its maximum production capacity (maximum gas consumption) per day as $F_{m}^{ij}$. Therefore, the utility $U^{ij}(\alpha^{ij},\beta^{ij},F_{m}^{ij})$ can be denoted by $U^{ij}(\alpha^{ij},\beta^{ij})$, because $F_{m}^{ij}$ is considered as a constant.

Based on [14] [18] [6], the natural logarithmic functions were adopted extensively in characterizing the satisfaction of comsuming energy. That is

where $k_{e}^{ij}$ (resp. $k_{h}^{ij}$) is a non-negative satisfaction coefficient for electricity (resp. heat) in community ${\rm C}_{ij}$, and $b_{e}^{ij}$ (resp. $b_{h}^{ij}$) is a non-negative adaption coefficient electricity (resp. heat) in this community as well. The adaption coefficients were proposed in [13] first, which aimed to control the variation range of the term $\ln(1+\cdot)$, avoid it growing infinitely. Generally, we let $\ln(1+b_{e}^{ij}\cdot E_{use}^{ij})=1$ (resp. $\ln(1+b_{h}^{ij}\cdot Q_{use}^{ij})=1$) when we choose $\alpha^{ij}=1$ (resp. $\beta^{ij}=1$) by setting a valid adaption coefficient $b_{e}^{ij}$ (resp. $b_{h}^{ij}$) [13]. Base on that, thereby we can formulate $b_{e}^{ij}$ and $b_{h}^{ij}$ as follows:

For aggregators in this city, power grid and heat station are the retailers for electricity and heat, however they do not have pricing power, because the retail prices of electricity and heat subject to government’s regulation. Hence, we define a retail price $r_{e}$ (resp. $r_{h}$) of electricity (resp. heat). As a selfish participant, it requires that $p_{e}^{i}\in[c_{e},r_{e}]$ and $p_{h}^{i}\in[c_{h},r_{h}]$. From (6), if $p_{e}^{i}$ (resp. $p_{h}^{i}$) offered by ${\rm EA}_{i}$ (resp. ${\rm HA}_{i}$) is too low, each ${\rm DES}_{ij}\in{\rm S}_{i}$ will respond with raising its dispatching factor $\alpha^{ij}$ (resp. $\beta^{ij}$), and sell less energy to aggregators. If the aggregators offer a high price to purchase energy, their profitable spaces are reduced even if DESs are willing to sell more energies to them. Both of these cases will cause aggregators’ profit to be cut down. Therefore, it is important for aggregators to offer a optimal price such that not only encourage DESs to sell more energies, but also ensure sufficient profitability. If ${\rm EA}_{i}$ (resp. ${\rm HA}_{i})$) offers a price $p_{e}^{i}$ (resp. $p_{h}^{i}$), its profit function can be defined as

where $V_{e}^{i}$ (resp. $V_{h}^{i}$) is the profit function of the ${\rm EA}_{i}$ (resp. ${\rm HA}_{i}$) that collects electricity (resp. heat) from DESs in its city, and $E_{exc}^{ij}$ and $Q_{exc}^{ij}$ are defined in (3) and (5).

A smart contract is a collection of programmable digital agreement that every participant commit to comply. Under our blockchain-based energies trading ecosystem, a transaction can only happen between aggregators and DESs in the same city. Thereby, consider a city ${\rm S}_{i}$, a smart contract can be decided together by its participants, which consist of an aggregator $k\in\{{\rm EA}_{i},{\rm HA}_{i}\}$ and a ${\rm DES}_{ij}\in{\rm S}_{i}$. We denote such a smart contract by $Contract(k,{\rm DES}_{ij},STime)$. Between anonymous and untrusted entities in a city, the smart contract is able to execute credible transactions without third institutions. Then, the procedure of its smart contract $Contract(k,{\rm DES}_{ij},STime)$ is presented as follows:

1) System initialization: At the beginning, each ${\rm DES}_{ij}\in{\rm S_{i}}$ needs to acquire a unique identification $ID_{ij}$ by registering in the designated institution authorized by government. It will be assigned with its public/private key pair $(PK_{ij},SK_{ij})$ and a $Account_{ij}$. That is

where each account is associated with its wallet address and balance, $Account_{ij}\leftarrow\{Address_{ij},Balance_{ij}\}$. Then, for each aggregator $k\in\{{\rm EA}_{i},{\rm HA}_{i}\}$ in this city, it has those necessary information $\{ID_{k},PK_{k},SK_{k},Account_{k}\}$ as well. But there is a credit value $Credit_{k}$ that represent the reputation of aggregator $k$, thereby we have $Account_{k}\leftarrow\{Address_{k},Balance_{k},Credit_{k}\}$. Here, technologies of asymmetric encryption are usually adopted by current blockchain system for the sake of security, privacy, and data integrity. Given a massage $msg$ encrypted by ${\rm DES}_{ij}$, we have $Hash(msg)=PK_{ij}(SK_{ij}(Hash(msg)))$, where the unforgeability and integrity is guaranteed.

2) Creation: An aggregator $k\in\{{\rm EA}_{i},{\rm HA}_{i}\}$ offers a price $p_{k}$ to buy energy from communities in its city, then ${\rm DES}_{ij}$ responds it with the amount of energy $x\in\{E_{exc}^{ij},Q_{exc}^{ij}\}$ that can be sold to $k$. Like this, a new smart contract $Contract(k,{\rm DES}_{ij},STime)$ is generated by signing with their private key respectively. Then, this contract will be broadcasted to all authorized participants (aggregators) in the ecosystem $\mathbb{S}$. After reaching a consensus, this smart contract will be deployed and executed automatically. Each smart contract between aggregator and DES, $Contract(k,{\rm DES}_{ij},STime)$, is associated with several variables, which include account information $(Account_{k},Account_{ij})$, offered price $p_{k}$, amount of energy $x$, expected transaction time $TransTime$, and timestamp $STime$. To guarantee this contract can be executed successfully, it needs to verify whether aggregator $k$ has sufficient balance such that ${\rm Balance}_{k}\geq p_{k}\cdot x$ and whether ${\rm DES}_{ij}$ has enough production capacity to supply $x$ amount of corresponding energy on time.

3) Execution: The $Contract(k,{\rm DES}_{ij},STime)$ will be executed if current time $t\geq{\rm TransTime}$ after reaching a consensus among aggregators in blockchain network. From now on, it begins to trade energy and finish payment. The smart meter in aggregator $k$ verifies whether the amount of energy has been transported to the designated location. Then, fed this result from smart meter into the smart contract, if yes, it will execute the payment process automatically, that is

Here, we design a mechanism that the balance is permitted to be negative. At the moment of payment, the smart contract will complete payment as usual if the $k$’s balance is not enough to pay. In this way, the $k$’s balance will become negative. Then, any contract that aggregator $k$ participants in will not be executed until its balance back to be positive.

Generally speaking, the energies trading between aggregators and DESs can be summarized as follows: In a smart city, a DES begins a smart contract with an aggregator by responding it according to its offered price. This contract needs to be verified by the consensus process in blockchain network. Then, it will execute the predefined procedure automatically once the trading conditions are met, which achieves the digital currency and energy exchange specified by contract between participants in a secure manner.

As mentioned early, a consensus process is necessary to be performed so as to ensure the consistency of blockchain stored in every authorized node. Castro et al. [19] proposed a practical byzantine fault tolerance (PBFT) algorithm, which has been used in consortium blockchain system widely. Based on it and combined with the characteristics of energies trading, we design a new byzantine-based consensus mechanism (BCM). Our consensus process is performed among all aggregators in the ecosystem $\mathbb{S}$, and each of them has a local transaction pool (LTP) to store all transactions it receives. The consensus process is executed round by round, and the time interval of block generation is given by $\Delta T$. There are three main stages, shown in Fig. 3, that is pre-prepare, prepare, and commit.

First, let us introduce the credit model, which will be used in leader election and to decide whether to reach a consensus. Let $M=\{{\rm EA}_{1},{\rm HA}_{1},{\rm EA}_{2},{\rm HA}_{2},\cdots\}$ be the collection of consensus nodes. We have known that there is an attribute $Credit_{k}\in[0,1]$ for each $k\in M$, where a larger $Credit_{k}$ implies node $k$ is more trustworthy. We denote by $Credit_{k}(i)$ the credit of node $k$ after finishing the $i$-th round consensus. Then, we can define $Credit_{k}(i+1)$ according to the result of consensus in the $(i+1)$-th round, where this result is whether to add the leader’s block into the blockchain. That is: (1) when $k$ is the leader, we have $Credit_{k}(i+1)=\min\{1,Credit_{k}(i)+\Delta_{1}\}$ if its block is accepted to be added into the blockchain, else $Credit_{k}(i+1)=\max\{0,Credit_{k}(i)-\Delta_{1}\}$ if its block is rejected; and (2) when $k$ is not the leader, we have $Credit_{k}(i+1)=\min\{1,Credit_{k}(i)+\Delta_{2}\}$ if its decision is consistent with consensus result; else $Credit_{k}(i+1)=\max\{0,Credit_{k}(i)-\Delta_{2}\}$ if it disagrees with the majority. We usually give $\Delta_{1}>\Delta_{2}>0$ and initialize the credit of each consensus node as $credit_{k}(0)=0.5$. Consider entering the $(i+1)$-th round consensus, the detailed process of BCM is represented as follows:

1) Leader election: The first step in this round is to select a leader from all consensus nodes. This leader election is based on node’s credit. Generally speaking, the better the credit value of a node, the more likely it is to be elected as the leader. Thus, the result of leader selection is unpredictable. For a node $k\in M$, the probability that it is elected as the leader of the $(i+1)$-th round consensus is $\Pr[L(i+1)=k]$,

where $L(i+1)$ represents the leader of the $(i+1)$-th round consensus. Obviously, there is no chance to select a node whose credit is zero as the leader.

2) Broadcast: Each aggregator in $M$ broadcasts all transactions which happen in current $\Delta t$ and co-signed with a DES in its city to the blockchain network. All the consensus nodes will verify whether their received transactions are valid. Those valid transactions will be stored in their LTP, and invalid transactions will be discarded.

3) Pre-prepare: After all non-leader consensus nodes in $M\backslash L(i+1)$ have completed above verification process for received transactions, the leader will package those selected valid transactions in its LTP into a block $B_{L}$. Then, the leader signs this block and broadcasts pre-prepare message $SK_{L}(SK_{L}(B_{L}),pre$-$prepare)$ to the blockchain network.

4) Prepare: For each non-leader node $k\in M\backslash L(i+1)$, it will check the identity of leader and verify the pre-prepare message from the leader. The block verification needs to confirm the pointer to the previous block, mercle root is correct and compare the transactions in $B_{L}$ with the corresponding transactions in its LTP. If node $k$ believes $B_{L}$ is valid, it broadcasts this prepare message $SK_{k}(SK_{L}(B_{L}),prepare)$ to the blockchain network. All consensus nodes must make decisions in this step, whether to agree or disagree with adding block $B_{L}$ into the blockchain. Then for each node $k\in M$, it gathers all prepare massages from other consensus node, checks their identities and counts the weighted sum of received prepare messages. Let $A_{k}(i+1)\subseteq M$ be the set of nodes from which node $k$ receives prepare messages, including itself. If satisfying the following inequality

where $\Pr[a]={Credit_{a}(i)}/{\sum_{j\in M}Credit_{j}(i)}$, we say node $k$ will accept block $B_{L}$ and broadcast commit message $SK_{k}(SK_{L}(B_{L}),commit)$ to the blockchain network.

5) Commit: After sending their commit messages, they should waiting commit messages from other consensus node. For each node $k\in M$, its consensus process is completed until it recieve sufficient commit messages such that $\sum_{a\in B_{k}(i+1)}\Pr[a]\geq(2\lfloor(|M|-1)/{3}\rfloor+1)/|M|$, where $B_{k}(i+1)\subseteq M$ is the set of nodes from which node $k$ receives commit message, including itself.

5) Add a block and update credits: If a consensus node accepts the new block $B_{L}$, it will be appended into the blockchain in a linear and chronological order, which includes a pointer to the previous block. Any failure occurs in these three stages will terminate the consensus of current round (do not add the new block). Besides, before finishing this round, we need to update the credits of all the consensus nodes according to the credit model. In next round, the consensus process will perform based on their new credits.

Failures that terminate the current round and do not add a new block mainly include the following reasons: (1) the leader sends invalid block or do not send its packaged block before the deadline; and (2) too many malicious nodes do not breadcast prepare messages even though this block is valid. Shown as node ${\rm HA}_{1}$ in Fig. 3, it is a faulty node. The credit of those nodes that make mistakes in this consensus process will be reduced. Let $f$ be the number of malicious nodes. According to [19], supposing $f\leq\lfloor(|M|-1)/3\rfloor$, the faults can be tolerated by the consensus system with $|M|$ nodes. In our BCM, each consensus node’s credit is initialized as a constant, thereby $2\lfloor(|M|-1)/3\rfloor+1$ good nodes can make sure that (13) is satisfied. As the consensus process performs more and more times, the good nodes’ credit increase but malicious nodes’ credit decrease gradually. Therefore, the credits in system will be more accumulative in good nodes. According to (13), the number of prepare message and commit message required to reach a consensus declines, which helps reduce latency and improve throughput. In summary, secure and traceable energies trading and digital currency exchange can be guaranteed by our proposed B-MET system.

Given a price strategy $\{p_{e}^{i},p_{h}^{i}\}\in\mathbb{P}$ offered by two aggregators in city ${\rm S}_{i}$, each ${\rm DES}_{ij}\in{\rm S}_{i}$ decides the amount of electricity $E_{use}^{ij}$ (resp. heat $Q_{use}^{ij}$) that sold to the ${\rm EA}_{i}$ (resp. ${\rm HA}_{i}$) by adjusting its dispatching factor $\alpha^{ij}$ (resp. $\beta^{ij}$). Thus, each ${\rm DES}_{ij}\in{\rm S}_{i}$ aims to choose its optimal dispatching factors $\{\alpha^{ij},\beta^{ij}\}$ according to $\{p_{e}^{i},p_{h}^{i}\}$ by solving the following optimization problem (${\rm OP_{DES}}$), that is

where $M_{min}^{ij}$ is the minimum amount of energy that is required to maintain the basic life for those residents living in this community. The objective function, shown in (6), is strictly concave and continuously differentiable, this ${\rm OP_{DES}}$ is a convex optimization problem, which will be proved later. Thus, the stationary solution is unique and optimal.

To ensure reasonableness, the $E_{min}^{ij}$ should be in a valid range, thus we have $M_{min}^{ij}\in(\max\{X^{ij},Y^{ij}\},X^{ij}+Y^{ij})$. It means that this minimum requirement is larger than the production capacity of electricity or heat separately, which implies that $\alpha^{ij}$ and $\beta^{ij}$ are impossible to approach zero definitely. Hence, the restrictions on (16) can be converted equivalently to constraint (17), that is

Then, its first-order derivatives is

Let $\partial U^{ij}/\alpha^{ij}=0$ and $\partial U^{ij}/\beta^{ij}=0$, we have

Here, we need to note that the setting of parameter $k_{e}^{ij}$ (resp. $k_{h}^{ij}$) must be in a valid range such that $\alpha^{ij}_{\circ}\in(0,1)$ (resp. $\beta^{ij}_{\circ}\in(0,1)$) for any offered price $p_{e}^{i}\in[c_{e},r_{e}]$ (resp. $p_{h}^{i}\in[c_{h},r_{h}]$). Or else, this utility function is monotone, and it is meaningless to adjust its dispatching factors. Base on that, thereby we can restrict $k_{e}^{ij}$ and $k_{h}^{ij}$ as follows:

where it assume $r_{e}<e\cdot c_{e}$ (resp. $r_{h}<e\cdot c_{h}$), or else no such $k_{e}^{ij}$ (resp. $k_{h}^{ij}$) can keep $\alpha^{ij}_{\circ}\in(0,1)$ (resp. $\beta^{ij}_{\circ}\in(0,1)$) satisfied for any offered prices.

Sequentially, we use Lagrange’s multipliers $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ for constraint (17), thereby the ${\rm OP_{DES}}$, shown as (15) and (17), can be converted to the following form $L^{ij}(\alpha^{ij},\beta^{ij},\lambda_{1},\lambda_{2},\lambda_{3})$, that is

where we denote $Z^{ij}=X^{ij}+Y^{ij}-M_{max}^{ij}$. Then, Based on (22), the complementary slackness conditions (KKT conditions) of ${\rm OP_{DES}}$ are demonstrated as follows:

The optimal solutions of ${\rm OP_{DES}}$, shown as (15) and (17), can take one of the following four cases, that is

1) Case 1: For $\alpha^{ij}<1$ and $\beta^{ij}<1$, we have $\lambda_{2}=\lambda_{3}=0$. Look at (25), if $\lambda_{1}=0$, substitute it into (23) and (24), we can get a solution $\{\alpha^{ij}_{\circ},\beta^{ij}_{\circ}\}$ according to (20). Then, we need to check whether constraint (17) can be satisfied. If yes, the optimal solution is $\{\alpha^{ij}_{\circ},\beta^{ij}_{\circ}\}$. If no, it means $\lambda_{1}>0$ and $X^{ij}\cdot\alpha^{ij}+Y^{ij}\cdot\beta^{ij}-M_{min}^{ij}=0$. At this time, by solving (23) and (24), we have

Substitute (29) and (30) into (25),

where $A^{ij}=M_{min}^{ij}+1/b_{e}^{ij}+1/b_{h}^{ij}$, $B^{ij}=k_{e}^{ij}+k_{h}^{ij}-A^{ij}(p_{e}^{i}+p_{h}^{i})$, and $C^{ij}=A^{ij}p_{e}^{i}p_{h}^{i}-k_{e}^{ij}p_{h}^{i}-k_{h}^{ij}p_{e}^{i}$. By solving (31), we have two solutions, they are

Here, it is easy to verify $B^{ij}<0$ and $C^{ij}>0$ based on (9), (21), and $M_{min}^{ij}\in(\max\{X^{ij},Y^{ij}\},X^{ij}+Y^{ij})$, thus it is possible that $\Delta^{ij}={B^{ij}}^{2}-4A^{ij}C^{ij}<0$. If $\Delta^{ij}<0$, there is no real solution; else we need to check whether the $\lambda_{1}$ defined on (32) satisfies $\lambda_{1}>0$. If $\lambda_{1}>0$, substitute (32) into (29) and (30), we obtain a solution $\{\alpha^{ij}_{\Diamond},\beta^{ij}_{\Diamond}\}$. If it is feasible, namely $\alpha^{ij}_{\Diamond},\beta^{ij}_{\Diamond}<1$, the optimal solution can be determined by $\{\alpha^{ij}_{\Diamond},\beta^{ij}_{\Diamond}\}$.

2) Case 2: For $\alpha^{ij}=1$ and $\beta^{ij}<1$, we have $\lambda_{3}=0$. Look at (25), if $\lambda_{1}=0$, substitute it into (24), we can get a solution $\{1,\beta^{ij}_{\circ}\}$ according to (20). Then, we need to check whether constraint (17) can be satisfied and $\lambda_{2}={\partial U^{ij}}/{\partial\alpha^{ij}}\geq 0$. If yes, the optimal solution $\{1,\beta^{ij}_{\circ}\}$. If no, it means $\lambda_{1}>0$ and $Y^{ij}\cdot\beta^{ij}+X^{ij}-M_{min}^{ij}=0$. According to (24), we have $\beta^{ij}_{\square}$ which is shown as (30). Substitute (30) into (25),

By solving (33), we have

If $\lambda_{1}>0$ and $\lambda_{2}={\partial U^{ij}}/{\partial\alpha^{ij}}+\lambda_{1}X^{ij}\geq 0$, substitute (34) into (30), we obtain a solution $\{1,\beta^{ij}_{\square}\}$. If we have $\beta^{ij}_{\square}<1$, the optimal solution can be determined by $\{1,\beta^{ij}_{\square}\}$.

3) Case 3: For $\alpha^{ij}<1$ and $\beta^{ij}=1$, we have $\lambda_{2}=0$. Look at (25), if $\lambda_{1}=0$, substitute it into (23), we can get a solution $\{\alpha^{ij}_{\circ},1\}$ according to (18). Then, we need to check whether constraint (17) can be satisfied and $\lambda_{3}={\partial U^{ij}}/{\partial\beta^{ij}}\geq 0$. If yes, the optimal solution is $\{\alpha^{ij}_{\circ},1\}$. If no, it means $\lambda_{1}>0$ and $X^{ij}\cdot\alpha^{ij}+Y^{ij}-M_{min}^{ij}=0$. According to (23), we have $\alpha^{ij}_{\square}$ which is shown as (29). Substitute (29) into (25),