An Equilibrium Chance-Constrained Multiobjective Programming Model with Birandom Parameters and Its Application to Inventory Problem

Abstract

An equilibrium chance-constrained multiobjective programming model with birandom parameters is proposed. A type of linear model is converted into its crisp equivalent model. Then a birandom simulation technique is developed to tackle the general birandom objective functions and birandom constraints. By embedding the birandom simulation technique, a modified genetic algorithm is designed to solve the equilibrium chance-constrained multiobjective programming model. We apply the proposed model and algorithm to a real-world inventory problem and show the effectiveness of the model and the solution method.

1. Introduction

As a key issue in supply chain management, inventory management plays an important role in controlling operation cost and improving management efficiency. As a result, inventory problems have attracted many researchers from various fields. Classical inventory problem research discussed single-item cases [1–5]. However, multiitem inventory problems are more common in the real-world. In fact, for many enterprises, especially supermarkets, it is necessary to prepare hundreds of goods to meet the needs of different consumers. Therefore, multiitem inventory problem research has become more attractive for researchers [6–9].

Apart from the multiitem inventory, uncertainty is another characteristic of inventory problems. For example, the price of a product usually varies due to the market fluctuation. One kind of common uncertainty is randomness. Many researchers have studied random inventory models. Baker and Ehrhardt analyzed a periodic-review, random-demand inventory model under the assumption that replenishment quantities are random fractions of the amounts ordered [10]. Bera et al. dealt with a multiitem mixture inventory model in which both demand and lead time are random [8]. Fuzziness is another type of uncertainty in inventory problems. Wang et al. [11] developed a novel joint replenishment problem model with fuzzy minor replenishment cost and fuzzy inventory holding cost. Roy et al. [12] considered an inventory model for a deteriorating item (seasonal product) with linearly displayed stock dependent demand in an imprecise environment (involving both fuzzy and random parameters) under inflation and the time value of money. M. K. Maiti and M. Maiti [13] proposed a multiitem inventory model with advertising costs, price, and displayed inventory level-dependent demand in a fuzzy environment (purchase cost, investment amount, and storehouse capacity were considered imprecise). Besides these, twofold uncertain inventory models have also been studied as more complex; imprecise information is considered. Dutta et al. [14] presented a single-period inventory problem in an imprecise and uncertain mixed environment, in which the demand is assumed as a fuzzy random variable. Xu and Zhao [15, 16] formulated inventory models under fuzzy rough environments.

However, an inventory model with twofold random phenomenon is seldom discussed. In reality, inventory problems in the real-world may be subjected to twofold randomness with incomplete or uncertain information. Consequently, developing an inventory strategy in a more complete stochastic environment takes place. As a case in point, it is widely accepted that the price of a product is a normal distributed variable, denoted by 𝑁(𝜇,𝜎²) from the viewpoint of probability theory, but the values of 𝜇 and 𝜎 may be still uncertain variables. If there is statistical information about 𝜇 and 𝜎, it is possible to specify realistic distributions for 𝜇 and 𝜎 by utilizing statistical methods. When the values of 𝜇 and 𝜎 are provided as random variables, the price then is not a conventional random variable but a so-called birandom variable. A birandom variable, which plays a role analogous to a random variable in probability theory, is appropriate to describe this kind of twofold randomness. In this paper, we discuss a class of multiobjective programming models with birandom parameters. Chance-constrained programming proposed by Charnes and Cooper [17] is an effective technique to deal with uncertain optimization problem [18]. By using the equilibrium chance-constrained programming technique, the initial model is meaningful mathematically.

The rest of the paper is organized as follows. Section 2 develops a general equilibrium chance-constrained programming model with birandom parameters. A linear model is converted into its crisp equivalent model. In order solve a general model, the birandom simulation-based genetic algorithm is deigned in Section 3. Section 4 applies the theoretical results into a real-world inventory problem. Further discussions in Section 5 illustrate the effectiveness of the proposed model and algorithm. Finally, concluding remarks are outlined in Section 6.

2. Equilibrium Chance-Constrained Multiobjective Programming Model with Birandom Parameters

In order construct a general equilibrium chance-constrained multiobjective programming model with birandom parameters, we first state some basic concepts and theorems on birandom theory which is presented firstly.

2.1. Birandom Variable and Equilibrium Chance

Birandom variable, which is proposed by Peng and Liu [19], is a mathematical tool to describe twofold random phenomena. An n-dimensional birandom vector 𝜉 is a map from the probability space (Ω,𝒜,Pr) to a collection of n-dimensional random vectors such that Pr{𝜉(𝜔)∈ℬ} is a measurable function with respect to 𝜔 for any Borel set ℬ of the real space ℛ^𝑛. In particular, 𝜉 is called a birandom variable as 𝑛=1.

Example 1. A birandom variable 𝜉 is said to be uniform, if for each 𝜔, 𝜉(𝜔) is a random variable with uniform distribution, denoted by 𝒰[𝑎(𝜔),𝑏(𝜔)], where 𝑎(𝜔) and 𝑏(𝜔) are random variables defined on the probability space (Ω,𝒜,Pr).

Example 2. A birandom variable 𝜉 is said to be normal, if for each 𝜔, 𝜉(𝜔) is a random variable with normal distribution, denoted by 𝒩(𝜇(𝜔),𝜎²(𝜔)), where 𝜇(𝜔) and 𝜎(𝜔) are random variables defined on the probability space (Ω,𝒜,Pr).

Example 3. A birandom variable 𝜉 is said to be exponential, if for each 𝜔, 𝜉(𝜔) is a random variable with exponential distribution, denoted by exp(𝜆(𝜔)), where 𝜆(𝜔) are random variables defined on the probability space (Ω,𝒜,Pr).

Example 4. Let (Ω,𝒜,Pr) be a probability space. If Ω={𝜔₁,𝜔₂,…,𝜔_𝑛}, 𝜉(𝜔)=𝜉₁,𝜔=𝜔₁𝜉₂,𝜔=𝜔₂𝜉⋮,⋮,_𝑛,𝜔=𝜔_𝑛, (1) where 𝜉₁,𝜉₂,…,𝜉_𝑛 are random variables with density function 𝑝₁(𝑥),𝑝₁(𝑥),…,𝑝_𝑛(𝑥), respectively. Birandom variable 𝜉 is illustrated by Figure 1.

Figure 1 

A birandom variable.

By birandom event we mean that 𝑔(𝜉)≤0. In order to compare the degrees of occurrence of two birandom events, quantitative measures of the chance of a birandom event are necessary. In the literature, the first attempt to develop the definition of the chance of a birandom event is primitive chance, which is a function from [0,1] to [0,1].

Definition 5 (see [19]). Let 𝜉=(𝜉₁,𝜉₂,…𝜉_𝑛) be a birandom vector defined on Ω, and 𝑔∶ℜ^𝑛↦ℜ is Borel measurable function. Then the primitive chance of a birandom event characterized by 𝑔(𝜉)≤0 is a function from [0,1] to [0,1], defined as Ch𝑔(𝜉)≤0(𝛼)=sup_[]𝛼∈0,1𝛽∣Pr𝜔∈Ω∣Pr𝑔(𝜉(𝜔))≤0≥𝛽≥𝛼, (2) where 𝛼 is a prescribed probability level. The value of primitive chance at 𝛼 is called 𝛼-chance.

Remark 6. It should be noted that the symbol Pr appears twice in the right side of (2). In fact, they represent different meanings. In other words, the overloading allows us to use the same symbol Pr for different probability measures, because we can deduce the exact meaning in the context.

In the case of fuzzy random programming problems, the equilibrium chance of a fuzzy random event was introduced to measure the degree of the occurrence of a fuzzy random event [20]. Rather than a function, the equilibrium chance is a scalar value, like the probability of a random event and the possibility of a fuzzy event. Thus it is easy for the decision maker to rank the decisions via equilibrium chance using the natural order of real numbers, rather than requiring a preference order from the decision maker. Motivated by the idea, we introduce the equilibrium chance of a birandom event as follows.

Definition 7. Let 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛) be a birandom vector, and 𝑔∶ℜ^𝑛↦ℜ is a Borel measurable function. Then the equilibrium chance of birandom event 𝑔(𝜉)≤0 is defined as Ch^𝑒𝑔(𝜉)≤0=sup_[]𝛼∈0,1𝛼∧Pr𝜔∈Ω∣Pr𝑔(𝜉(𝜔))≤0≥𝛼. (3)

Remark 8. Peng and Liu [19] introduced the equilibrium chance of a birandom event as Ch^𝑒𝑔(𝜉)≤0=sup_𝛼∈[0,1]𝛼∣Ch𝑔(𝜉)≤0(𝛼)≥𝛼, (4) where Ch is the primitive chance. It is easy to verify the forms (3) and (4) are equivalent.

Remark 9. If 𝜉 degenerates to a random vector, Pr{𝜔∈Ω∣Pr{𝑔(𝜉(𝜔))≤0}≥𝛼} implies Pr{𝑔(𝜉(𝜔))≤0}≥𝛼, and then (3) is equivalent to Pr{𝑔(𝜉(𝜔))≤0}, which is the probability measure.

For any 𝛼∈[0,1], denote 𝐹(𝛼)𝑔(𝜉(𝜔,𝐹()𝑔(𝜉(𝜔.=Pr𝜔∈Ω∣Pr))≤0≥𝛼𝛼+=Pr𝜔∈Ω∣Pr))≤0>𝛼 (5) It follows from the definition of birandom variable that Pr{(𝜉(𝜔))≤0} is a random variable. Then 𝐹(𝛼) is a left-continuous and nonincreasing function with respect to 𝛼, and for any given 𝛼, 𝐹(𝛼) represents the following random event occurring: 𝜔∈Ω∣Pr𝑔(𝜉(𝜔))≤0≥𝛼. (6) Therefore, the equilibrium chances defined above measure the twofold maximum probability simultaneously.

Generally, if 𝐹(𝛼)≤𝛼≤𝐹(𝛼+), (7) then 𝛼 is just the value of the equilibrium chance. As Pr{𝑔(𝜉(𝜔))≤0} is a continuous random variable, the left side and right side of relation (7) are equal. Thus the equilibrium chance is available at the fixed point of 𝐹; that is, 𝐹(𝛼)=𝛼.

It follows from the definitions of equilibrium chance and primitive chance of a birandom event that they have the following differences.(i)The equilibrium chance of a birandom chance of a birandom event is a scalar value, just like the probability of a random event and the possibility of a fuzzy event, while the primitive chance of a birandom event is a function from [0,1] to [0,1].(ii)The equilibrium chance measures the twofold probabilities at the same time, while the value of the primitive chance at 𝛼 measures the single maximum probability of a birandom event occurring under the given value of the other probability.

However, the connections of the equilibrium chance and the primitive chance can be summarized as follows.(i)Equilibrium chance (3) can be represented as Ch^𝑒𝑔(𝜉)≤0=sup_[]𝛼∈0,1𝛼∧Ch𝑔_𝑟(𝜉(𝜔))≤0(𝛼), (8) where Ch{𝑔(𝜉(𝜔))≤0}(𝛼) is the value of primitive chance at 𝛼.(ii)The primitive chance of a birandom event is the pseudo-inverse function of the function 𝐹(𝛼).

In order to generate the notation of the equilibrium chance of a birandom event defined by (3), we give another equivalent form as follows: Ch^𝑒𝑔(𝜉)≤0=sup_{(𝛼,𝛽)∈[0,1]2}𝑔(𝜉(𝜔},𝛼∧𝛽∣Pr𝜔∈Ω∣Pr))≤0≥𝛼≥𝛽 (9) where the parameters 𝛼 and 𝛽 represent two kinds of probability. It is easy to see from (3) that we use the min/max operator, which is a special triangular norm, to define the equilibrium chance of a birandom event. In fact, by using a common triangular norm 𝑇, the equilibrium chance can be extended as Ch^𝑒_𝑇𝑔(𝜉)≤0=sup_{𝛼,𝛽∈[0,1]2}𝑇∣𝑓𝛼,𝛽Pr𝜔∈Ω∣Pr_𝑗(𝜉(𝜔))≤0≥𝛼≥𝛽, (10) where 𝑇∶[0,1]↦[0,1] is a triangular norm. In practice, we may also use various triangular norms such as 𝑇₁(𝛼,𝛽)=𝛼𝛽, 𝑇₂(𝛼,𝛽)=𝛼𝛽/[1+(1−𝛼)(1−𝛽)], 𝑇₃(𝛼,𝛽)=𝛼𝛽/(𝛼+𝛽−𝛼𝛽), or 𝑇₄=max{0,𝛼+𝛽+1} according to a decision maker’s philosophy of modeling uncertainty.

The following theorem implies a basic property of the equilibrium chance.

Theorem 10. Let 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛) be a birandom vector, and let 𝑔∶ℜ^𝑛↦ℜ be Borel measurable function, and 𝛼∈[0,1]. Then Ch^𝑒𝑔(𝜉)𝑔(𝜉(𝜔≤0≥𝛼⟺Pr𝜔∈Ω∣Pr))≤0≥𝛼≥𝛼. (11)

Proof. “⇒” Assume that Ch^𝑒{𝑔(𝜉)≤0}≥𝛼. It follows from the definition of Ch^𝑒{⋅} that there exists a real number 𝛼₀∈[0,1] such that 𝛼₀∧Pr𝜔∈Ω∣Pr𝑔(𝜉(𝜔))≤0≥𝛼₀≥𝛼. (12) Thus we have 𝛼₀𝑔(𝜉(𝜔≥𝛼,Pr))≤0≥𝛼₀≥𝛼. (13) So we get that 𝑔(𝜉(𝜔⊇𝑔(𝜉(𝜔𝜔∈Ω∣Pr))≤0≥𝛼𝜔∈Ω∣Pr))≤0≥𝛼₀. (14) It follows that 𝑔(𝜉(𝜔𝑔(𝜉(𝜔Pr𝜔∈Ω))≤0≥𝛼≥Pr𝜔∈Ω))≤0≥𝛼₀≥𝛼. (15) “⇐” If Pr{𝜔∈Ω∣Pr{𝑔(𝜉(𝜔))≤0}≥𝛼}≥𝛼, then we have 𝛼=𝛼∧Pr𝜔∈Ω∣Pr𝑔(𝜉(𝜔))≤0≤sup_[]𝛽∈0,1𝛽∧Pr𝜔∈Ω∣Pr𝑔(𝜉(𝜔))≤0≥𝛽=Ch^𝑒𝑔(𝜉(𝜔))≤0. (16) The theorem is proved.

The independence of birandom variables is defined as follows.

Definition 11. Birandom variables 𝜉₁,𝜉₂,…,𝜉_𝑛, which are defined on the probability space (Ω,𝒜,Pr), are said to be independent if and only if 𝜉₁(𝜔),𝜉₂(𝜔),…,𝜉_𝑛(𝜔) are independent random variables for all 𝜔∈Ω.

2.2. Model Formulation

A general equilibrium chance-constrained multiobjective programming model with birandom parameters is formulated as max𝑓₁,𝑓₂,…,𝑓_𝑚s.t.Ch^𝑒𝑓_𝑖(𝑥,𝜉)≥𝑓_𝑖≥𝛼_𝑖,𝑖=1,2,…,𝑚,Ch^𝑒𝑔_𝑟(𝑥,𝜉)≤0≥𝛽_𝑟,𝑟=1,2,…,𝑝,𝑥∈𝐷, (17) where 𝑥 is an 𝑛-ary decision vector, 𝜉𝑚-ary is a birandom vector, 𝐷 is a fixed set that is usually determined by a finite number of inequalities involving functions of 𝑥, 𝑓_𝑖 and 𝑔_𝑟 are (𝑚+𝑛)-ary real-valued continuous functions, and 𝛼_𝑖 and 𝛽_𝑟 are predetermined confidence levels, 𝑖=1,2,…,𝑚, 𝑟=1,2,…,𝑝.

It follows from Remark 9 that if 𝜉 degenerates to random vector, then model (17) degenerates to max𝑓₁,𝑓₂,…,𝑓_𝑚s.t.Pr𝑓_𝑖(𝑥,𝜉)≥𝑓_𝑖≥𝛼_𝑖𝑔,𝑖=1,2,…,𝑚,Pr_𝑟(𝑥,𝜉)≤0≥𝛽_𝑟,𝑟=1,2,…,𝑝,𝑥∈𝐷, (18) which is a stochastic chance-constrained programming model.

Definition 12 (birandom efficient solution at 𝛼_𝑖-levels). Suppose a feasible solution 𝑥^∗ of problem (17) satisfies Ch^𝑒𝑓_𝑖𝑥^∗,𝜉≥𝑓_𝑖𝑥^∗≥𝛼_𝑖,𝑖=1,2,…,𝑚, (19) where confidence levels 𝛼_𝑖∈[0,1].  𝑥^∗ is said to be a birandom efficient solution at 𝛼_𝑖-levels to problem (17) if and only if there exists no other feasible solution 𝑥 such that Ch^𝑒𝑓_𝑖(𝑥,𝜉)≥𝑓_𝑖(𝑥)≥𝛼_𝑖,𝑖=1,2,…,𝑚, (20) and 𝑓_𝑖(𝑥)≥𝑓_𝑖(𝑥^∗) for all 𝑖 and 𝑓_𝑖0(𝑥)>𝑓_𝑖0(𝑥^∗) for at least one 𝑖₀∈{1,2,…,𝑚}.

Specially, we consider the linear form of model (17): max𝑓₁,𝑓₂,…,𝑓_𝑚s.t.Ch^𝑒𝑐^𝑇_𝑖̃̃𝑥≥𝑓_𝑖≥𝛼_𝑖,𝑖=1,2,…,𝑚,Ch^𝑒𝑎^𝑇_𝑟𝑥≤𝑏̃̃_𝑟≥𝛽_𝑟,𝑟=1,2,…,𝑝,𝑥∈𝐷, (21) where 𝑐_𝑖̃̃=(𝑐̃̃_𝑖1,𝑐̃̃_𝑖2,…,𝑐̃̃_𝑖𝑛)^𝑇 and 𝑎_𝑟=(𝑎_𝑟1,𝑎_𝑟2,…,𝑎_𝑟𝑛)^𝑇 are birandom vectors, and 𝑏̃̃_𝑟 are birandom variables, 𝑖=1,2,…,𝑚, 𝑟=1,2,…,𝑝.

It follows from Theorem 10 that model (21) can be rewritten as max𝑓₁,𝑓₂,…,𝑓_𝑚s.t.Pr𝜔∈Ω∣Pr𝑐^𝑇_𝑖̃̃(𝜔)𝑥≥𝑓_𝑖≥𝛼_𝑖≥𝛼_𝑖,𝑎𝑖=1,2,…,𝑚,Pr𝜔∈Ω∣Pr^𝑇_𝑟(𝜔)𝑥≥𝑏̃̃_𝑟≥𝛽_𝑟≥𝛽_𝑟,𝑟=1,2,…,𝑝,𝑥∈𝐷. (22)

Lemma 13. Assume that birandom vector 𝑐_𝑖̃̃(𝜔)=(𝑐̃̃_𝑖1(𝜔),𝑐̃̃_𝑖2(𝜔),…,𝑐̃̃_𝑖𝑛(𝜔))^𝑇 follows normal distribution with mean vector 𝑐_𝑖̃(𝜔) and positive definite covariance matrix 𝑉^𝑐_𝑖, denoted by 𝑐_𝑖̃̃(𝜔)∼𝒩(𝑐_𝑖̃(𝜔),𝑉^𝑐_𝑖). 𝑐_𝑖̃(𝜔) is a normal random vector with mean vector 𝜇^𝑐_𝑖 and positive definite covariance matrix 𝑉_𝑖^′𝑐, written as 𝑐_𝑖̃(𝜔)∼𝒩(𝜇^𝑐_𝑖,𝑉_𝑖^′𝑐). If 𝑐̃̃_𝑖1(𝜔),𝑐̃̃_𝑖2(𝜔),…,𝑐̃̃_𝑖𝑛(𝜔) are independent birandom variables, then Pr{𝜔∣Pr(𝑐_𝑖̃̃(𝜔)^𝑇𝑥≥𝑓_𝑖)≥𝛼_𝑖}≥𝛼_𝑖 holds if and only if 𝜇_𝑖^𝑐𝑇𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐′_𝑖𝑥≥𝑓_𝑖, (23) where Φ is the standardized normal distribution.

Proof. Let 𝑦_𝑖=𝑐^𝑇_𝑖̃̃(𝜔)𝑥−𝑓_𝑖; then 𝑦_𝑖 follows normal distribution with the following expected value and variance: 𝐸𝑦_𝑖=𝑐_𝑖̃(𝜔)^𝑇𝑥−𝑓_𝑖,Var𝑦_𝑖=𝑥^𝑇𝑉^𝑐_𝑖𝑥, (24) where 𝑐_𝑖̃(𝜔)=(𝑐̃_𝑖1(𝜔),𝑐̃_𝑖2(𝜔),…,𝑐̃_𝑖𝑛(𝜔))^𝑇. We note that 𝑐_𝑖̃̃(𝜔)^𝑇𝑥−𝑓_𝑖−𝑐_𝑖̃(𝜔)^𝑇𝑥−𝑓_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥 (25) must be standardized normally distributed. The inequality 𝑐_𝑖̃̃(𝜔)^𝑇𝑥≥𝑓_𝑖 is equivalent to 𝑐_𝑖̃̃(𝜔)^𝑇𝑥−𝑓_𝑖−𝑐_𝑖̃(𝜔)^𝑇𝑥−𝑓_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥𝑐≥−^𝑖(𝜔)^𝑇𝑥−𝑓_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥. (26) Let 𝜂=((𝑐_𝑖̃̃(𝜔)^𝑇𝑥−𝑓_𝑖)−(𝑐_𝑖̃(𝜔)^𝑇𝑥−𝑓_𝑖))/𝑥^𝑇𝑉^𝑐_𝑖𝑥, which is standardized normally distributed. Then we have Pr𝑐̃̃_𝑖𝑗(𝜔)^𝑇𝑥≥𝑓_𝑖≥𝛼_𝑖⟺Pr𝜂≥𝑓_𝑖−𝑐_𝑖̃(𝜔)^𝑇𝑥𝑥^𝑇𝑉^𝑐_𝑖𝑥≥𝛼_𝑖⟺1−Pr𝜂≤𝑓_𝑖−𝑐_𝑖̃(𝜔)^𝑇𝑥𝑥^𝑇𝑉^𝑐_𝑖𝑥≥𝛼_𝑖⟺Pr𝜂≤𝑓_𝑖−𝑐_𝑖̃(𝜔)^𝑇𝑥𝑥^𝑇𝑉^𝑐_𝑖𝑥≤1−𝛼_𝑖⟺Φ𝑓_𝑖−𝑐_𝑖̃(𝜔)^𝑇𝑥𝑥^𝑇𝑉^𝑐_𝑖𝑥≤1−𝛼_𝑖⟺𝑓_𝑖−𝑐_𝑖̃(𝜔)^𝑇𝑥𝑥^𝑇𝑉_𝑖𝑥≤Φ⁻¹1−𝛼_𝑖⟺𝑐_𝑖̃(𝜔)^𝑇𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥≥𝑓_𝑖. (27) Thus Pr𝜔∣Pr𝑐̃̃_𝑖𝑗(𝜔)^𝑇𝑥≥𝑓_𝑖≥𝛼_𝑖≥𝛼_𝑖⟺Pr𝑐_𝑖̃(𝜔)^𝑇𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥≥𝑓_𝑖≥𝛼_𝑖⟺Pr𝑐_𝑖̃(𝜔)^𝑇𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥−𝑓_𝑖−𝜇_𝑖^𝑐𝑇𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥−𝑓_𝑖×𝑥^𝑐𝑇𝑉^𝑐′_𝑖𝑥⁻¹𝜇≥−^𝑖𝑐𝑇𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥−𝑓_𝑖𝑥^𝑇𝑉^𝑐′_𝑖𝑥≥𝛼_𝑖⟺𝜇_𝑖^𝑐𝑇𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥+Φ⁻¹1−𝛼_𝑖×𝑥^𝑇𝑉^𝑐′_𝑖𝑥≥𝑓_𝑖. (28) This completes the proof.

Lemma 14 (see [21]). Assume that birandom vector 𝑎_𝑟(𝜔)=(𝑎_𝑟1(𝜔),𝑎_𝑟2(𝜔),…,𝑎_𝑖𝑛(𝜔))^𝑇 follows normal distribution with mean vector 𝑎_𝑟(𝜔) and positive definite covariance matrix 𝑉^𝑎_𝑟, written as 𝑎_𝑟(𝜔)∼𝒩(𝑎_𝑟(𝜔),𝑉^𝑎_𝑟). 𝑎_𝑟(𝜔) is a normally distributed random variable, written as 𝑎_𝑟(𝜔)∼𝒩(𝜇^𝑎_𝑟,𝑉_𝑟^′𝑎). Birandom variable 𝑏̃̃_𝑟(𝜔) follows normal distribution with mean value 𝑏̃_𝑟(𝜔) and variance (𝜎^𝑏_𝑟)², denoted by 𝑏̃̃_𝑟(𝜔)∼𝒩(𝑏̃_𝑟(𝜔),(𝜎^𝑏_𝑟)²). 𝑏̃_𝑟(𝜔) is a normally distributed random variable, written as 𝑏̃_𝑟(𝜔)∼𝒩(𝜇^𝑏_𝑟,(𝜎_𝑟^′𝑏)²). If 𝑎_𝑟1(𝜔),𝑎_𝑟2(𝜔),…,𝑎_𝑟𝑛(𝜔),𝑏̃̃_𝑟(𝜔) are independent birandom variables, then Ch^𝑒{𝑎^𝑇_𝑟𝑥≤𝑏̃̃_𝑟}≥𝛽_𝑟 holds if and only if 𝑢_𝑟^𝑎𝑇𝑥+Φ⁻¹𝛽_𝑟𝑥^𝑇𝑉^𝑎_𝑟𝑥+𝜎^𝑏_𝑟²+Φ⁻¹𝛽_𝑟𝑥^𝑇𝑉^𝑎′_𝑟𝑥+𝜎^𝑏′_𝑟²≤𝜇^𝑏_𝑟,𝑟=1,2,…,𝑝, (29) where 𝛽_𝑟 are predetermined confidence levels, 𝑟=1,2,…,𝑝.

It follows from Lemmas 13 and 14 that model (21) is equivalent to the following crisp multiobjective programming model: max𝑓₁,𝑓₂,…,𝑓_𝑚s.t.𝑓_𝑖≤𝜇_𝑖^𝑐𝑇𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐_𝑖𝑥+Φ⁻¹1−𝛼_𝑖𝑥^𝑇𝑉^𝑐′_𝑖𝑥,𝑖=1,2,…,𝑚,𝑥∈𝑋, (30) or equivalently max_𝑥∈𝑋𝐹₁(𝑥),𝐹₂(𝑥),…,𝐹_𝑚(𝑥), (31) where 𝐹_𝑖(𝑥)∶=𝜇_𝑖^𝑐𝑇𝑥+Φ⁻¹(1−𝛼_𝑖)𝑥^𝑇𝑉^𝑐_𝑖𝑥+Φ⁻¹(1−𝛼_𝑖)𝑥^𝑇𝑉^𝑐′_𝑖𝑥, 𝑖=1,2,…,𝑚, and 𝑋∶={𝑥∣𝑢_𝑟^𝑎𝑇𝑥+Φ⁻¹(𝛽_𝑟)𝑥^𝑇𝑉^𝑎_𝑟𝑥+(𝜎^𝑏_𝑟)²+Φ⁻¹(𝛽_𝑟)𝑥^𝑇𝑉^𝑎′_𝑟𝑥+(𝜎^𝑏′_𝑟)²≤𝜇^𝑏_𝑟,𝑥∈𝐷,𝑟=1,2,…,𝑝}.

3. Solution Method

If model (17) satisfies the conditions for Lemmas 13 and 14, then model (17) can be converted into a crisp multiobjective programming model. Many classical methods, such as the goal programming method, the interactive method, and the weighted sum method, can be used to solve it. However, for a general case, it is a difficult to convert the general model into its deterministic equivalent for the predetermined confidence levels. In order to handle the birandom objective functions and to check the birandom equilibrium chance constraints, we used a birandom simulation technique as this method is similar to a stochastic simulation but more complicated and time consuming.

For the following constraints, Pr𝜔∈Ω∣Pr𝑐_𝑖̃̃(𝜔)^𝑇𝑥≥𝑓_𝑖≥𝛼_𝑖≥𝛼_𝑖,𝑖=1,2,…,𝑚; (32) in view of the purpose of maximizing 𝑓_𝑖, we should find the maximal 𝑓_𝑖 such that (32) holds for a given 𝑥. It suffices to estimate the maximal value of 𝑓_𝑖 such that the probability of the following random event 𝜔∈Ω∣Pr𝑐_𝑖̃̃(𝜔)^𝑇𝑥≥𝑓_𝑖≥𝛼_𝑖 (33) is not less than 𝛼_𝑖.

First, we generate 𝑁 independent vectors 𝜔^𝑘=(𝜔^𝑘₁,𝜔^𝑘₂,…,𝜔^𝑘_𝑛) from Ω according to distribution function. Then 𝑐_𝑖̃̃(𝜔^𝑘) are random vectors, 𝑘=1,2,…,𝑁. So we can apply the stochastic simulation to handle Pr𝑐_𝑖̃̃𝜔^𝑘𝑇𝑥≥𝑓_𝑖,𝑖=1,2,…,𝑚. (34) We define ℎ𝑥,𝑐_𝑖̃̃𝜔^𝑘=1,if𝑐_𝑖̃̃𝜔^𝑘𝑇𝑥≥𝑓_𝑖,0,otherwise, (35) for 𝑘=1,2,…,𝑁, which are random variables, and the expected value 𝐸[ℎ(𝑥,𝑐_𝑖̃̃(𝜔^𝑘))]=𝛼_𝑖 for all 𝑘. By the strong law of large numbers, we obtain ∑^𝑁_𝑘=1ℎ𝑥,𝑐_𝑖̃̃𝜔^𝑘𝑁⟶𝛼_𝑖 (36) as 𝑁→∞. Note that ∑^𝑁_𝑘=1ℎ(𝑥,𝑐_𝑖̃̃(𝜔^𝑘)) is just the number of 𝑐_𝑖̃̃(𝜔^𝑘) satisfying 𝑐_𝑖̃̃(𝜔^𝑘)^𝑇𝑥≥𝑓_𝑖 for 𝑖=1,2,…,𝑁. Thus 𝑓_𝑖 is just the 𝑁^′th largest element in the sequence {𝑐_𝑖̃(𝜔¹)^𝑇𝑥,𝑐_𝑖̃̃(𝜔²)^𝑇𝑥,…,𝑐_𝑖̃̃(𝜔^𝑁)^𝑇𝑥}, where 𝑁^′ is the integer part of 𝛼_𝑖𝑁. The process to estimate the maximal 𝑓_𝑖 such that (32) holds is summarized as follows.

Step 1. Sample 𝜔¹,𝜔²,…,𝜔^𝑁 from Ω according to distribution function.

Step 2. Find the maximal values 𝑓_𝑖𝑛 such that Pr{𝑐^𝑇_𝑖̃̃(𝜔^𝑘)𝑥≥𝑓_𝑖𝑛}≥𝛼_𝑖 for 𝑛=1,2,…,𝑁, respectively, by stochastic simulation.

Step 3. Set 𝑁^′ as the integer part of 𝛼_𝑖𝑁.

Step 4. Return the 𝑁^′th largest element in {𝑓_𝑖1,𝑓_𝑖2,…,𝑓_𝑖𝑁} as 𝑓_𝑖, 𝑖=1,2,…,𝑚.

For a fixed 𝑥, we check whether the constraint Pr𝜔∣Pr𝑎_𝑟(𝜔)^𝑇𝑥≤𝑏̃̃_𝑟(𝜔)≥𝛽_𝑟≥𝛽_𝑟,𝑟=1,2,…,𝑝, (37) holds by following process.

Step 1. Set 𝑁^′=0.

Step 2. Generate 𝜔=(𝜔₁,𝜔₂,…,𝜔_𝑛)^𝑇 from Ω according to distribution function.

Step 3. Calculate Pr{𝑎_𝑟(𝜔)^𝑇𝑥≤𝑏̃̃_𝑟(𝜔)} by stochastic simulation.

Step 4. If Pr{𝑎_𝑟(𝜔)^𝑇𝑥≤𝑏̃̃_𝑟(𝜔)}≥𝛽_𝑟, then 𝑁^′=𝑁^′+1.

Step 5. Repeat Step 2 to Step 4𝑁 times.

Step 6. If 𝑁^′_𝑟/𝑁≥𝛽_𝑟, return 𝑥 is feasible, or else 𝑥 is infeasible.

Genetic algorithms (GAs) are stochastic search methods for optimization problems based on the mechanics of natural selection and natural genetics. They have been applied to different sectors for both technical and management problems and have shown good performance [22–26]. In this paper, the birandom simulation technique is embedded into a GA to develop the birandom simulation-based GA. The overall procedure of the birandom simulation-based GA for solving the birandom programming models is shown in Figure 2. The main parts of the algorithm are stated in more detail as follows.

Figure 2 

The flowchart of the birandom simulation-based GA.

(1) Representation: a vector 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛) is chosen as a chromosome to represent a solution to the optimization problem.

(2) Handling the constraints: to ensure the chromosomes generated by genetic operators are feasible, we can use the technique of birandom simulation to check them.

(3) Initializing process: suppose that the decision maker is able to predetermine a region which contains the feasible set. Randomly generate a vector 𝑥 from this region until a feasible one is accepted as a chromosome. Repeat the above process 𝑁_pop-size times; then we have 𝑁_pop-size initial feasible chromosomes 𝑥¹,𝑥²,…,𝑥^𝑁pop-size. In general, the larger the number of population size 𝑁_pop-size is, the better the result is. However, the computation time will increase with 𝑁_pop-size increasing. In the paper, we set 𝑁_pop-size from 20 to 40.

(4) Evaluation function: let eval(𝑥) equal weighted sum of all objectives; that is, eval(𝑥)=^𝑚_𝑖=1𝑤_𝑖𝑓_𝑖, (38) where 𝑤_𝑖 are random numbers in [0,1], representing the weight for objective 𝑓_𝑖,  𝑖=1,2,…,𝑚. The random weight-sum approach explores the entire solution space in order to avoid local optima and thus gives a uniform chance to search all the possible solutions. All chromosomes are arranged from large to small according to their objective function values. In other words, after rearrangement, 𝑥¹ is the best chromosome, and 𝑥^𝑁pop-size is the worst one.

(5) Selection process: the selection process is based on spinning the roulette wheel 𝑁_pop-size times. Each time a single chromosome for a new population is selected in the following way: calculate the cumulative probability 𝑞_𝑖 for each chromosome 𝑥^𝑖: 𝑞₀=0,𝑞_𝑖=^𝑖_𝑗=1eval𝑥^𝑗,𝑖=1,2,…,𝑁_pop-size. (39) Generate a random number 𝑟 in [0,𝑞_𝑁pop-size] and select the 𝑖th chromosome 𝑥_𝑖 such that 𝑞_𝑖−1<𝑟≤𝑞_𝑖,  1≤𝑖≤𝑁_pop-size. Repeat the above process 𝑁_pop-size times and we obtain 𝑁_pop-size copies of chromosomes.

(6) Crossover operation: generate a random number 𝑐 from the open interval (0, 1) and the chromosome 𝑥^𝑖 is selected as a parent provided that 𝑐<𝑃_𝑐, where parameter 𝑃_𝑐 is the probability of crossover operation. Repeat this process 𝑁_pop-size times and 𝑃_𝑐⋅𝑁_pop-size chromosomes are expected to be selected to undergo the crossover operation. The crossover operator on 𝑥¹ and 𝑥² will produce two children 𝑦¹ and 𝑦² as follows: 𝑦¹=𝑐𝑥¹+(1−𝑐)𝑥²,𝑦²=𝑐𝑥²+(1−𝑐)𝑥². (40) If both children are feasible, then we replace the parents with them, or else we keep the feasible one if it exists. Repeat the above operation until two feasible children are obtained or a given number of cycles is finished.

Let us consider the setting of probability of crossover 𝑃_𝑐. It is obvious that larger crossover probability results in reaching larger solution space, and then it helps reduce the chance of stopping at nonoptimal solution. However, too large 𝑃_𝑐 will result in considerable time consuming because of too much searching in unnecessary solution space. In the paper, we set 𝑃_𝑐 from 0.2 to 0.4.

(7) Mutation operation: similar to the crossover process, the chromosome 𝑥^𝑖 is selected as a parent to undergo the mutation operation provided that random number 𝑚<𝑃_𝑚, where parameter 𝑃_𝑚 is the probability of mutation operation. 𝑃_𝜆𝑖⋅𝑁_pop-size are expected to be selected after repeating the process 𝑁_pop-size times. Suppose that 𝑥¹ is chosen as a parent. Choose a mutation direction 𝐝∈𝐑^𝐧 randomly. Replace 𝑥 with 𝑥+𝑀⋅𝐝 if 𝑥+𝑀⋅𝐝 is feasible; otherwise we set 𝑀 as a random number between 0 and 𝑀 until it is feasible or a given number of cycles are finished. Here, 𝑀 is a sufficiently large positive number.

𝑃_𝑚 controls the proportion of new genes generating in population. If 𝑃_𝑚 is too small, it will be difficult for some effective genes to be selected. On the contrary, if 𝑃_𝑚 is too large, that is, there exists too much random change, then offsprings may lose good characteristics inherited from their parents. Thus, the algorithm will lose the learning ability from the past searching. In the paper, we set 𝑃_𝑚 from 0.2 to 0.4.

After running a given number of cycles of above birandom simulation-based genetic algorithm, the best chromosome can be regarded as the optimal solution.

4. Practical Application for H Chain Co., Ltd.

A real-world inventory problem is discussed in this section to illustrate the effectiveness of the theoretical results in this paper.

4.1. Key Problem Statement and Model Formulation

The problem is from H Chain Co., Ltd., which is one of largest chain enterprises in southwest of China. It operates a variety of products, such as eggs, grains, milk, and meat. Inventory management is an important part of company management. The inventory system of H Chain Co., Ltd. is illustrated by Figure 3.

Figure 3 

The inventory system of H Chain Co., Ltd.

A new executive is assigned to the management of four product items: eggs, dairy products, grains, and meat products. The executive needs to make an effective inventory strategy to reduce costs and increase profit.

4.1.1. Assumptions

In order to develop the mathematical model for the inventory problem, there are some assumptions made, which are as follows. (1) This is an inventory model with a finite time horizon. (2) There are multiple items in this inventory system. (3) Average demand rates are continuous and constant. (4) Deterioration rates are constant. (5) Shortage is allowed. (6) Initial inventory quantities are not zeros; (7) The inventory system uses a continuous review fixed-order-quantity strategy. (8) The lead time is proportional to the order quantity. (9) Fixed-ordering costs are constant. (10) Unit price, unit acquisition cost, holding cost per unit per unit time and shortage cost per unit per unit time are birandom variables. (11) Available total storage space is limited. (12) Available total budget is limited. (13) The inventory system uses nonintegrated management.

4.1.2. Notations

The following notations are used to describe this inventory model:𝑛 is the number of items; 𝑊_𝑖 is the available storage space of warehouse of product 𝑖, 𝑖=1,2,…,𝑛; 𝐵 is the available total budgetary cost; 𝐾_𝑖 is the required storage area per unit quantity 𝑖, 𝑖=1,2,…,𝑛; 𝐴_𝑖 is the fixed-ordering cost of product 𝑖, 𝑖=1,2,…,𝑛; 𝐷_𝑖 is the average demand rate of product 𝑖, 𝑖=1,2,…,𝑛; 𝐼_𝑖 is the initial demand rate of product 𝑖, 𝑖=1,2,…,𝑛; 𝑎_𝑖 is the average deteriorating rate of product 𝑖, 𝑖=1,2,…,𝑛; 𝑙_𝑖 is the lead time for unit product 𝑖, 𝑖=1,2,…,𝑛; 𝑝_𝑖 is the unit selling price of product 𝑖 (birandom variable), 𝑖=1,2,…,𝑛; 𝑐̃̃_𝑖1 is the unit acquisition cost of product 𝑖 (birandom variable), 𝑖=1,2,…,𝑛; 𝑐̃̃_𝑖2 is the unit shortage cost of product 𝑖 (birandom variable), 𝑖=1,2,…,𝑛; 𝑠_𝑖 is the reorder point of product 𝑖 (decision variable), 𝑖=1,2,…,𝑛; 𝑄_𝑖 is the order quantity of product 𝑖 (decision variable), 𝑖=1,2,…,𝑛; 𝑁_𝑖 is the net revenue of product 𝑖, 𝑖=1,2,…,𝑛; 𝐶_𝑖1 is the accquisition cost of product 𝑖, 𝑖=1,2,…,𝑛; 𝐶_𝑖2 is the holding cost of product 𝑖, 𝑖=1,2,…,𝑛; 𝐶_𝑖3 is the shortage cost of product 𝑖, 𝑖=1,2,…,𝑛; 𝑇𝑃_𝑖 is the average profit of product 𝑖, 𝑖=1,2,…,𝑛.

4.2. Model Formulation

It follows from the assumptions above that the inventory level 𝑞_𝑖(𝑡) of product 𝑖 at time 𝑡 can be illustrated by Figure 4.

Figure 4 

Inventory level of product 𝑖 at time 𝑡.

In Figure 4, 𝑡_𝑖1 is the time of order point for product 𝑖; 𝑡_𝑖2 is the time when inventory level of product 𝑖 reduces to zero; 𝑡_𝑖3 is the time when replenishment of product 𝑖 arrives; 𝑡_𝑖4 is the time when inventory level of product 𝑖 reduces to reorder point.

From Figure 4, we have 𝑡_𝑖1=𝐼_𝑖−𝑠_𝑖𝐷_𝑖+𝑎_𝑖,𝑡_𝑖2=𝐼_𝑖𝐷_𝑖+𝑎_𝑖,𝑡_𝑖3=𝑡_𝑖1+𝑙_𝑖𝑄_𝑖=𝐼_𝑖−𝑠_𝑖𝐷_𝑖+𝑎_𝑖+𝑙_𝑖𝑄_𝑖,𝑡_𝑖4=𝑡_𝑖3+𝑄_𝑖−𝑠_𝑖−𝑡_𝑖3−𝑡_𝑖2𝐷_𝑖𝐷_𝑖+𝑎_𝑖=𝐼_𝑖−𝑠_𝑖𝐷_𝑖+𝑎_𝑖+𝑙_𝑖𝑄_𝑖+𝑄_𝑖−𝑠_𝑖−−𝑠_𝑖/𝐷_𝑖+𝑎_𝑖+𝑙_𝑖𝑄_𝑖𝐷_𝑖𝐷_𝑖+𝑎_𝑖=𝐼_𝑖+𝑄_𝑖−2𝑠_𝑖−−𝑠_𝑖/𝐷_𝑖+𝑎_𝑖+𝑙_𝑖𝑄_𝑖𝐷_𝑖𝐷_𝑖+𝑎_𝑖+𝑙_𝑖𝑄_𝑖. (41)

For product 𝑖, the total profit from initial time 0 to 𝑡_𝑖4 equals the net revenue minus total cost, including fixed-ordering cost, acquisition cost, holding cost, and shortage cost. The net revenue is equal to unit price multiplied by sale quantity, that is, 𝑁_𝑖=𝑝_𝑖𝑡_𝑖2+𝑡_𝑖4−𝑡_𝑖3𝐷_𝑖=𝑝_𝑖𝐼_𝑖+𝑄_𝑖−𝑠_𝑖−𝑙_𝑖𝑄_𝑖𝐷_𝑖𝐷_𝑖+𝑎_𝑖𝐷_𝑖. (42)

The acquisition cost is equal to unit acquisition cost multiplied ordering quantity; that is, 𝐶_𝑖1=𝑐̃̃_𝑖1𝑄_𝑖. (43)

The total holding cost consists of holding cost from 0 to 𝑡_𝑖2 and holding cost from 𝑡_𝑖3 to 𝑡_𝑖4; that is, 𝐶_𝑖2=𝑐̃̃_𝑖2^𝑡𝑖2₀𝑞_𝑖(𝑡)𝑑𝑡+^𝑡𝑖4_𝑡𝑖3𝑞_𝑖(𝑡)=𝑐̃̃𝑑𝑡_𝑖2𝐼_𝑖2𝑡_𝑖2+𝑄_𝑖−𝑡_𝑖3−𝑡_𝑖2𝐷_𝑖+𝑠_𝑖2𝑡_𝑖4−𝑡_𝑖3=𝑐̃̃_𝑖2𝐼²_𝑖+𝑄_𝑖−𝑙_𝑖𝑄_𝑖𝐷_𝑖²−𝑠²_𝑖2𝐷_𝑖+𝑎_𝑖. (44)

The shortage cost occurs from 𝑡_𝑖2 to 𝑡_𝑖3; that is, 𝐶_𝑖3=𝑐̃̃_𝑖3^𝑡𝑖3_𝑡𝑖3𝐷_𝑖𝑑𝑡=𝑐̃̃_𝑖3𝑡_𝑖3−𝑡_𝑖2𝐷_𝑖2=𝑐̃̃_𝑖3𝑙_𝑖𝑄_𝑖𝐷_𝑖2. (45)

The total profit of product 𝑖 from 𝑡_𝑖2 to 𝑡_𝑖4 is 𝑁_𝑖0−𝐴_𝑖−𝐶_𝑖1−𝐶_𝑖2−𝐶_𝑖3. Then the average total profit of product 𝑖 is formulated as 𝑇𝑃_𝑖=𝑁_𝑖−𝐴_𝑖−𝐶_𝑖1−𝐶_𝑖2−𝐶_𝑖3𝑡_𝑖4. (46)

It is noted that 𝑇𝑃_𝑖 can not be maximized due to birandomness. By utilizing the idea of Charnes and Cooper [17], 𝑓_𝑖 is maximized which satisfies Ch^𝑒𝑇𝑃_𝑖≥𝑓_𝑖≥𝛼_𝑖, (47) where Ch^𝑒{⋅} is the equilibrium chance and 𝛼_𝑖 is given confidence level.

The space of product 𝑖 ordered should not exceed the available storage space of warehouse; that is, 𝐾_𝑖𝑄_𝑖≤𝑊_𝑖. (48)

All the costs should not exceed the available total budgetary cost; that is, ∑^𝑚_𝑖=1(𝐶_𝑖1+𝐶_𝑖2+𝐶_𝑖3+𝐶_𝑖4)≤𝐵. It is noted that the constraint is meaningless mathematically due to the existence of birandom variable. We use equilibrium chance-constrained technique to make it meaningful mathematically, that is, Ch^𝑒𝑛_𝑖=1𝐴_𝑖+𝐶_𝑖1+𝐶_𝑖2+𝐶_𝑖3≤𝐵≤𝛽, (49) where 𝛽 is the confidence level predetermined by the decision maker.

Since shortage is allowed, the order quantity of product 𝑖 should not be less than reorder point 𝑠_𝑖; that is, 𝑄_𝑖≥𝑠_𝑖. (50)

In addition, It is natural to require the nonnegativity of decision variable; that is, 𝑄_𝑖≥0,𝑠_𝑖≥0. (51)

By integration of (47)–(51), we can formulate the programming model of the inventory problem as max𝑓₁,𝑓₂,…,𝑓_𝑚s.t.Ch^𝑒𝑇𝐶_𝑖≥𝑓_𝑖≥𝛼_𝑖𝐾,𝑖=1,2,…,𝑛,_𝑖𝑄_𝑖≤𝑊_𝑖𝑄,𝑖=1,2,…,𝑛,_𝑖≥𝑠_𝑖,𝑖=1,2,…,𝑛,Ch^𝑒𝑛_𝑖=1𝐴_𝑖+𝐶_𝑖1+𝐶_𝑖2+𝐶_𝑖3𝑄≤𝐵≤𝛽,_𝑖≥0,𝑠_𝑖≥0,𝑖=1,2,…,𝑛. (52)

4.3. Data and Compaction Result

All the data are from the marketing section of H Chain Co., Ltd. These data can be divided into two different groups: determined parameters and birandom parameters. Determined parameters include 𝑊_𝑖,  𝐵,  𝐾_𝑖,  𝐴_𝑖,  𝐷_𝑖,  𝐼_𝑖,  𝑎_𝑖, and 𝑙_𝑖. The available total budgetary cost 𝐵=250000 yuan. Except for 𝐵, all determined parameters are listed in Table 1. Birandom variables including 𝑝_𝑖 (unit selling price of product 𝑖), 𝑐̃̃_𝑖1 (unit acquisition cost of product 𝑖), and 𝑐̃̃_𝑖2 (unit shortage cost of product 𝑖). These birandom variables are listed in Table 2. The decision maker set confident levels 𝛼₁=𝛼₂=𝛼₃=𝛼₄=𝛽=0.8 and weighs 𝑤₁=𝑤₂=𝑤₃=𝑤₄=0.25.

Table 1 

Determined parameters.

Table 2 

Birandom variables.

We implement the birandom simulation-based genetic algorithm in Matlab 7.0 (the population size 𝑁_pop-size=30, the probability of crossover 𝑃_𝑐=0.3, and the probability of mutation 𝑃_𝑚=0.2) and ran on a PC, 2.40 GHz with 1024 MB memory. After running 500 generations, the final solution is listed in Table 3. In addition, the objective function values are (𝑓^∗₁,𝑓^∗₂,𝑓^∗₃,𝑓^∗₄)=(1878.8,1256.6,2890.3,3097.4).

Table 3 

Optimal order quantities and reorder points.

5. Further Discussions

To show the effectiveness of the model and algorithm better, evaluation, analysis, and comparison from various aspects are presented.

5.1. Algorithm Evaluation

We evaluate the birandom simulation-based algorithm from two aspects.

First, we will show that the birandom simulation-based algorithm is robust to genetic parameters. We compare the results by the algorithm with different parameters, including population size 𝑁_pop-size, probability of crossover 𝑃_𝑐, probability of mutation 𝑃_𝑚, cycles in birandom simulation, and generations in GA. These results are listed in Table 4. It appears that almost all the objective values differ little from each other. In fact, each relative error does not exceed 3% when different parameters are selected, which implies that the proposed algorithm is robust to the parameters setting and effective to solve the problem in this paper.

Table 4 

Comparison of the results obtained with different parameters.

Second, since the functions in model (52) are linear, by Lemmas 13 and 14, model (52) can be transformed into its crisp equivalent model. Assume that all the parameters are the same and the final objective function values are (𝑓^∗₁,𝑓^∗₂,𝑓^∗₃,𝑓^∗₄)=(1881.2,1253.6,2895.3,3093.8) by using the weighted sum method. The results obtained by the birandom simulation genetic algorithm and by the traditional weighted sum method are close to each other, which implies the effectiveness of the birandom simulation-based genetic algorithm proposed in this paper.

5.2. Sensitivity Analysis

However, under a different decision making environment and different conditions, the decision maker may alter the predetermined confidence levels. Then the optimal solution will alter accordingly. In order to show how the optimal solutions change along with the predetermined confidence levels, we conducted a sensitivity analysis as shown in Table 5. As shown from Table 5, a bigger 𝛽 induces a better result, because a bigger 𝛽 extends the feasible region. Conversely, a bigger 𝛼_𝑖 induces a worse result since a bigger 𝛼_𝑖 narrows the feasible region.

Table 5 

Sensitivity analysis.

5.3. Model Comparison

Birandom system is a generation for stochastic system. For illustrating the advantage of birandom system, all the birandom variables in (52) are assumed to be random variables, and other parameters keep original values. Each birandom variable in (52) has the following form: ̃̃̃𝜉∼𝒩𝜉,𝜎²,𝜉̃∼𝒩𝜇,𝜎^′2. (53) Replace 𝜉̃ with 𝜇, which is a constant. Then ̃̃𝜉 degenerates into a random variable. For example, we use 𝑝₁₁(∼𝒩(16,1)) to replace 𝑝₁₁. As all the birandom variables are replaced by random variables, equilibrium chance Ch^𝑒 degenerates into probability Pr. Then (52) can be written as max𝑓₁,𝑓₂,…,𝑓_𝑚s.t.Pr𝑇𝐶_𝑖≥𝑓_𝑖≥𝛼_𝑖𝐾,𝑖=1,2,…,𝑛,_𝑖𝑄_𝑖≤𝑊_𝑖𝑄,𝑖=1,2,…,𝑛,_𝑖≥𝑠_𝑖,𝑖=1,2,…,𝑛,Pr^𝑛_𝑖=1𝐴_𝑖+𝐶_𝑖1+𝐶_𝑖2+𝐶_𝑖3𝑄≤𝐵≤𝛽,_𝑖≥0,𝑠_𝑖≥0,𝑖=1,2,…,𝑛. (54) This stochastic system is a special case of a birandom system, which can be solved using a stochastic simulation-based GA. Similar to the birandom simulation-based GA, the stochastic simulation-based GA is formed by embedding stochastic simulation into a GA. The robustness of the stochastic simulation-based GA can be verified as similar to the birandom simulation-based GA.

In order to make a comparison with the birandom system, we solve problem (54) with following parameter: 𝑁_pop-size=30, 𝑃_𝑐=0.3, and 𝑃_𝑚=0.2, 500 cycles in stochastic simulation, 500 generations in GA. The final result is (𝑓^∗₁,𝑓^∗₂,𝑓^∗₃,𝑓^∗₄)=(1873.1,1231.5,2790.2,3016.9), which is worse than the result of the birandom system. Thus it shows that birandom system is more flexible and can generate better objective values than the corresponding stochastic system.

5.4. Managerial Insight

Some managerial insights can be derived from the above discussions. From the sensitivity analysis, we can see the following. (1) A pessimistic decision maker will set lower confidence levels to avoid higher risk. As a result, the values for the objective functions are worse than those at higher confidence levels. (2) An optimistic decision maker will set higher confidence levels to obtain better values for the objective functions. However, there will be the cost of higher risk. To sum up, the values for the objective functions depend on the attitude of the decision maker towards risk. From a comparison between the stochastic model and the birandom model, the decision maker can obtain more flexible results if they can construct a birandom model rather than a stochastic model in a complex stochastic decision environment. However, depicting the complex stochastic decision environment as birandom variables costs more, so the decision maker must make a trade-off between the cost and the benefit.

6. Conclusions

In this paper, we have formulated an equilibrium chance-constrained multiobjective programming model with birandom parameters, which extends the general multiobjective programming model. We converted a special linear model into a crisp multiobjective programming model which can be solved using traditional techniques. A modified genetic algorithm was designed by embedding the birandom simulation technique to deal with the general model. By assuming that some parameters (unit selling price, unit acquisition cost, etc.) are birandom variables, we developed an equilibrium chance-constrained multiobjective programming model with birandom parameters for an inventory problem and solved it using the birandom simulation-based GA. Further discussions show the effectiveness of the proposed model and algorithm. Some managerial insights may help decision makers improve their decision making.

Though the inventory problem considered is specific, the theoretical results can be extended to various inventory problems. In the future, detailed analysis, further research, and practical application of the model and algorithm proposed in this paper will be considered.

Acknowledgments

The authors would like to give their great appreciates to the editors and anonymous referees for their helpful comments and valuable suggestions which improved the content and composition substantially. This research was supported by the Programs of NSFC (Grant nos. 70831005 and 71273036), Projects of International Cooperation and Exchanges NSFC (Grant no. 71011140076), “985” Program of Sichuan University (Innovative Research Base for Economic Development and Management), Sichuan University Young Teachers Scientific Research Start Funds (Grant no. 2012SCU11016), and the Key Program of Sichuan System Science and Enterprise Development Research Center (Grant no. Xq13B04).