# An Uncertain Aggregate Production Planning Model Considering Investment in Vegetable Preservation Technology

Authors: Yufu Ning; Na Pang; Xiao Wang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, we study the aggregate production planning problem for vegetables within the framework of uncertainty theory. In detail, preservation technology investment is taken into consideration to reduce the deterioration rate and improve the freshness of the vegetables. Meanwhile, an expected profit model considering preservation technology investment under the capacity constraints is built, whose objective is to find the optimal yield, workforce, and preservation investment strategies. Moreover, the proposed model can be transformed into its crisp equivalent form. Finally, a numerical example is carried out to illustrate the effectiveness of the proposed uncertain aggregate production planning model.


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## Body

## 1. Introduction

Aggregate production planning (APP) is a large-scale and top-level production strategy, whose aim is to meet the market demand and achieve the maximum profit or minimum cost by adjusting the output, workforce, and other controllable factors for all kinds of products over a finite planning horizon. Since Holt et al. [1] proposed a linear decision rule for production and employment scheduling in 1955, researchers have developed a large number of models to solve the aggregate production planning problem, such as [2, 3]. Flights and lodgings, bread and milk, vegetables and fruit are all classified as perishable products with the properties of low salvage value and volatile markets, which makes every step of the entire supply chain face greater risks. As the initial link of the supply chain, the practical production planning problem will be considered. As a special category of perishable goods, APP problem for fresh vegetables can learn from the perishable goods.For the fresh vegetables, the market demand and deterioration rate are two important factors in production process. The demand is an important driving force for product flow in the supply chain. Then the deterioration rate is used to indicate the proportion that the amount of the perished products accounts for the inventory level and it is the best interpreter of the perishable nature for vegetables. Scholars usually considered the deterioration rate and the market demand to be constant or time-dependent. Ghare and Schrader [4] firstly assumed a constant rate of deterioration following the basic procedure for determination and studied the perishable inventory problems with a deterministic demand.However, some scholars presumed the deterioration rate and market demand might change over time. Hence, an economic order quantity (EOQ) model for items with Weibull distribution deterioration was proposed by Covert and Philip [5]. In addition, Deb and Chaudhuri [6] proposed a perishable goods production planning model which was permitted a shortage inventory and defined the demand changing over time and obeying the linear trend, and then a heuristic procedure was come up with to solve the production planning. In fact, influenced by quantities of uncertain factors, the market demand may be indeterministic. So the scholars handled the APP problem under uncertain environment and a production planning model for a deteriorating rate with stochastic demand and consumer choice was presented by Lodree and Uqochukwu [7]. Then, a production planning model considering uncertain demand using two-stage stochastic programming in a fresh vegetable supply chain context was presented by Jordi Mateo et al. [8]. Meanwhile, some scholars also studied the APP problem on the background of possibility and established lots of APP models [9, 10]. The problem of integrated production planning with uncertain parameters such as multiobjective, multiproduct, multiplanning period and demand, production cost, and production capacity was studied by Zhu et al [11].From the previous paragraph, the study on deterioration rate is one of the key points and a large number of researchers studied this item by various forms. However, most of these looked upon the deterioration rate as an inherent property of vegetables. In the real management process, many enterprises have studied the causes of deterioration and developed preservation technologies to control it and increase the profit. Hence, the deterioration rate can be controlled and reduced by means of effective capital investment in warehouse like procedural changes and specialized equipment acquisition. Taso and Sheen [12] analyzed the sensitivity of the parameters in numerous studies and revealed that a lower deterioration rate was considered beneficial from an economic viewpoint.To describe the practical inventory situation, Hsu et al. [13] proposed a deteriorating inventory with a constant deterioration rate and time-dependent partial backlogging, whose objective is to find the replenishment and preservation technology investment strategies. Assume that the preservation technology cost is a function of the length of replenishment cycle and incorporating time-varying deterioration and reciprocal time-dependent partial backlogging rates, Dye and Hsieh [14] presented an extended model. Wang and Dan [15] developed a time-varying consumer choice model influenced by the greenness and price of the fresh agricultures product. Li et al. [16] considered a joint ordering, pricing, and preservation technology investment decision problem for noninstantaneously deteriorating items with generalized price sensitive demand rate, time-varying deterioration rate, and no shortage.The above literature considered the deterioration rate to be controllable factors, and they studied the problem from the respects of increasing preservation investment to reduce the deterioration rate. In reality, preservation technology investment not only can reduce losses, but also will make the products being in a fresh condition. However, little attention has been paid to these two aspects at present. In this paper, we will introduce the preservation investment into this uncertain APP model and help the manufactures make reasonable decisions through establishing the relationship between the preservation investment and the deterioration rate and the relationship between the preservation investment and the freshness.As a predetermined production strategy for the future production, the aggregate production planning not only involves massive data over a long planning horizon, but also may be undergone by various unpredictable disruptions in the actual production, which makes the deterministic models perform very poorly and encourages us to cope with this problem with some uncertain methods. However, the characteristics of perishable product can make the actual data not be received directly. So belief degrees given by experts might be employed to estimate the distributions. Since surveys have shown that these human beings estimations generally possessed much wider range of values than the real ones, these belief degrees cannot be treated as random variables or fuzzy variables. To study the behaviour of uncertain phenomena, uncertainty theory was founded by Liu [17], where uncertain measure in uncertainty theory is used to indicate the belief degree. Now uncertainty theory has been developed steadily and applied widely, such as uncertain programming [18, 19], risk analysis [20, 21], uncertain differential equations [22, 23], uncertain portfolio [24], and finance [25, 26].With respect to the production planning problem, a multiproduct aggregate production planning model based on uncertainty theory was presented by Ning et al. [27] in 2013, whose studying object was the general products. Then Pang and Ning [28] built an uncertain aggregate production planning model according to the characteristic of sensitivity to storage time and overproduction; this model was based on the price discount affected by the freshness and the overproduction punishment subject to the penalty function under the capacity constraints. So, on the basis of Ning and Pang’s model and aiming at the particularities of vegetables, an uncertain APP model for the vegetables considering investment in preservation technology is built.The structure of the paper is organized as follows. Section2 introduces some basic concepts and theorems of uncertainty theory. In Section 3, we describe the uncertain APP problem for vegetables. In Section 4, an uncertain APP model is built and then it is transformed into the equivalent crisp form based on uncertainty theory. Then, we give a numerical example to illustrate the proposed model in Section 5. Finally, some conclusions are covered in Section 6.


## 2. Preliminary

Some foundational concepts and theorems of uncertainty theory are introduced in this part.Definition 1 (Liu [17, 29]).
LetL be a σ-algebra on a nonempty set Γ. A set function M is called anuncertain measure if it satisfies four axioms:(1)
MΓ=1;(2)
MΛ+MΛc=1 for any Λ∈L;(3)
for every countable sequence of{Λi}∈L, we have M⋃i=1∞Λi≤∑i=1∞MΛi;(4)
let(Γk,Lk,Mk) be uncertainty spaces for k=1,2,⋯; the product uncertain measure M is an uncertain measure satisfying (1)M∏k=1∞Λk=⋀k=1∞MkΛkwhere Λk are arbitrarily chosen events from Lk for k=1,2,⋯, respectively.Definition 2 (Liu [17]).
Anuncertain variableξ is a measurable function from an uncertainty space (Γ,L,M) to the set of real numbers.Definition 3 (Liu [17]).
Theuncertainty distributionΦ of an uncertain variable ξ is defined by (2)Φx=Mξ≤x,∀x∈R.Definition 4 (Liu [17]).
An uncertain variablesξ is called linear if it has a linear uncertainty distribution (3)Φx=0,ifx≤ax-ab-a,ifa<x≤b1,ifx>bdenoted by L(a,b), where a and b are real numbers with a≤b.Definition 5 (Liu [17]).
An uncertain variableξ is called zigzag if it has a zigzag uncertainty distribution (4)Φx=0,ifx≤ax-a2b-a,ifa<x≤bx+c-2b2c-b,ifb<x≤c1,ifx>cdenoted by Z(a,b,c), where a,b,c are real numbers with a<b<c.Definition 6 (Liu [17]).
An uncertain variableξ is callednormal if it has a normal uncertainty distribution (5)Φx=1+exp⁡πe-x3σ-1,x∈Rdenoted by ξ~N(e,σ), where e and σ are real numbers with σ>0.Theorem 7 (Liu and Ha [30]).
Assumeξ1,ξ2,…,ξn are independent uncertain variables with regular uncertainty distributions Φ1,Φ2,…,Φn, respectively. If the function f(x1,x2,…,xn) is strictly increasing with respect to x1,x2,⋯,xm and strictly decreasing with respect to xm+1,xm+2,…,xn, then the uncertain variable ξ=fξ1,ξ2,…,ξn has an expected value (6)Eξ=∫01fΦ1-1α,…,Φm-1α,Φm+1-11-α,…,Φn-11-αdα.Theorem 8 (Liu [20]).
Assumeξ1,ξ2,…,ξn are independent uncertain variables with regular uncertainty distributions Φ1,Φ2,…,Φn, respectively. If the function g(x1,x2,…,xn) is strictly increasing with respect to x1,x2,…,xm and strictly decreasing with respect to xm+1,xm+2,…,xn, then (7)Mgξ1,ξ2,…,ξn≤0≥αholds if and only if (8)gΦ1-1α,Φ2-1α,…,Φm-1α,Φm+1-11-α,…,Φn-11-α≤0.


## 3. Problem Description

Assume that a manufacturer intends to produceN kinds of vegetables over a planning horizon T including t periods. After production, the ripe vegetables will be stored in the inventory to wait to be bought by the distributors. The manufacturer should take account of a variety of uncertain factors and ensure that the output can keep up with the market demand. The deterioration rate θnt and the market demand Dnt will be affected by the nature of vegetables and the storage conditions as well as other factors, which makes them usually obtained on the basis of the belief degree form experienced experts instead of the historical data. Hence, we employ uncertain variable to denote these two factors. And we define the inventory cost cnt occupied per unit in period t as an uncertain variable.Meanwhile, the production planning will be restricted by limited resources, such as capital levelCtmax, workforce production capacity Wtmax, and machine production capacity Mtmax. So how to arrange the production reasonably to achieve maximal profit should be paid attention to by the decision maker. Then there may be various interference factors which might affect the growth of vegetables, like plant diseases and insect pests and poor weather conditions which might interfere with us to make an accurate judgment for the working hours of employee int and the running hours of machine mnt occupied. The same situation also appears on the largest capacity constraints Wtmax, Mtmax, and Ctmax, which will make the manufacturer unable to set a deterministic capacity level accurately for the production. Above all, we employ uncertain variables to denote these capacity constraints occupied per unit of vegetables and the largest capacity constraints.In addition, the freshness is a significant factor that affects customer requirements in sales. The fresh products can not only improve the market demand, but also diminish the possibility of metamorphism and decrease the superfluous deterioration cost. Thus taking the effective vegetables preservation measures and halting the speed at which freshness decreases with storage time is a good choice for manufacturer to raise enterprise profit and competition. However, additional preservation measures will inevitably cause the corresponding investment cost. Hence, how to deal with the relationship of preservation costs and profit reasonably has become a problem worth considering for decision makers. The other assumptions and simplifications are stated as follows.( 1 ) The vegetables will be stored in the inventory after harvest and begin deteriorating at that time. Once being sold or going bad, this part of vegetables will leave the storage and no longer expand the inventory cost.( 2 ) The characteristics of freshness and deterioration make vegetables difficult to cross-cycle sale. Hence, there is no beginning inventory in each period.( 3 ) To simplify, vegetables are not allowed out of stock and we take no account of inventory capacity constraint.( 4 ) Initial sales price, production cost, processing cost, labor cost, and original freshness attenuation index are deterministic and constant.The parameters of this model are shown in Table1. To sum up, we set Dnt, θnt, cnt, int, mnt, Wtmax, Mtmax, Ctmax as uncertain variables. And Qnt, Ht, xnt, ωnt are set as decision variables.Table 1
Parameters of the APP model.
Notation Meaning N Types of vegetables T Planning horizon f Profit function overT D n t Demand for thenth product in period t (units) r n t Unit initial selling price of thenth product in period t ($/unit) x n t Storage time for thenth product in period t μnt Original freshness attenuation index of thenth product in period t ωnt Freshness attenuation index of thenth product taken preservation measures in period t A n t Freshness function of thenth product in period t, Ant∈(0,1] v n t Unit discount price based on freshness of thenth product in period t ($/unit) θ n t Original deterioration rate of thenth product in period t, θnt∈(0,1) β n t Deterioration rate of thenth product after being taken preservation measures in period t, βnt∈(0,1) K n t Preservation technology cost of thenth product in period t b n t Unit processing cost for the perished products of thenth product in period t ($/unit) g n t Unit production cost of thenth product in period t ($/unit) Q n t Production of thenth product in period t (units) c n t Unit inventory cost of thenth product in period t ($/unit) H t Workers hired in periodt (man-hour) h t Cost to hire one worker in periodt ($/man-hour) i n t Hours of labor occupied by per unit of thenth product in period t (man-hour/unit) m n t Hours of machine occupied by per unit of thenth product in period t (machine-hour/unit) W t m a x Largest workforce level available in periodt (man-hour) M t m a x Largest machine capacity in periodt (machine-hour) C t m a x Largest capital level in periodt ($)


## 4. Model Formulation

### 4.1. Preservation Technology Investment and Deterioration Rate

Vegetables in storage are easy to decay. Once going bad, the manufacturer not merely loses the production cost, but also needs to handle the perished parts. Consequently, the vegetables are highly susceptible to degeneration. Assume that the manufacturer invests on equipments to reduce the deterioration rate to extend the product expiration date, such as refrigeration or temperature controlling equipment. The preservation technology costKnt is used for preserving the products and we define a function between Knt and βnt as follows:(9)βnt=θnte-λKntwhere θnt is the original deterioration rate, βnt is the deterioration rate after preservation technology investment, and λ>0 is the attenuation factor representing the sensitivity of the original deterioration rate θnt for the preservation technology investment Knt. In formula (9), the exponential function shows that there is a negative correlation between Knt and βnt.


### 4.2. Preservation Technology Investment and Freshness Degree

The preservation measures taken in stock can delay the attenuation rate, where freshness decreases over the storage time, and contribute to more fresher vegetables than before. In order to depict the relationship between the preservation technology investment and the attenuation rate of freshness over time better, Dan [31] defined a function as follows:(10)Knt=ρμnt-ωnt2where ρ>0 represents the intensity of the preservation technology investment, μnt is the original freshness attenuation index, ωnt is the attenuation index. In formula (10), there is a negative correlation between Knt and ωnt and quadratic form shows that the decrease of ωnt will lead to increase for Knt. We can observe these characteristics clearly from the graph about the preservation investment function with ρ=1000 and μnt=0.3 in Figure 1.Figure 1
Preservation investment function withρ=1000 and μnt=0.3.


### 4.3. Uncertain Aggregate Production Planning Model

To depict the characteristics of the freshness changing over the storage time, we refer to a freshness function from Chen [32]. That is, Ant=1/1+ωntxnt2, Ant∈(0,1] and there is a discount price vnt=Antrnt based on this freshness function. Because of being not allowed out of stock, there is Qnt(1-θnt)≥Dnt and the sales are equal to the demand. Hence, the profit function can be defined as follows:(11)f=∑n=1N∑t=1TvntDnt-∑n=1N∑t=1TgntQnt+cntQnt+gntβntQnt+bntβntQnt+Knt-∑t=1ThtHtwhere vnt=Antrnt=1/1+ωntxnt2rnt,Knt=ρ(μnt-ωnt)2,βnt=θnte-λKnt, Qnt(1-θnt)≥Dnt, Ht,xnt≥0,0≤ωnt≤μnt and n=1,2,…,N, t=1,2,…,T.In formula (11), ∑n=1N∑t=1TvntDnt is denoted as total revenue for all variety of vegetables over the planning horizon T, and ∑n=1N∑t=1T(gntQnt+cntQnt+gntβntQnt+bntβntQnt+Knt)+∑t=1ThtHt is the total cost, including production cost, inventory cost, deterioration cost, preservation cost, and labor cost. ∑t=1TgntβntQnt+∑t=1TbntβntQnt represents the deterioration cost.Different managers have different attitudes towards the risk in the decision-making process. Assume that the decision maker take a neutral attitude. So an uncertain expected value model about total revenue is built for all variety of vegetables as follows:(12)maxEfsubjectto:M∑n=1NintQnt≤Wtmax≥ζM∑n=1NmntQnt≤Mtmax≥δM∑n=1NgntQnt+cntQnt+gntβntQnt+bntβntQnt+Knt+htHt≤Ctmax≥τQnt1-θnt≥Dnt,Ht,xnt≥0,0≤ωnt≤μntn=1,2,3,…,N;t=1,2,3,…,Twhere f is determined by formula (11) and its objective function is to maximize expected profits. Being subject to various uncertainties, the production planning cannot be predicted accurately. And production decision can only meet the constraints at a certain confidence level. So three chance constraints are constructed for this model. The first constraint ensures that the uncertain measure where the hours of labor used by all products do not exceed the biggest workforce level is not less than ζ in period t, where ζ is the confidence level. The second guarantees that the belief degree where the hours of machine taken up by all products do not exceed the biggest machine capability is not less than δ in period t, where δ is the confidence level. And the third one notes that the manufacturer expects the chance where the sum of all charges does not exceed the largest capital level is not less than τ in period t. τ is also the confidence level.


### 4.4. Equivalent Crisp Form

In uncertainty theory, belief degree obtained from experienced experts might be employed to estimate uncertainty distributions, and then uncertainty distributions are usually used to depict uncertain variables and play the role of a carrier for incomplete information of uncertain variables. And the objective function and constraints can be transformed into equivalent crisp form by some theorems of uncertainty theory.In formula (11), vnt=rnt/1+ωntxnt2 is defined as a nonnegative real value function. Qnt,Knt are nonnegative variables and gnt,bnt,λ are positive constant; thus the profit function f increases with respect to Dnt and decreases with regard to cnt and θnt. At the same time, all of these uncertain variables are independent; then the expected value model can be transformed into the following form by Theorem 7 and the objective function can be expressed as (13)Ef=∫01Ψ-1αdαwhere Ψ-1(α) is the inverse function of the profit function f and it can be denoted as follows:(14)Ψ-1α=∑n=1N∑t=1Trnt1+ωntxnt2ΦDnt-1α-∑n=1N∑t=1TQntgnt+QntΦcnt-11-α+ρμnt-ωnt2+gnte-λρμnt-ωnt2QntΦθnt-11-α+bnte-λρμnt-ωnt2QntΦθnt-11-α-∑t=1ThtHt.The objective function can be further simplified into the following form:(15)Ef=∑n=1N∑t=1Trnt1+ωntxnt2∫01ΦDnt-1αdα-∑t=1ThtHt-∑n=1N∑t=1TQntgnt+Qnt∫01Φcnt-11-αdα+ρμnt-ωnt2+gnte-λKntQnt∫01Φθnt-11-αdα+bnte-λKntQnt∫01Φθnt-1αd1-α.By Theorem8, the constraint (16)M∑n=1NintQnt≤Wtmax≥ζis equivalent to (17)∑n=1NΦint-1ζQnt≤ΦWtmax-11-ζ.By the same method, the second constraint can be obtained:(18)∑n=1NΦmnt-1δQnt≤ΦMtmax-11-δ.And the third constraint can be converted into (19)∑n=1NQntgnt+QntΦcnt-1τ+gnt+bnte-λKntQntΦθnt-1τ+ρμnt-ωnt2+htHt≤ΦCtmax-11-τ.Because of not being allowed to be out of stock,Qnt(1-θnt)≥Dnt, where Dnt~L(aDnt,bDnt),θnt~Z(aθnt,bθnt,cθnt). To ensure that the enterprise can grab the market shares, the manufacturer will guarantee no shortage and make the minimum amount of the unmetamorphosed vegetables. That is, Qnt(1-cθnt)≥bDnt. Hence, model (12) can be transformed into an equivalent crisp model as follows:(20)maxEfsubjectto:∑n=1NΦint-1ζQnt≤ΦWtmax-11-ζ∑n=1NΦmnt-1δQnt≤ΦMtmax-11-δ∑n=1NQntgnt+QntΦcnt-1τ+gnt+bnte-λKntQntΦθnt-1τ+Knt+htHt≤ΦCtmax-11-τQnt1-cθnt≥bDnt,xnt,Ht≥0,0≤ωnt≤μntn=1,2,3,…,N,t=1,2,3,…,Twhere E[f] is determined by formula (15) and Knt=ρ(μnt-ωnt)2.


## 4.1. Preservation Technology Investment and Deterioration Rate

Vegetables in storage are easy to decay. Once going bad, the manufacturer not merely loses the production cost, but also needs to handle the perished parts. Consequently, the vegetables are highly susceptible to degeneration. Assume that the manufacturer invests on equipments to reduce the deterioration rate to extend the product expiration date, such as refrigeration or temperature controlling equipment. The preservation technology costKnt is used for preserving the products and we define a function between Knt and βnt as follows:(9)βnt=θnte-λKntwhere θnt is the original deterioration rate, βnt is the deterioration rate after preservation technology investment, and λ>0 is the attenuation factor representing the sensitivity of the original deterioration rate θnt for the preservation technology investment Knt. In formula (9), the exponential function shows that there is a negative correlation between Knt and βnt.


## 4.2. Preservation Technology Investment and Freshness Degree

The preservation measures taken in stock can delay the attenuation rate, where freshness decreases over the storage time, and contribute to more fresher vegetables than before. In order to depict the relationship between the preservation technology investment and the attenuation rate of freshness over time better, Dan [31] defined a function as follows:(10)Knt=ρμnt-ωnt2where ρ>0 represents the intensity of the preservation technology investment, μnt is the original freshness attenuation index, ωnt is the attenuation index. In formula (10), there is a negative correlation between Knt and ωnt and quadratic form shows that the decrease of ωnt will lead to increase for Knt. We can observe these characteristics clearly from the graph about the preservation investment function with ρ=1000 and μnt=0.3 in Figure 1.Figure 1
Preservation investment function withρ=1000 and μnt=0.3.


## 4.3. Uncertain Aggregate Production Planning Model

To depict the characteristics of the freshness changing over the storage time, we refer to a freshness function from Chen [32]. That is, Ant=1/1+ωntxnt2, Ant∈(0,1] and there is a discount price vnt=Antrnt based on this freshness function. Because of being not allowed out of stock, there is Qnt(1-θnt)≥Dnt and the sales are equal to the demand. Hence, the profit function can be defined as follows:(11)f=∑n=1N∑t=1TvntDnt-∑n=1N∑t=1TgntQnt+cntQnt+gntβntQnt+bntβntQnt+Knt-∑t=1ThtHtwhere vnt=Antrnt=1/1+ωntxnt2rnt,Knt=ρ(μnt-ωnt)2,βnt=θnte-λKnt, Qnt(1-θnt)≥Dnt, Ht,xnt≥0,0≤ωnt≤μnt and n=1,2,…,N, t=1,2,…,T.In formula (11), ∑n=1N∑t=1TvntDnt is denoted as total revenue for all variety of vegetables over the planning horizon T, and ∑n=1N∑t=1T(gntQnt+cntQnt+gntβntQnt+bntβntQnt+Knt)+∑t=1ThtHt is the total cost, including production cost, inventory cost, deterioration cost, preservation cost, and labor cost. ∑t=1TgntβntQnt+∑t=1TbntβntQnt represents the deterioration cost.Different managers have different attitudes towards the risk in the decision-making process. Assume that the decision maker take a neutral attitude. So an uncertain expected value model about total revenue is built for all variety of vegetables as follows:(12)maxEfsubjectto:M∑n=1NintQnt≤Wtmax≥ζM∑n=1NmntQnt≤Mtmax≥δM∑n=1NgntQnt+cntQnt+gntβntQnt+bntβntQnt+Knt+htHt≤Ctmax≥τQnt1-θnt≥Dnt,Ht,xnt≥0,0≤ωnt≤μntn=1,2,3,…,N;t=1,2,3,…,Twhere f is determined by formula (11) and its objective function is to maximize expected profits. Being subject to various uncertainties, the production planning cannot be predicted accurately. And production decision can only meet the constraints at a certain confidence level. So three chance constraints are constructed for this model. The first constraint ensures that the uncertain measure where the hours of labor used by all products do not exceed the biggest workforce level is not less than ζ in period t, where ζ is the confidence level. The second guarantees that the belief degree where the hours of machine taken up by all products do not exceed the biggest machine capability is not less than δ in period t, where δ is the confidence level. And the third one notes that the manufacturer expects the chance where the sum of all charges does not exceed the largest capital level is not less than τ in period t. τ is also the confidence level.


## 4.4. Equivalent Crisp Form

In uncertainty theory, belief degree obtained from experienced experts might be employed to estimate uncertainty distributions, and then uncertainty distributions are usually used to depict uncertain variables and play the role of a carrier for incomplete information of uncertain variables. And the objective function and constraints can be transformed into equivalent crisp form by some theorems of uncertainty theory.In formula (11), vnt=rnt/1+ωntxnt2 is defined as a nonnegative real value function. Qnt,Knt are nonnegative variables and gnt,bnt,λ are positive constant; thus the profit function f increases with respect to Dnt and decreases with regard to cnt and θnt. At the same time, all of these uncertain variables are independent; then the expected value model can be transformed into the following form by Theorem 7 and the objective function can be expressed as (13)Ef=∫01Ψ-1αdαwhere Ψ-1(α) is the inverse function of the profit function f and it can be denoted as follows:(14)Ψ-1α=∑n=1N∑t=1Trnt1+ωntxnt2ΦDnt-1α-∑n=1N∑t=1TQntgnt+QntΦcnt-11-α+ρμnt-ωnt2+gnte-λρμnt-ωnt2QntΦθnt-11-α+bnte-λρμnt-ωnt2QntΦθnt-11-α-∑t=1ThtHt.The objective function can be further simplified into the following form:(15)Ef=∑n=1N∑t=1Trnt1+ωntxnt2∫01ΦDnt-1αdα-∑t=1ThtHt-∑n=1N∑t=1TQntgnt+Qnt∫01Φcnt-11-αdα+ρμnt-ωnt2+gnte-λKntQnt∫01Φθnt-11-αdα+bnte-λKntQnt∫01Φθnt-1αd1-α.By Theorem8, the constraint (16)M∑n=1NintQnt≤Wtmax≥ζis equivalent to (17)∑n=1NΦint-1ζQnt≤ΦWtmax-11-ζ.By the same method, the second constraint can be obtained:(18)∑n=1NΦmnt-1δQnt≤ΦMtmax-11-δ.And the third constraint can be converted into (19)∑n=1NQntgnt+QntΦcnt-1τ+gnt+bnte-λKntQntΦθnt-1τ+ρμnt-ωnt2+htHt≤ΦCtmax-11-τ.Because of not being allowed to be out of stock,Qnt(1-θnt)≥Dnt, where Dnt~L(aDnt,bDnt),θnt~Z(aθnt,bθnt,cθnt). To ensure that the enterprise can grab the market shares, the manufacturer will guarantee no shortage and make the minimum amount of the unmetamorphosed vegetables. That is, Qnt(1-cθnt)≥bDnt. Hence, model (12) can be transformed into an equivalent crisp model as follows:(20)maxEfsubjectto:∑n=1NΦint-1ζQnt≤ΦWtmax-11-ζ∑n=1NΦmnt-1δQnt≤ΦMtmax-11-δ∑n=1NQntgnt+QntΦcnt-1τ+gnt+bnte-λKntQntΦθnt-1τ+Knt+htHt≤ΦCtmax-11-τQnt1-cθnt≥bDnt,xnt,Ht≥0,0≤ωnt≤μntn=1,2,3,…,N,t=1,2,3,…,Twhere E[f] is determined by formula (15) and Knt=ρ(μnt-ωnt)2.


## 5. Numerical Example

In this section, we intend to illustrate the proposed model through a numerical example that shows that a manufacturer plans to produce two kinds of vegetables during two periods under uncertain environment. The information of the numerical instance including uncertain variables and various deterministic costs is shown in Table2.Table 2
Information of the numerical example.
Item Period 1 Period 2 Item Period 1 Period 2 D1t L(80,150) L(60,100) Mtmax N(40000,4) N ( 30000,5 ) D2t L ( 60,80 ) L(60,90) C t m a x Z ( 20000,50000,80000 ) Z ( 20000,60000,100000 ) θ1t Z(0,0.1,0.2) Z(0,0.1,0.15) r 1 t 40 50 θ2t Z(0,0.1,0.15) Z ( 0,0.1,0.2 ) r 2 t 50 60 c 1 t N ( 1,0.2 ) N(2,0.2) g 1 t 6 8 c 2 t N ( 2,0.2 ) N ( 1,0.2 ) g 2 t 8 6 i 1 t L ( 2,4 ) L ( 3,6 ) b1t 1 2 i 2 t L ( 2,4 ) L(3,6) b 2 t 2 1 m 1 t N ( 4,1 ) N ( 5,2 ) ht 4 5 m 2 t N(5,2) N ( 4,1 ) μ1t 0.2 0.3 W t m a x L(20000,80000) L ( 30000,90000 ) μ 2 t 0.3 0.2Moreover, we consider other parameters with the following data,ρ=1100,λ=0.09,ζ=0.7,δ=0.8,τ=0.6,N=2,T=2.Based on Table2, model (20) can be further converted into the following form:(21)maxE-7+0.7e-0.09K11Q11-10+0.875e-0.09K12Q12-10+0.875e-0.09K21Q21-7+0.7e-0.09K22Q22-K11-K12-K21-K22-4H1-5H2subjectto:3.4Q11-3.4Q21≤380005.1Q12-5.1Q22≤480004+3πln⁡4Q11+5+23πln⁡4Q21≤40000-43πln⁡45+23πln⁡4Q12+4+3πln⁡4Q22≤30000-53πln⁡47+0.23πln⁡32+0.98e-0.09K11Q11+10+0.23πln⁡32+2.4e-0.09K21Q21+K11+K21≤4400010+0.23πln⁡32+2.4e-0.09K12Q12+7+0.23πln⁡32+0.98e-0.09K22Q22+K12+K22≤52000Q11≥1500.8,Q12≥1000.85,Q21≥800.85,Q22≥900.80≤ω11≤μ11,0≤ω12≤μ12,0≤ω21≤μ21,0≤ω22≤μ22x11,x12,x21,x22,H1,H2≥0where(22)E=46001+ω11x112+40001+ω12x122+35001+ω21x212+45001+ω22x222K11=1400(0.2-ω11)2,K12=1400(0.3-ω12)2,K21=1400(0.3-ω21)2 and K22=1400(0.2-ω22)2.From model (21), we can know that the deterministic form is a nonlinear programming model, which can be solved with the method of some traditional algorithms. However, unlike linear programming with universal algorithms, we need to face more challenges to solve the nonlinear programming because the traditional methods tend to be trapped in local optima and the optimal solutions always depend on the initial values. So these traditional algorithms can not guarantee the optimality of the solution. And the experiment results demonstrated that the Genetic Algorithm is a global optimization algorithm and is adopted to avoid local optima. Hence, we use Genetic Algorithm and Direct Search Toolbox of MATLAB 8.5 to search for the optimal solutions for this model. In order to get more accurate feasible solution and increase the diversity of the population, we set ‘PopulationSize’ = 35, ‘CrossoverFraction’ =0.35, ‘PopInitRange’ = [0; 10] and we set ‘rng(0, ‘twister’)’ for reproducibility in calculating process. Through calculation, we obtain the optimal objective value 12208.2 and the values of the decision variables are shown in Table 3.Table 3
Decision variables of the optimal production planning.
Item Period 1 Period 2 Q1t 187.5687 117.7052 Q2t 100.0792 106.1172 x1t 0.2254 0.0271 x2t 0.0132 0.1432 Ht 0.6538 0.3627 ω 1 t 0.0180 0.1562 ω 2 t 0.1599 0.0540To demonstrate the effectiveness of the factorλ for this APP problem, we assume ρ=1100 and assign fourteen different values to observe the changes of the maximal profit. The objective values under different λ are shown in Table 4 and we can find the changing trend of the optimal values from Figure 2.Table 4
The changes of optimal values under differentλ.
λ Optimal Objective Value λ Optimal Objective Value 0.00 11883.0 0.07 12196.2 0.01 11957.7 0.08 12208.4 0.02 12027.3 0.09 12208.2 0.03 12078.5 0.10 12221.0 0.04 12123.4 0.11 12225.6 0.05 12151.6 0.12 12229.5 0.06 12172.7 0.13 12256.2Figure 2
The changing trend of optimal objective value.From Figure2, the optimal profit of the same kind of vegetables in the same period always changes over the attenuation factor λ. And it plays a large role in revealing that the optimal results of the objective function improve with the increase of λ on the whole. This indicates that the bigger the factor λ is, the more sensitive the deterioration rate is to the preservation investment Knt and the more obvious the preservation effect is in the case of the same preservation cost, which will contribute to a more sharply falling deterioration cost and improve the total profit level to a certain degree. Hence, the attenuation factor λ has a significant effect on this uncertain APP model and for the manufacturer, a bigger value of λ is more beneficial to improve the profits. Therefore, the manufacturer should strive to choose the preservation technology with the better preservation effect through market research and field test.To demonstrate the effectiveness of the factorρ, we assume the attenuation factor λ=0.09 and assign fourteen different values to the factor ρ to observe the changes of the maximal profit. The objective results under different ρ are shown in Table 5 and Figure 3 shows the changing trend of the optimal objective value.Table 5
The changes of optimal values under differentρ.
ρ Optimal Objective Value ρ Optimal Objective Value 0 11963.1 600 12213.5 50 12030.3 700 12221.4 100 12100.7 800 12219.5 200 12168.6 900 12221.2 300 12200.8 1000 12205.5 400 12208.8 1100 12208.2 500 12214.7 1200 12193.5Figure 3
The changing trend of optimal objective value.From Figure3, we can find that maximal value of objective function increases significantly along with the growth of ρ at the very beginning. However, after reaching a certain threshold, the range of growth gradually slows and becomes stable and then even begins falling, which implies that factor ρ has profound effects on the optimal objective values. In addition, we can know that a certain degree of fresh-keeping investment can improve the freshness of vegetables and contribute to satisfactory revenue. But if the manufacturers just invest on substantial fresh-keeping cost, they not only cannot receive the sustainability of earnings growth, but also suffer from the losses, because the values of revenues and costs balance out along with the increase of ρ. Finally, the revenue produced by the preservation investment is gradually less than the investment cost when the factor ρ increases to a certain extent. Therefore, the manufacturers should choose the appropriate investment from the preservation technology as far as possible in order to save costs and realize the profit maximization.Based on the above analysis, we can draw the conclusion that the optimal values of the objective function increase along with the increase of the the attenuation factorλ and as the factor ρ goes up, the maximal values increase in large amplitude and then grow slowly and even gradually decline. Obviously, the manufacturer should strive to choose the preservation technology with the better preservation effect. In the meantime, a certain degree of fresh-keeping investment could effectively reduce the deterioration rate and improve the freshness of vegetables to generate better incomes. Furthermore, the manufacture should avoid the high preservation investment costs which have a negative effect on profit.


## 6. Conclusion

In this paper, we built an uncertain aggregate production planning model considering the characteristics of vegetables. In the proposed model, investment in vegetable preservation technology was taken into consideration. Meanwhile, this proposed model can be transformed into a crisp equivalent form. By a numerical example, we finally concluded that the attenuation factors have a significant impact on the proposed APP model. Meanwhile, to save costs and realize the profit maximization, the manufacturer had better choose the preservation measures with good effect and take the appropriate preservation cost. In future, other different factors for different kinds of vegetables in different periods can be taken into consideration.



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