Analysis on Dynamic Decision-Making Model of the Enterprise Technological Innovation Investment under Uncertain Environment

Abstract

Under the environment of fuzzy factors including the return of market, performance of product, and the demanding level of market, we use the method of dynamic programming and establish the model of investment decision, in technology innovation project of enterprise, based on the dynamic programming. Analysis of the influence caused by the changes of fuzzy uncertainty factors to technological innovation project investment of enterprise.

1. Introduction

The enterprise technological innovation is a creative process. The uncertainty risk mainly includes the environment, technology, market, and risk management. At the same time, the process of enterprise technological innovation is a dynamic process. In the initial stage of the technical innovation, the enterprise must evaluate and select the innovation project and also consider the social and economic benefits and the development of technology with the combination of their own development strategies; at the end select the most suitable for the development of innovative investment projects. Sarkar [1] had studied market uncertainty and corporate investment relationship in consideration of system risk conditions, and he thinks that increasing the uncertainty may increase the probability of investment of enterprises to some low growth and low risk of investment project. In fact, a technical innovation project can be regarded as embedded in a series of options chain, and each option gives investors investment rights, so every decision stage contains an “improvement option”; when the difficult technology problem definitely is solved, we can make further investment in product prototype development and innovative design and then continue investing to enter the testing phase. At the same time, in every decision stage there is an abandonment option, so the flexibility of project management not only increases the value of the project, but also reduces the investment risk of the technical innovation project. Weeds [2] described if research and development is successful as a Poisson process, and it is used to describe the uncertainty of technology and of Hershey’s bad market opportunity arrival timing. He builds two stages’ R&D investment pricing model and discusses the reality of "Sleeping Patent” rationality; that is, the enterprise is willing to develop a technology, but after the success of the technology research and development, will put it away unheeded and not put it into the market. Mitchell and Hamilton [3] point out that due to R&D plays an important role in creating competitive advantage and, therefore, should be treated from a strategic perspective, discussing in detail the multiple features of the R&D strategic options and then taking three steps: clear strategic aim, evaluation strategy options, and select influence strategic target to study R&D strategy option. Lint and Pennings [4, 5] mainly studied the innovation of real option in the process of marketization and pointed out that there were two choices, disposable rapid advancing, and slow advancing and related options opportunities and options value hid in slow advancing. Huchzermeier and Loch [6] proposed a decision model of multiple stages and considered in each phase of this model that managers had three solutions: continue to invest in the project, improve the project, and give up the project. From the market returns, assumed total return consists of two levels: a basic income can be relatively easy to be obtained; only in the project performance exceeding market demands becomes uncertain. They also identified several different sources of the flexibility and uncertainty and made analysis of the impact problem increased by uncertainty and flexibility. From the characteristics that the state variable of R&D project is a nonfinancial parameter, using an equivalent method, dynamic planning of the option evaluation to build the dynamic programming model of R&D project, without the need for asset replication, Sheng [7] had solved the flexible problem of R&D project well.

The aforementioned is the study conducted under random environment (some discrete environmental). In fact, the essence of fuzzy real option is that tolerance in the same information shows diversification before the rationality and that the complexity of the human mind is admitted, namely, the introduction of nonuniform rational in the value assessment. At the same time, there are often still some realities that we cannot accurate valuation or expect net cash flow situation and due to objective factors some variables cannot be estimated by the exact data, and some actual situation etc (Liu [8]). So evaluation results often deviate from the actual if we use the accurate values to determine model input parameters. In this paper, we mainly combine the dynamic programming method and option analysis method, in a fuzzy environment making an analysis of some flexible decision problems of enterprise technology innovation and innovation investment. Firstly the Huchzermeier and Loch [6] model is extended. Secondly a discussion of dynamic programming model of the second stage under fuzzy environment is made. Thirdly analysis is made about technological innovation project decision model under fuzzy environment. Then the elastic value of the project of technology innovation is discussed. It focuses on the analysis of changes of the fuzzy uncertainty factors (including market returns and demanding level).

2. The Dynamic Programming Decision Model of the Technological Innovation Investment under Fuzzy Environment

2.1. Extending of Huchzermeier and Loch Model

Under the fuzzy environment, we consider a technical innovation project; the success of the project depends mainly on the performance of the products during commercialization in the market, and the fuzzy uncertainty of market performance is caused by market and technology risk, denoted by (𝑖,𝑡) project at time 𝑡 expected market performance (the market performance can be expected through the simulation test, get). Typically managers have difficulty to predict the distribution of 𝑖 during the stage of outcome of the market commercialization, assuming 𝑖 as fuzzy variables and 𝑖 obeying credibility distribution; the expected profits of the maximum and minimum values are 𝑄 and 𝑞, respectively, actual process of project meets no aftereffect, and the state transfer of market performance obeys two distribution; if the products of the project development reach to the performance state 𝑖, the expected profit is 𝜋_𝑖=𝑞+Φ(𝑖)𝑄−𝑞Φ(𝑖)iscredibilitydistributionfunction. (2.1)

By Liu [8], in the processing of fuzzy event, credibility measure plays a similar role in the probability measure on random events. So, modeled as stochastic events in the transition probability, in the evaluation model of two-fork tree option, assuming the state transfer of market performance obeys two distributions, namely, the condition of market upgraded by the credibility of Cr{𝐴}, and under adverse conditions turning for the worse with probability (1−Cr{𝐴}), we generalize this process as the market’s performance improvement and deterioration and then easily conclude that the transition probability 𝑃_𝑖𝑗 is expressed as follows under fuzzy environment: 𝑃_𝑖𝑗=Cr{𝐴}𝑁1,𝑗∈𝑖+2𝑁,…,𝑖+2,1−Cr{𝐴}𝑁1,𝑗∈𝑖−2𝑁,…,𝑖−2,0,other. (2.2)

At the same time, while expanding the scale of investment, under fuzzy environment the transition probability is 𝑃_𝑖𝑗=Cr{𝐴}𝑁1,𝑗∈𝑖+1+2𝑁,…,𝑖+1+2,1−Cr{𝐴}𝑁1,𝑗∈𝑖+1−2𝑁,…,𝑖+1−2,0,other. (2.3)

Under fuzzy environment, the project management dynamic programming optimal value function, in the last stage of commercial stage project: 𝑉_𝑖(𝑇)(𝑇)+∑=max−𝐶^𝑁𝑗Cr{𝐴}𝜋_𝑖+𝑗/2+(1−Cr{𝐴})𝜋_{𝑖−𝑗/2}𝑁()(𝑇)(𝑇)+∑1+𝑟continue,−𝐶−𝐴^𝑁𝑗Cr{𝐴}𝜋_{𝑖+1+𝑗/2}+(1−Cr{𝐴})𝜋_{𝑖+1−𝑗/2}𝑁(1+𝑟)improvement,0giveup, (2.4) where 𝐶(𝑇) is the cost of project's continuation and 𝐴(𝑇) are the cost when the project is increased in size, including the net present value of option strategic: 𝑉=𝑉(0)−𝐼; theoretically speaking, fuzzy environment is closer to the reality of the technical innovation. So, this model has certain practical significance, of course, according to the difference of the actual investment situation; we also can extend the model to the fuzzy random environment, fuzzy environment, rough fuzzy environment, and so forth.

2.2. The Dynamic Programming Model of the Two Stages under Fuzzy Environment

Assuming 𝐼 is sunk cost, the interest rate without the risk is 𝑟>0, and 𝑢, 𝑑 are parameters; hypothesize; the price of the product in stage 0 is 𝑃₀, from the beginning of stage 1, the feasibility of the prices (1+𝑢)𝑃₀ is Cr{𝐴}, the feasibility of the prices of (1−𝑑)𝑃₀ is (1−Cr{𝐴}), assuming that the investment opportunity lies only in stage 0; if the technical innovation enterprises do not invest at this stage, so in stage 1, it will not change the decision forever; we use 𝑉₀ to be the symbol for the expectation value obtained from its investment of technology innovation enterprise, then 𝑉₀=𝑃₀+Cr{𝐴}(1+𝑢)𝑃₀+(1−Cr{𝐴})(1−𝑑)𝑃₀1+11+𝑟(1+𝑟)²=𝑃+⋯₀[1+𝑟+Cr{𝐴}(𝑢+𝑑)−𝑑]𝑟. (2.5)

Now, we consider the reality of the situation; in any future stage, the investment opportunities still exist. So, at this time, in stage 0, we can choose to invest or wait to select until stage 1; from stage 1 forward conditions will not change; if in phase 1 waiting, the price becomes 𝑃₁=(1+𝑢)𝑃₀{𝐴},()𝑃,whenthefeasibilityofCr1−𝑑₀,whenthefeasibilityof1−Cr{𝐴}. (2.6)

To either possibilities (price changing in stage 0 and stage 1), if 𝑉₀>𝐼, then enterprise invests, we can get the net return: 𝐹₁=max{𝑉₁−𝐼,0}.

The discounted value is 𝑉₁=𝑃₁(1+𝑟)/𝑟; from stage 0, the price is 𝑃₁ in stage 1, the value is 𝑉₁, 𝐹₁ is the random variables, and 𝐸₀ is the expected value calculated by feasible weighted average in stage 0, then 𝐸₀𝐹₁{𝐴}()𝑃=Crmax1+𝑢₀(1+𝑟)𝑟+({𝐴()𝑃−𝐼,01−Cr})max1−𝐷₀(1+𝑟)𝑟−𝐼,0. (2.7)

Back to stage 0, the enterprise has two kinds of choices. If it invests, the income is 𝑉₀−𝐼; if not, the enterprise has continuous value 𝐸₀(𝐹₁), but the value is obtained in stage 1, so it should use 1/(1+𝑟) to discount; therefore, the whole investment opportunities are the net present value of the investment profit arranged optimally, credited as 𝐹₀𝐹₀=max𝑉₀1−𝐼,𝐸𝐹1+𝑟¹. (2.8)

Previously we discussed the dynamic programming model in Sheng [7] under fuzzy environment. Theoretically speaking, the fuzzy environment is closer to the technical innovation in reality, so this model has certain practical significance. Below we start from Liu [8] model, combine Dixit and Pindyck [9], and make discussion about the multistage model.

2.3. The Decision Model of Technological Innovation Project during Multistages under Fuzzy Environment

2.3.1. Fuzzy Uncertainty of Technological Innovation Project during Development Stage

Investment management of technical innovation project is a decision process of multistages. Each stage has decision points of project evaluation. Each decision point includes the project evaluation at present, investment decision making of the current state, and future earnings evaluation based on each kind of decision-making choice. In this section, we use the fuzzy theory proposed by Liu [8] to handle uncertainty of technical innovation project’s development phase. Firstly the decision model of technology innovation project during multistage is described [7].

Santiago and Vakili Model
Assuming totally there are 𝑇 stages of decision making of technology innovation project, 𝑡=0,1,2,…,𝑇−1, in each stage 𝑡, decision makers will face three alternatives, “continue” “improvement,” and “give up”. The success of technical innovation project depends on the performance of the product put into market; we use the state variables of the project to show the product performance in the process of the development. Let the 𝑋_𝑡 project be the state variables in the initial stage 𝑡, and assume when 𝑡=0,𝑋₀=0; 𝜉_𝑡 is the fuzzy uncertainty of the project inside and outside during 𝑡 stage in the process of innovation, and {𝜉₀,𝜉₁,…,𝜉_𝑇−1} are independent from each other; 𝑢_𝑡 is the choice decision of the project in stage of starting time. So, when the 𝑡 stage is completed, the state variable of project can be expressed as [10] 𝑋_𝑡+1=𝑋_𝑡+𝑘𝑢_𝑡+𝜉_𝑡,if𝑢_𝑡choose“continue”or“improvement”,𝑋_𝑡,if𝑢_𝑡chooseto“giveup”, (2.9) where the feasibility of 𝑘(continue)=0 and 𝑘(improvement)=1 is Cr{𝐴}/𝑁, the feasibility of 𝜉_𝑡=−𝑖/2 is (1−Cr{𝐴})/𝑁,𝑖=1,2,…,𝑁, 𝑁 is regarded as a measure of the uncertainty and fuzziness. In other words, if the item “continues," then in the next stage, the expected performance will be present together with some fuzzy uncertainty; if the item “improves,” the project of state variable will be plused one more improved unit and fuzzy uncertain effects; if the project “gives up”, the project stops at current state variables and remains unchanged. According to the discussion of Sheng [7], we have the following

Nature 1
A decision-making problem of a technical innovation project is considered under fuzzy environment: 𝑋_𝑡=𝑋^′_𝑡,ifchoose“continue”or“improvement”,𝑋_𝑡,ifchooseto“giveup”. (2.10) According to hypothesis at the initial time of 𝑡 stage, the state variables of project in two cases are 𝑋_𝑡 and 𝑋^′_𝑡, respectively, and 𝑋_𝑡 and 𝑋^′_𝑡 are fuzzy variables on possibility space (Θ,𝑃(Θ),Pos), then, 𝑋_𝑡≥𝑋^′_𝑡.

Proof. Assume in two cases that the initial state variables of project equal, that is, 𝑋₀=𝑋^′₀=0, because 𝑋_𝑡 and 𝑋^′_𝑡 are fuzzy variables of (Θ,𝑃(Θ),Pos) on possibility space. According to hypothesis in 𝑡−1 phase, for all 𝜃∈Θ, we have 𝑋_𝑡−1(𝜃)≥𝑋^′_𝑡−1(𝜃). Assume in 𝑡−1 phase situation 1, decision makers take 𝑢^∗ as the optimal decision, and then in the 𝑡 stage by Santiago and Vakili [10] model, we can get 𝑋_𝑡(𝜃)=𝑋_𝑡−1(𝜃)+𝑘𝑢^∗+𝜉_𝑡−1(𝜃)≥𝑋^′_𝑡−1(𝜃)+𝜉_𝑡−1(𝜃)=𝑋^′_𝑡(𝜃),then𝑋_𝑡≥𝑋^′_𝑡. (2.11)

2.3.2. Fuzzy Expected Value, Variance

The fuzzy uncertainty of technical innovation project 𝜉_𝑡 (𝑡=0,…,𝑇−1) includes the technical risk of the project development process inside and the fuzziness influenced by the external environment, though there is statistical data of the project of the same type for reference, because technical innovation project is unique and singular, which makes subjective judgment of decision makers essential. We describe this kind of fuzzy uncertainty as fuzzy variable, which is independent and identically distributed, and its expected value is 0. It indicates that the project performance may be improved due to favorable fuzzy uncertain events, and also may be worse due to the occurrence of adverse; by assumptions of Yi Chang sheng, the fuzzy variable 𝜉_𝑡 expectations of the technological innovation project are expressed as 𝐸𝜉_𝑡=₀^+∞Cr𝜉_𝑡≥𝑟𝑑𝑟−⁰_−∞Cr𝜉_𝑡≤𝑟𝑑𝑟=0. (2.12)

According to Liu [8] and 𝐸(𝜉)=0, then the variance is 𝑉[𝜉_𝑡]=𝐸[(𝜉_𝑡−𝐸[𝜉_𝑡])²]. The variance of 𝑉[𝜉_𝑡] can be used as a measure of fuzzy degree of uncertainty of technical innovation project. If 𝑉[𝜉_𝑡] is smaller, then the fuzzy uncertainty is smaller. The solvent of fuzzy uncertainty can be reflected by the cumulative value of technical innovation project. Assumed in the of the initial state item variables is 𝑋_𝑡, at the end of 𝑡 phase and the initial time of 𝑡+1 stage, the fuzzy uncertain factors 𝜉_𝑡 of 𝑡 phase are solved, and the state variables of project will change to 𝑋_𝑡+𝑘(𝑢_𝑡)+𝜉_𝑡. 𝜉_𝑡 shows the fuzzy uncertainty of the project in the phase under internal and external environment. It includes the technical risk during the process of project development, the evaluation of the project's profitability made by the project team, the external market information of project, and other aspects.

2.3.3. Development Costs and Market Returns

According to the model of Huchzermeier and Loch [6], because during the development process of each stage of the development cost and decision-making choices, we can assume the project in phase 𝑡 development costs as 𝐶_𝑡(𝑢_𝑡); if the decision maker chooses “give up,” then the cost of project development is 0. If the decision maker selects “continue,” then the cost of project development is 𝑐(𝑡) (the “continue” cost of 𝑐(𝑡) for stage 𝑡); if the decision maker selects “improvement,” then the cost of project development is 𝑐(𝑡)+𝑎(𝑡) (𝑎(𝑡) is the additional cost invested when taking corrective action and does not need to extend the project schedule, such as processing engineers and the experimental equipment; it can make the project status variables improve a unit). If project’s initial investment is 𝐼 while 𝑡=0, the revenue and cost of project discount according to the nonrisk free rate 𝑟. At the end of 𝑡−1 phase, the project is completed and products are put into market, enterprises will get market gains closely related to product performance, and 𝑅(𝑋_𝑇) expressed as follows: 𝑅𝑋_𝑇=𝑚,if𝑋_𝑇<𝜂,𝑀,if𝑋_𝑇≥𝜂, (2.13) where 𝜂 is the market's demand level of the product and 𝑋_𝑇 is the state variable after 𝑡−1 stage is over, that is, the final product performance obtained by enterprise. If the product's performance meets or exceeds 𝜂, then the business has more advantages than its competitors in product performance and will gain a perfect profit of 𝑀; conversely, the enterprise can only get a small profit of 𝑚 (clearly, 𝑚<𝑀 ); because the 𝜂 is unknown before the product entering market, we postulate it is the fuzzy variable. For any 𝜃∈Θ, assums the variable is 𝑋_𝑇(𝜃) during project phase, we use Cr{𝑋_𝑇(𝜃)≥𝜂} to show that project state variable 𝑋_𝑇(𝜃) reaches or exceeds the credibility of 𝜂; then, when the state variable is 𝑋_𝑇(𝜃), the expected value of fuzzy market returns is 𝐸Π𝑋_𝑇=₀^+∞Cr𝑅𝑋_𝑇(𝜃)=≥𝑟𝑑𝑟_𝑚0Cr𝑅𝑋_𝑇(𝜃)≥𝑟𝑑𝑟+^𝑀_𝑚Cr𝑅𝑋_𝑇(𝜃)≥𝑟𝑑𝑟=𝑚+^𝑀_𝑚Cr𝑋_𝑇(𝜃)𝑋≥𝜂𝑑𝑟=𝑚+Cr_𝑇(𝜃)≥𝜂^𝑀_𝑚𝑋𝑑𝑟=𝑚+Cr_𝑇(𝜃)≥𝜂(𝑀−𝑚). (2.14)

We use 𝜙(⋅) to show credibility distribution function of the fuzzy variable 𝜂, then 𝐸Π𝑋_𝑇=𝑚+𝜙𝑋_𝑇⋅(𝑀−𝑚). (2.15)

Huchzermeier and Loch [6] show the benefit function: if the performance level is 𝑋_𝑇 at the moment of 𝑇, it will generate the expected market return Π_𝑖. According to the different actual investment situations, we can also extend the conclusion to fuzzy random environment, rough fuzzy environment, and so on. For example, under fuzzy random environment, when the variables of product state are 𝑋_𝑇(𝜃), the expectation obtained by fuzzy stochastic market returns Π(𝑋_𝑇) can also be presented in this way.

Definition 2.1. Let 𝜉 be a fuzzy random variable, and then one has a finite expected value 𝐸(𝜉), 𝑉[𝜉]=𝐸[(𝜉−𝐸[𝜉])²] that is called fuzzy variable 𝜉 variance.
From the previous discussion, when the product state variable is 𝑋_𝑇(𝜃), the expectation of fuzzy stochastic market returns Π(𝑋_𝑇) is obtained by 𝐸Π𝑋_𝑇=₀^+∞Pr𝜉∈Ω∣𝑅𝑋_𝑇(𝜃)=≥𝑟𝑑𝑟_𝑚0Pr𝜉∈Ω∣𝑅𝑋_𝑇(𝜃)≥𝑟𝑑𝑟+^𝑀_𝑚Pr𝜉∈Ω∣𝑅𝑋_𝑇(𝜃)≥𝑟𝑑𝑟=𝑚+^𝑀_𝑚Pr𝜉∈Ω∣𝑅𝑋_𝑇(𝜃)≥𝑟𝑑𝑟=𝑚+Pr𝜉∈Ω∣𝑋_𝑇(𝜃)≥𝑟^𝑀_𝑚𝑑𝑟=𝑚+Pr𝜉∈Ω∣𝑋_𝑇(𝜃)≥𝑟(𝑀−𝑚). (2.16)
Therefore, a double-fuzzy environment and rough fuzzy environment are similar to be launched.

2.3.4. The Dynamic Programming Model of Technological Innovation Project

We consider the decision behavior of the enterprise technology innovation investment; the project current state variable is indicated by 𝑥, it will affect opportunities of the enterprise's decision-making and expansion, at any stage of 𝑡, and the value of variable 𝑥_𝑡 is known, let the future value 𝑥_𝑡+1,𝑥_𝑡+2,… be random variables. {𝑥_𝑡} is Markov process; we use the 𝑉_𝑡(𝑥) to be the whole decision results of company from 𝑡; when selecting 𝑢_𝑡 as the control variables, its cash flow is Π_𝑡(𝑥_𝑡,𝑢_𝑡), at the phase of 𝑡+1, the state is 𝑥_𝑡+1, the result of optimal decision result is 𝑉_𝑡+1(𝑥_𝑡+1); at the phase of 𝑡, this result is a random variable, so, we take 𝐸(𝑉_𝑡+1(𝑥_𝑡+1)) as expected value, discounting to the stage of 𝑡; the plus of sight cash flow and the continuous value is Π_𝑡𝑥_𝑡,𝑢_𝑡,𝜉_𝑡+1𝐸𝑉1+𝜌^𝑡+1𝑥_𝑡+1. (2.17)

Enterprises will choose 𝑢_𝑡 to be maximum, and the result just is 𝑉_𝑡(𝑥_𝑡); then we have 𝑉_𝑡𝑥_𝑡=max_𝑢𝑡Π_𝑡𝑥_𝑡,𝑢_𝑡+1𝐸𝑉1+𝜌^𝑡+1𝑥_𝑡+1. (2.18) This equation is the optimal basic equation (see [9]).

If multiple phase problems have limited stage 𝑇, the ultimate returns of enterprises are Ω_𝑇(𝑥_𝑡); then in the previous stage, we have 𝑉_𝑇−1𝑥_𝑇−1=max_𝑢𝑇−1Π_𝑇−1𝑥_𝑇−1,𝑢_𝑇−1+1𝐸Ω1+𝜌^𝑇−1𝑥_𝑇−1, (2.19) which provides theoretical basis to simulate by using computer.

In fact, the 𝑥_𝑡, 𝑥_𝑡+1 may be in any state, and it can be generally denoted as 𝑥, 𝑥′; then, for any 𝑥, we have Bellman equation of infinite duration dynamic programming: 𝑉(𝑥)=max_𝑡Π()+1𝑥,𝑢𝐸𝑉𝑥1+𝜌^′∣𝑥,𝑢. (2.20) In the following we put this problem to further discussion under fuzzy environment.

Technical innovation project is on the stage of development spending is by the final market returns to compensate for an evaluation, according to the Santiago and Vakili model, since each choose temporary investment costs are known, therefore, management decision based primarily on the final market assessment. We use the value of the function 𝑉_𝑡(𝑥) to be this evaluation, fuzzy uncertainty 𝜉_𝑡 (𝑡=0,…,𝑇−1) of technical innovation project includes the technical inside risk of the project during the development process, and the uncertainty influenced by external environment, and 𝜉_𝑡 is fuzzy variable independent and identically distributed, and its expected value is 0. Assuming in the stage that the development cost is 𝐶_𝑡(𝑢_𝑡), according to Santiago and Vakili model, it can be described by dynamic programming equation: 𝑉_𝑡𝑥_𝑡=max_𝑢𝑡−𝐶_𝑡𝑢_𝑡+1𝐸𝑉1+𝑟^𝑡+1𝑋_𝑡+1𝑥_𝑡,𝑢_𝑡,𝜉_𝑡. (2.21)

Value of the function is 𝑉_𝑡(𝑥) at 𝑡 phase; initial project state variable is 𝑥. At the end of the project phase of 𝑡=𝑇, we have 𝑉_𝑡(𝑥)=𝐸[Π(𝑥)]. (2.22)

At the same time, under fuzzy environment, if multiple phase problems have limited stage 𝑇, the ultimate returns obtained by the enterprises are Ω_𝑇(𝑥_𝑡); at an earlier stage, we have 𝑉_𝑇−1𝑥_𝑇−1=max_𝑢𝑇−1−𝐶_𝑇−1𝑢_𝑇−1+1𝐸𝑉1+𝜌^𝑇Ω_𝑇𝑥_𝑇−1,𝑢_𝑇−1,𝜉_𝑇−1. (2.23)

In fact, 𝑥_𝑡, 𝑥_𝑡+1 may be any state, it can be generally written as 𝑥, 𝑥^′; and then for any 𝑥, we have Bellman equation of infinite duration dynamic programming under fuzzy environment: 𝑉(𝑥)=max_𝑡()+1−𝐶𝑥,𝑢,𝜉𝐸𝑉𝑥1+𝜌^′∣𝑥,𝑢,𝜉. (2.24)

The discussion of Santiago and Vakili [10] and Sheng [7] also gives a property of value function under the stochastic uncertainty environment; the following is the promotion under fuzzy environment.

Nature 2
Assuming the technology innovation project of state variables is indicated by 𝑥 under fuzzy environment, if the expected market return function 𝐸[Π(𝑥)] is nondecreasing, then the value of the function 𝑉_𝑡(𝑥)(𝑡=0,1,…,𝑇−1) of technical innovation project in any stage is non-decreasing too.

Proof. In reference Santiago and Vakili [10], if 𝑉_𝑡+1(𝑥) is non-decreasing, we assume that 𝑉_𝑡+1(𝑥) is nondecreasing; for two project state variables 𝑥₁ and 𝑥₂ of a given 𝑡 phase, if 𝑥₂>𝑥₁, then we only need to prove 𝑉_𝑡(𝑥₂)≥𝑉_𝑡(𝑥₁). Assuming that at 𝑡 phase, the project of state variable is 𝑥₁, selecting the optimal decision 𝑢^∗, makes the enterprise obtain the maximum 𝑉_𝑡(𝑥₁). When the state variable of 𝑡 phase is 𝑥₂, the taken decisions 𝑢^∗ make enterprises get the project value 𝑉^′_𝑡(𝑥₂); then consider the following;(1) If 𝑢^∗ is “continue” or “improved,” we have 𝑉^′_𝑡𝑥₂−𝑉_𝑡𝑥₁=max_𝑢𝑡−𝐶_𝑡𝑢_𝑡+1𝐸𝑉1+𝑟^𝑡+1𝑋_𝑡+1𝑥₂,𝑢_𝑡,𝜉_𝑡−max_𝑢𝑡−𝐶_𝑡𝑢_𝑡+1𝐸𝑉1+𝑟^𝑡+1𝑋_𝑡+1𝑥₁,𝑢_𝑡,𝜉_𝑡=1𝐸𝑉1+𝑟^𝑡+1𝑥₂+𝑘𝑢^∗+𝜉_𝑡−𝑉_𝑡+1𝑥₁+𝑘𝑢^∗+𝜉_𝑡. (2.25)(2) If 𝑢^∗ is “gives up.” 𝑉^′_𝑡(𝑥₂)−𝑉_𝑡(𝑥₁)=0. Because 𝜉_𝑡 is a fuzzy variable, so, 𝑥₁+𝑘(𝑢^∗)+𝜉_𝑡 and 𝑥₂+𝑘(𝑢^∗)+𝜉_𝑡 are fuzzy variables. If 𝑥₂>𝑥₁, we have 𝑥₁+𝑘(𝑢^∗)+𝜉_𝑡>𝑥₂+𝑘(𝑢^∗)+𝜉_𝑡. Because 𝑉_𝑡+1(𝑥) is monotonicity, we have 𝑉_𝑡+1𝑥₂+𝑘𝑢^∗+𝜉_𝑡−𝑉_𝑡+1𝑥₁+𝑘𝑢^∗+𝜉_𝑡≥0. (2.26)From the properties that the fuzzy variable is nonnegative, and its expected value is also non-negative, we have 𝐸𝑉_𝑡+1𝑥₂+𝑘𝑢^∗+𝜉_𝑡−𝑉_𝑡+1𝑥₁+𝑘𝑢^∗+𝜉_𝑡≥0. (2.27) Therefore, 𝑉^′_𝑡(𝑥₂)−𝑉_𝑡(𝑥₁)≥0. When the status of technical innovation project is 𝑥₂, the maximum value of technology innovation obtained, and the makers’ optimal decision is 𝑉_𝑡(𝑥₂), so we have 𝑉_𝑡(𝑥₂)≥𝑉^′_𝑡(𝑥₂). Therefore,𝑉_𝑡𝑥₂−𝑉_𝑡𝑥₁≥𝑉^′_𝑡𝑥₂−𝑉_𝑡𝑥₁≥0,thatis,𝑉_𝑡𝑥₂≥𝑉_𝑡𝑥₁. (2.28)

Nature 3
From the aforementioned, we also can get that under fuzzy environment, if the optimal decision selected by the decision maker is “give up,” when the state variable is 𝑥 during the state of 𝑡 and when the variable is less than 𝑥, the optimal decision is “give up” too.

3. Conclusion

Combining the dynamic programming method and option analysis method, we make analysis of flexible decision problems of enterprise technological innovation investment, under fuzzy environment, mainly introducing fuzzy factors based on the model of Huchzermeier and Loch [6], Santiago and Vakili [10], Dixit and Pindyck [9], Sheng [7], and so forth. We establish the model, focus on the promotion of the Huchzermeier and Loch model under fuzzy environment, establish models of two-phase, multi-stage dynamic programming decision and make some analysis, and then draw valuable conclusions. But it only extends the models of Huchzermeier and Loch and Santiago and Vakili [10] to the fuzzy environment; in fact, this kind of promotion can also be extended to the fuzzy random environment and rough fuzzy environment. Although some attempt has been made, it is still not enough. This is what we should try our best in during the next step.