Imperfect Production Inventory Model with Uncertain Elapsed Time

Most of the classical inventory control model assumes that all items received conform to quality characteristics. However, in practice, items may be damaged due to production conditions, transportation and environmental conditions. Modelling such real world problems involve various indeterminate phenomena which can be estimated through human beliefs. The uncertainty theory proposed by Liu (2015) is extensively regarded as an appropriate tool to deal with such uncertainty. This paper investigates the optimum production run time and optimum cost in an imperfect production process, where the rate of imperfect items are different in different states of the process. The process may be shifting from ‘in-control’ state to the ‘out-of-control’ state is an uncertain variable with certain uncertainty distribution. Some propositions are derived for the optimal production run time and optimized the expected total cost function per unit time. Finally, numerical examples have been illustrated to study the practical feasibility of the model.


Introduction
In some real uncertain situation, we have to depend on domain experts to represent the belief degree when no samples are available to estimate a probability distribution. To deal with uncertainty in human belief, which is neither random nor fuzzy, Liu (2009Liu ( ), (2015, (2016) introduced uncertainty theory. It deals with modeling of uncertainty, based on normality, monotonicity, self-duality, countable sub-additivityand product measure axioms. Uncertain variable, uncertain set and uncertain measure are the basic tools to describe the uncertain phenomenon. Ghosh and Dey/Decis. Mak. Appl. Manag. Eng. 3 (2) (2020) 1-18 2 Expected value operator for the uncertain variable has become asignificant role in both theory and practice. The expected value of a monotone function of an uncertain variable is a Lebesgue-Stieltjes integral of the function concerning its uncertainty distribution. Liu (2012) proposed the concept of expected value of uncertain variables to rank the variables. Liu (2016) also verifiedthe linearity of the expected value operator. Liu and Ha (2010) derived a useful formula for calculating the expected values of strictly monotone functions of independent uncertain variables. Liu (2016) founded uncertain programming involving uncertain variables, which has been used to model in practical view of the system reliability design, project scheduling problem, transportation problem (Gao and Kar (2017), )portfolio selection problem ; Qin et al. (2016); ) and facility location problem ( , Ke et al. (2015)). In financial mathematics, Liu (2009Liu ( ), (2016 gave an uncertain stock model and European option price formula. Zhou et al. (2014) studied a dynamic recruitment problem with enterprise performance in an uncertain environment and presented an optimal search strategy for the firms' employment decisions. Chen et al. [3] analyses the pricing and effort decisions of a supply chain with a single manufacturer and single retailer considering the demand expansion effectiveness of sales effort under uncertainty.
In most of the classical economic production quantity (EPQ) model, it is assumed that the production process is always in good condition and produces 100% perfect quality items. But this assumption may not true in a real production system. In most of the practical situations, the production process continuously deteriorates and produce a certain percentage of defective (imperfect) items. Rosenblatt and Lee (1986) studied the effect of production process deterioration on the EPQ model and considering the shifting of the process from 'In-Control' state to 'Out-of-Control' state, which is exponentially distributed. The deteriorating production system is an imperfect production system that has a threshold level of defectiveness to separate the system into in-control and out-of-control states. Khouja and Meherez (1994), have considered the elapsed time until the production process shifts to an 'out-ofcontrol' state to be an exponentially distributed random variable and shown the weak and strong relationship between the rate of production and process quality.  extended the model of Khouja and Meherez (1994), assuming the percentage of defective items varies not linearly with production rate and production run time. Hariga and Ben-Daya(1998) extended the model of Rosenbaltt and Lee(1986) considering the general shift distribution and optimal production run time to be unique. Yehet al. (2000Yehet al. ( ), (2007 has considered different defective rates in in-control and out-of-control state for the imperfect production process and investigate production run length with warranty policy. Chen and Lo (2006) have developed an imperfect production process with allowable shortages and the products are sold with free minimal warranty. The probabilities of imperfect items in both states are different.
Again the EPQ models are derived under very modern heuristics and soft computing techniques, especially usingprobabilistic reasoning, fuzzy logic, Rough-Fuzzy logic,uncertainty theory etc. Chen et al. (2005) derived a fuzzy EPQ model with fuzzy opportunity cost. Wang et al. (2009) investigate a model of the imperfect production process with fuzzy elapsed time. Qin and Kar (2013) investigate a newsboy model under uncertain environment. Wang et al. (2015) contributed a paper is to provide a more general framework for single-period inventory problem by considering single-item and multiple items with a budget constraint under uncertain and random environment. The proposed models consider both uncertain and random behavior of the demands and cover not only the random instance but also the single-fold uncertain situation.
In this paper, we investigate optimum production run time and optimum cost in an imperfect production process, where the rate of imperfect items are different in different states of the process. The process may be shifting from 'in-control' state to the 'out-of-control' state is an uncertain variable with certain uncertainty distribution proposed by Liu (2009). The rest of the manuscript is organized as follows. Some preliminary concepts related to our study are discussed in Section 2. Section 3, states the assumptions and notations of the model. Section 4 and 5 are for the mathematical modeling and solution respectively. Section 6 provides numerical examples and discuss the results. Some sensitivity analyses are provided in Section 7. The paper summarizes and concludes in Section 8.

Preliminaries
Before presenting the inventory model in an uncertain environment, in this section, we introduce some useful definitions and fundamental results of Liu's Uncertainty theory. Uncertainty theory is an extremely important feature of the real world. The interpretation of uncertainty measure is the personal belief degree of an uncertain event.
Definition 1. Let S be a non-empty set and  a  -algebra over S . Each element A in S is called an event. A set function M from  to [0, 1] is called an uncertain measure if it satisfies the following axioms.
Definition 3 (Liu, 2016). In practice, the uncertain variable is described by the concept of uncertainty distribution  . Which is defined by For any set B of real numbers, we have (Liu, 2016). Let be  an uncertain variable. Then the expected value provided at least one of the two integrals is finite.
Theorem 1 (Liu, 2016). Let  be an uncertain variable with uncertainty  , then Similarly, Imperfect production inventory model with uncertain elapsed time 5 1 1 1 11 Hence the theorem. Note that the Expected value of a monotone function nothing but the Lebesgue-Stieltjes integral of the function with respect to its uncertainty distribution.

Assumptions& Notations
The following notations and assumptions are used in developing the model.

Assumptions
1. The production process has two states 'in-control-state' and 'out-of-control state'. At the beginning of each production process, the system is in 'in-control-state' and produces defective items at a rate 11 (0 1)  . During the production run, the process may be shifted from "in-control-state" to "out-of-control-state" at any uncertain time in production period and produces re-workable defective items at a rate 2 2 2 1 (0< <1 ) and ( ) 2. The elapsed time until the production process shift is  assumed to be an uncertain variable with uncertainty distribution  . 3. The production rate and demand rate are constant and deterministic. 4. Full (100%) inspection is considered at a certain cost. 5. The re-workable defective products are reworked at the end of the screening process with negligible reworked time.
6. The process is brought back to its initial conformable state 'in-control-state' for each setup so, incurred more setup cost including restoration cost which is fixed.
7. In real life situation for the competition market, shortages are not allowed. 8. The time horizon is infinite.

Mathematical formulation
In this proposed model, under the above assumptions, we consider an imperfect production process, in which the production process is in two states 'in-control' and 'out of control' state. We consider the production system has production uptime up to time t. In between production starting point and production uptime, the system shifted from 'in control' state to 'out of control' state at any uncertain time point having uncertainty distribution  . An inspection section separates the perfect and re-workable defective quality items through 100% screening process and the screening process finishes after production end. The perfect items are kept for satisfying customer demand and re-workable defective items are reworked with a cost after screening and stored in the main inventory. In the 'in-control' and 'out-ofcontrol' states, two types of items are produced among which the re-workable defective are produced at the rate  is positive valued, then 0 1 2 1 00 As (0) 0  , it follows that, Hence the result follows.

 be an uncertain variable with uncertainty distribution
 whose support is ( , ) ab,then The optimum production run length t  exists and is unique, which optimizes the function Where, Imperfect production inventory model with uncertain elapsed time Here C(t) represents the expected average cost per unit time for the classical EPQ model with constant defective rate and rework.
has a unique minimum atˆ tt  . Here () Ct is the cost function for the standard EPQ model with a constant defective rate of product throughout the production period and it is convex for all 0 t  . Let Ht is a sum of the convex function of C(t) and G(t), which is bounded and non-decreasing, there exist a unique 0 t   such that tt   for which H(t) is minimum.
And in this case

Solution
We cannot find out closed form solution of the optimum production run from the above objective function. Using the Search procedure along with the bisection method and bound of the optimum production time, a search procedure is appliedtofind out

Algorithm of the Search Procedure
Step 1: Set Step 2: Compute 2 LU o tt t   and the value of 0 () Ft .
Step 3: If  error tolerance).Control goes to step 5 otherwise goes to next step.

Numerical Example
The numerical examples are given for illustrative and verification of the real world problem.

Case 1: Linear Uncertain Distribution
Consider the uncertain variable  as linear with support ( , ) ab , where a=2.5 and b=10.0 and other parameters are K = 2000, 1   0.10, 2   0.20, h Figure 1. Expected average cost function is convex with respect to production time    By the search procedure, the optimum production time is t  = 4.55557 and corresponding optimum cost = 3379.72.

Sensitivity Analysis
Since the shifting time point from in-control state to out-of-control state is uncertain variable in between beginning and end of the production run and it  depends on support set of uncertainty distribution, so, the optimum production time and optimum cost per unit time depending on that.
From Table-1, Figure-2 and Table-2, Figure-5, for both the cases, it is observed that optimum production time is decreased with increase of production rate and rate of reworkable defective items in 'out-of-control' state and it increases with the increase of other parameter demand rate, setup cost and rate of reworkable defective items in 'in-control' state. Similarly from Table-1, Figure-3 and Table-2, Figure-6, for both the cases, it follows that optimum cost increases with the increases of all parameters.
Optimum production time is sensitive to the change of parameters production rate (p) and demand rate (d) and optimum cost per unit time is sensitive to the change of parameter demand rate (d). Optimum production time is moderately sensitive with the change of a parameter (p), slightly sensitive to the change of a parameter (d) and insensitive to the change of all other parameters. In the same way, the optimum cost/per unit time is slightly sensitive to the changes of the parameter (d) and insensitive to all other parameters.

Conclusion
In this article, we have discussed an imperfect production inventory model in an uncertain environment. It is assumed that an imperfect production process has two states 'in-control' state and 'out-of-control' state. Here we also assumed that the elapsed time of the production process follow uncertain shift distribution, which is an uncertain variable follows an uncertainty distribution. The basic difference from the earlier research article is that our model is on the study of uncertain phenomena while the stochastic is about the study of stochastic phenomena. For the lack of historical data, the shifting of the production process is quantified by domain experts' belief degree and by the principle of least square, the manager should follow a particular uncertainty distribution. The optimum production time and optimum cost depend on the type of uncertainty distribution along with the support set of that.
Two case examples justify the numerical verification of theorem and propositions in this proposed model on linear and zigzag uncertainty distribution with the same support. It follows that the expected optimum cost is near about approximately the same for both the uncertainty distribution, though optimum production time is slightly different. Finally, for illustrating the procedure, an algorithm is designed to find out the optimum goal.
Inthe future, we would like to extend our modelfor the imperfect production process with random uncertain circumstances. Moreover, the possible extension of different variants of imperfect production inventory problem like demand variability and trade credit policy to uncertain single/multi-objective models will also be the area of our research interest.