A stochastic multiobjective multiconstraint inventory model under inflationary condition and different inspection scenarios

Abstract

This article presents inspection scenarios for the multiobjective multiconstraint mixed backorder and lost sales inventory model with imperfect items in which the order quantity, reorder point, ordering cost, lead time, and backorder rate are decision variables. The objectives are minimizing expected annual cost and variance of shortages. The number of imperfect items is assumed to be a beta-binomial random variable. There are two inspection scenarios: the imperfect items observed during inspection and screenings are either all reworked or all discarded. In order to fit some real environment, this study assumes the maximum permissible storage space and available budget are limited. Backorder rate is considered as a function of expected shortages at the end of cycle. Stochastic inflationary conditions with a probability density function are also considered in the presented model. This study assumes that the purchasing cost is paid when an order arrives at the beginning of the cycle, and the ordering cost is paid at the time of the order placing. The aggregate demand follows a normal distribution function. Finally, a solution procedure is proposed in order to solve the discussed multiobjective model. In addition, a numerical example is presented to illustrate the multiobjective model and its solution procedure for different inspection scenarios, and a sensitivity analysis is conducted with respect to the important system parameters.

Keywords

Stochastic inventory model defective items lead time inflation chance-constrained programming

Introduction

The effect of inflation and time value of money are vital in practical environment. Analysis of inventory system under inflationary condition is carried out using two procedures. The first one determines optimal values of the control variables by minimizing the average annual cost, and the second one determines the optimal ordering policy by minimizing the discounted value of all future costs. Hadley¹ showed that the detailed computations in the simplest case, corresponding to the familiar deterministic lot size model, and the ordering quantities computed by minimizing the average annual cost and by minimizing the discounted cost do not differ significantly. Mirzazadeh² extends Hadely’s work by minimizing the inflation and time value of money under uncertain conditions, shortages and the effects of deterioration. The above-mentioned system is formulated with two methods, which are derived under some assumptions that the objective of inventory management is to minimize the average annual cost and the discounted cost. These methods are compared to each other carefully. The results reveal that the mentioned methods (the average annual cost and the discounted cost) have a negligible difference to each other. This study determines optimal values by minimizing the average annual cost under an uncertain inflationary condition.

A number of articles have considered the effect of inflation on the inventory system since 1975. Buzacott³ dealt with an economic order quantity (EOQ) model with inflation subjects to different types of pricing policies. Misra⁴ developed a discounted cost model and included internal (company) and external (general economy) inflation rates for various costs associated with an inventory system. Sarker and Pan⁵ surveyed the effects of inflation and the time value of money on an order quantity with a finite replenishment rate. Some efforts were extended to consider variable demands, such as Uthayakumar and Geetha,⁶ Vrat and Padmanabhan,⁷ Datta and Pal,⁸ Hariga,⁹ Hariga and Ben-Daya,¹⁰ and Chaung.¹¹ Roy et al.¹² prepared an inventory model for a deteriorating item with displayed stock-dependent demand under fuzzy inflation and time discounting over a random planning horizon. Mirzazadeh et al.¹³ considered stochastic inflationary conditions with variable probability density functions (pdfs) over the time horizon, and the demand rate is dependent on inflation rates. The developed model also implicates to a finite replenishment rate and a finite time horizon with shortage. The objective is to minimize the expected present value of costs over the time horizon. Another research has been performed by Ameli et al.¹⁴ considering an EOQ model for imperfect items under fuzzy inflationary conditions. In addition, Taheri-Tolgari et al.¹⁵ presented a discounted cash-flow approach for an inventory model for imperfect item with considering inspection errors.

A lot of inventory models have been considered under the assumption that shortages are allowed. One important group of these models considers that all customers wait until the arrival of the next replenishment (i.e. full backlogging case) when there is shortage. Another situation is to admit that all the served customers leave the system (i.e. lost sale case). However, in many practical situations, there are customers (i.e. whose needs are not critical at that time) willing to wait for the next replenishment in order to satisfy their demands, while others do not want to or cannot wait and leave the system. These situations are modeled by considering partial backlogging in the information of mathematical models. Some studies^16,17 have considered backorder rate as a fixed constant. Some other authors have considered a backorder rate as a function of an expected shortage quantity at the end of the cycle. Their studies are based upon this assumption that the larger amount of the expected shortage at the end of cycle, the smaller amount of customer can wait and hence the smaller backorder rate would be. Ouyang and Chaung¹⁸ were first to introduce this assumption in their model and some other authors generalized this assumption in their models for a backorder rate.^19

–22

For the traditional inventory model, lead time is treated as a predetermined constant or a random variable (therefore, it is uncontrollable). Recently, inventory models considering lead time as a decision variable and can be decomposed into several components, each having a crashing cost function for the respective reduced lead time. Liao and Shyu²³ have initiated a study on lead time reduction by presenting an inventory model in which lead time is a decision variable and the order quantity is predetermined. Ben-Daya and Rauf²⁴ developed a model that considered both lead time and order quantity as decision variables. Moon and Gallego²⁵ assumed unfavorable lead time demand distribution and solved both continuous review and periodic review models with a mixture of backorders and lost sales using minimax distribution free approach. Ouyang et al.²⁶ have generalized Ben-Daya and Rauf²⁴ model in which the lead time demand is considered a normal distribution or distribution free. Later, Hariga and Ben-Daya¹⁶ extend Ouyang et al.’s²⁶ study and presented both continuous review and periodic review inventory models in partial and perfect lead time demand distribution information environment in which reorder point is treated as a variable. Lee¹⁹ considered an inventory model involving variable lead time with the mixture of distribution. Gholami-Qadikolaei et al.²⁷ considered an inventory problem with variable lead time, defective items and controllable backorder rate in which a nonlinear relationship exists between the crashing cost and lead time. They calculated optimal reorder point in their study, whereas the previous researchers do not obtain optimal safety factor when they consider backorder rate as a function of expected shortage quantity at the end of cycle in their studies. On the other hand, accompanying the growth of Just-In-Time (JIT) production, which has evidence that many benefits can be obtained from reducing setup cost, the issue of investment in setup cost reduction has received a great deal of attention. Porteus²⁸ first introduced a framework of the setup cost reduction in the classical EOQ, and then, many authors such as Billington,²⁹ Kim and Haya,³⁰ Paknejad et al.³¹ and Sarker and Coates³² have explored the more investigations in this area. Wu and Lin³³ studied on lead time and ordering cost reduction when the receiving quantity is different from the ordered quantity. Lee et al.²¹ considered an inventory model with backorder discount and ordering cost reduction including mixture of distribution. Wu and Ouyang³⁴ examined the effect of defective items on a mixture of backorders and lost sales inventory models, and using the result of a basic theorem from renewal reward processes,³⁵ they calculated expected annual cost (EAC). Annadurai and Uthayakumar³⁶ considered an inventory model combining ordering cost and lead time reduction with defective units. Moreover, Ben-Daya and Noman³⁷ proposed integrated inventory models for inspection policies in which order quantity and reorder point are decision variables.

Recently, Salameh and Jaber³⁸ proposed an EOQ for a buyer who receives imperfect lots. They extended the traditional EOQ model by accounting for imperfect quality items under the following assumptions: (1) a lot contains a probabilistic fraction of defective items that is independent of lot size and (2) screening rate is finite and it is more than demand rate. Maddah and Jaber³⁹ proposed an inventory model in which expected annual profit could be calculated using the renewal reward theorem. Khan et al.⁴⁰ extended the study by Salameh and Jaber³⁸ by assuming that stock outs (which are treated as both lost sales and backorders) may occur when screening rate is slower than demand rate. In the case of considering defective units in an order lot, our model formulation differs from the three mentioned articles since the number of defective items is assumed to be a beta-binomial random variable and inspection time is negligible.

There are some inventory models with shortages including restrictions on inventory investment, storage space, or reorder workload. Brown and Gerson⁴¹ proposed some models for stochastic inventory system with the limitation on total inventory investment. Schrady and Choe⁴² proposed a model with the total time weighted shortages with limitations on inventory investment and reorder workload. Gardner⁴³ develops models for minimizing expected approximate backordered sales with the restrictions on aggregate investment and replenishment workload. Schroeder⁴⁴ presents a model constrained by total expected annual ordering with an objective of minimizing the expected number of units backordered per year. Hariga⁴⁵ presented a stochastic full backlogging inventory system with space restriction in which the order quantity and reorder point are decision variables. Xu and Leung⁴⁶ propose an analytical model in a two-party vendor-managed system where the retailer restricts the maximum space allocated to the vendor. Bera et al.⁴⁷ presented a minimax distribution free procedure for stochastic lead time and demand inventory model under budget restriction when the purchasing cost payment is due at the time of order receiving in which order quantity and reorder point are decision variables. Moon et al.⁴⁸ proposed three extended models with variable capacity. First, they presented an EOQ model with random yields. Second, they developed a multiitem EOQ model with storage space and investment constraint and solved model with Lagrange multiplier method. Third, they applied a distribution free approach to the (Q,r) with variable capacity.

With today’s high inflation rate, it is important to take into account the effect of inflationary condition in the stochastic inventory system. Also, the cost of land acquisition is high in most of the countries, and one of the main concerns of inventory managers is to ensure that storage space is enough when an order arrives. Besides, in order to fit some real and practical environment, the amount received is assumed to be a random variable due to rejection during inspection. So, in the case of controllable lead time, this article, with considering several constraints on the stochastic inventory system, will support managers in decision making (DM).

This study proposes multiobjective stochastic mixed backorder and lost sale continuous review inventory model in which the order quantity, reorder point, lead time, ordering cost, and backordering rate are decision variables. The objectives are minimizing total EAC and variance of shortages. An average annual cost method with stochastic inflationary conditions and pdf is considered in the presented model. The storage space and the budget constraints are also considered in the presented model. Both constraints are random variables since X (i.e. lead time demand) and Y (i.e. number of defective units) are random variables. A chance-constrained programming technique is applied to make the random constraints crisp. This article considers the amount received as random variables due to rejection during inspection. Imperfect units are assumed to be beta-binomial random variable. Different inspection scenarios are assumed including the case when imperfect items are reworked and the case when imperfect items are discarded without reworking. The scenario adopted is the one leading to least overall cost. The piece-wise linear crashing cost function is considered, which is widely used in project management in which the duration of some activities can be reduced by assigning more resources to the activities. The demand for ith source is independent and identically distributed (IID) and normally distributed, and hence, the lead time demand follows a normal distribution function. A solution procedure is proposed to find effective solution of the respective multiconstraint multiobjective inventory model for different inspection scenarios. In addition, a numerical example is presented to illustrate the multiobjective model and its solution procedure for different inspection scenarios, and a sensitivity analysis is conducted with respect to the important system parameters.

This article is organized as follows: In section “Assumptions,” assumptions are given. In section “Inventory inspection model formulation,” we present a mathematical model and explain about two inventory inspection models, in which inflationary condition is not considered. In section “Inventory inspection models with inflation,” we present a mathematical model of two inventory inspection approaches, in which inflationary condition is considered. Solution method is proposed in section “Solution method.” In section “Numerical example,” a numerical example is provided to illustrate the models. Finally, we conclude this article.

Assumptions

The developed model is based on the following assumptions.

Shortages are allowed and partially backlogged.

Inventory position is continuously monitored, and order of size Q is made whenever the inventory level hits the reorder point $r$ and replenishment rate is infinite.

The purchasing cost is paid when order arrives (at the beginning of the cycle).

Ordering cost is paid at the time an order is placed.

Shortage cost is calculated at the end of cycle.

The demand rate at the ith source, $D_{i} (i = 1, 2, \dots, u)$ , is identically distributed (IID) and normally distributed with mean $μ_{D_{i}}$ and standard deviation $σ_{D_{i}}$ , and hence, aggregate demand rate $D = D_{1} + D_{2} + \dots + D_{u}$ is normally distributed with mean $μ_{D}$ and standard deviation $σ_{D}$ , which are obtained as follows

μ_{D} = \sum_{i = 1}^{u} μ_{D_{i}}

(1)

σ_{D} = \sqrt{\sum_{i = 1}^{u} {σ_{D_{i}}}^{2} + 2 \sum_{i = 1}^{u - 1} \sum_{j = i + 1}^{u} σ_{D_{i}} σ_{D_{j}} ρ_{ij}}

(2)

where $ρ_{ij}$ is the correlation coefficient of $D_{i}$ and $D_{j}$ ( $i = 1, 2, \dots, u, j = 1, 2, \dots, u$ )

Lead time demand $x$ is convolution of aggregate demand rate $D$ and lead time $L$ . If lead time is fixed, then mean and variance of lead time demand are obtained as follows⁴⁹

E (x) = L μ_{D} and σ_{x} = σ_{D} \sqrt{L}

(3)

The reorder point $r$ is the expected demand during lead time plus safety stock (SS) and $SS = k \times (standard deviation of lead time demand)$ (i.e. $r = E (x) + k σ_{x}$ , where $k$ is the safety factor satisfying $P (X > r) = P (Z > k) = α$ , $Z$ represents the standard normal random variable and $α$ represents the allowable stock out probability during lead time).

Backordering rate is dependent on expected shortage quantity with a negative exponential function as given as follows

β = ν e^{- ϑ E {(x - r)}^{+}}

(4)

where

(0 \leq ν \leq 1, ϑ \geq 0)

are backorder parameters.

Units are demanded in small quantities, so that overshooting of the reorder point is not appreciable.

An arrival order may contain some defective items. The number of defective items in an arriving order of size $Q$ is a beta-binomial random variable, where $p (0 \leq p \leq 1)$ represents the defective rate in an order lot. When an order is arrived, all the items are inspected and all defective items are assumed to be found out and is returned to the vendor at the same time.

Shortage cost does not depend on time.

Planning horizon is infinite.

X (i.e. lead time demand), Y (i.e. number of defective items), and I (i.e. inflation rate) are independent.

The inflation rate is a random variable with a known distribution function.

The cost equations are approximations because inventory levels and demands are treated as continuous instead of discrete quantities.

The time the system is out of stock during a cycle is small compared to the cycle length.

There are no orders outstanding at the time the reorder point is reached.

The reorder level is larger than the mean of the lead time demand.

Inspection is error free and inspection time is negligible.

The lead time consists of m mutually independent components. The zth component has a minimum duration $a_{z}$ , the normal duration $b_{z}$ , and a crashing cost $c_{z}$ per unit time. Furthermore, for convenience, we rearrange $c_{z}$ such that $c_{1} \leq c_{2} \leq \dots \leq c_{m}$ .

If we let $L_{0} = \sum_{j = 1}^{m} b_{j}$ and $L_{z}$ be the length of lead time with components $1, 2, \dots, z$ crashed to their minimum duration, then $L_{z}$ can be expressed as $L_{z} = \sum_{j = 1}^{m} b_{j} - \sum_{j = 1}^{z} (b_{j} - a_{j}), z = 1, 2, \dots, m$ ; the lead time crashing cost $C (L)$ per cycle for a given $L ϵ [L_{z}, L_{z - 1}]$ is given by $C (L) = c_{z} (L_{z - 1} - L) + \sum_{j = 1}^{z - 1} c_{j} (b_{j} - a_{j})$ .

The components of lead time are crashed one at a time starting with component 1 (because it has a minimum unit crashing cost) and then component 2 and so on.

Inventory inspection model formulation

It is considered that defective rate, p, follows a beta-distribution with probability distribution function, $f (p)$ , with parameters ( $α, β$ ). The production process is assumed to be under binomial control. The distribution of defective items in an order lot, Y, is beta-binomial distribution with parameters ( $Q, α, β$ ). In this case, the expected value and variance of defective items are as follows (refer to Appendix 2)

E (y) = QE (p)

(5)

\to Var (y) = QE (p (1 - p)) + Q^{2} Var (p)

(6)

Inspection model without reworking imperfect units

The inventory level of an item will be diminished due to the random demand. The management places an order with the amount of $Q$ , in which $Y$ items may be defective, when the inventory level reaches the reorder level. The number of cycle per year, n, is a random variable and equals $D / (Q - y)$ , and the cycle time is $(Q - y) / D$ . The amount of shortages per cycle is a random variable since X (i.e. demand during lead time) is a random variable and can be expressed by

{(x - r)}^{+} = Max (x - r, 0) = {\begin{matrix} x - r, x \geq r \\ 0, x \leq r \end{matrix}

(7)

So, the number of backorders per cycle is $β (x - r)^{+}$ and the number of lost sales is $(1 - β) (x - r)^{+}$ . The net inventory level just before order arrives is $r - x + (1 - β) (x - r)^{+}$ , and the maximum inventory level at the beginning of the cycle (considering y defective items) is $Q - y + r - x + (1 - β) (x - r)^{+}$ . The average inventory level is a random variable and it can be calculated as follows

(\frac{Q - y}{2} + r - x + (1 - β) {(x - r)}^{+})

(8)

Therefore, in the new stochastic environment, the EAC for partial backlogging policy can be written as

\begin{matrix} EAC (Q, r, L) = (A + c_{z} (L_{z - 1} - L) + \sum_{j = 1}^{z - 1} c_{j} (b_{j} - a_{j})) \\ DE (\frac{1}{Q - y}) \\ + h [E (\frac{Q - y}{2}) + r - E (x) + (1 - β) E {(x - r)}^{+}] \\ + DE (\frac{1}{Q - y}) (π + π_{\circ} (1 - β)) E {(x - r)}^{+} + C_{p} D, L ϵ [L_{z}, L_{z - 1}] \end{matrix}

(9)

The underlying assumption in the above model (9) is that ordering cost, A, is a fixed constant and not subject to control. In this study, we consider that the ordering cost can be reduced through capital investment and it can be treated as a decision variable. Therefore, we seek to minimize the sum of capital investment cost of reducing ordering cost A and the inventory-related costs by optimizing over Q, A, r, and L constrained $0 \leq A \leq A_{0}$ . That is, according to our model, the objective of our problem is to minimize the following expected total annual cost

EAC (Q, A, r, L) = θ I (A) + EAC (Q, r, L)

(10)

Over $A \in [0, A_{0})$ , where $θ$ is the fractional opportunity cost of capital per year, and $I (A)$ follows a logarithmic function given by

I (A) = b \ln (\frac{A_{0}}{A}) for A \in (0, A_{0}]

(11)

where $1 / b$ is the fraction of the reduction in A per dollar increase in investment. This logarithmic function is widely utilized since it is consistent with the Japanese experience as reported in the study by Hall⁵⁰ and has been used by Porteus²⁸ and others. Thus, equation (10) becomes as follows

\begin{matrix} EAC (Q, A, r, L) = θ b ln (\frac{A_{0}}{A}) \\ + (A + c_{z} (L_{z - 1} - L) + \sum_{j = 1}^{z - 1} c_{j} (b_{j} - a_{j})) DE (\frac{1}{Q - y}) \\ + h [E (\frac{Q - y}{2}) + r - E (x) + (1 - β) E {(x - r)}^{+}] \\ + DE (\frac{1}{Q - y}) (π + π_{\circ} (1 - β)) E {(x - r)}^{+} + C_{p} D \end{matrix}

(12)

where

L ϵ [L_{z}, L_{z - 1}], A ϵ (0, A_{0}]

As we know, the following expected value is calculated⁵¹

E (g (x)) ≅ g (μ) + \frac{g ″ (μ)}{2} {σ_{x}}^{2}

(13)

The above approximation for calculating expected value of a random quantity is accurate when standard deviation is small. Considering this assumption, the expected value of $1 / (Q - y)$ is calculated by

\begin{matrix} g (y) = \frac{1}{Q - y}, g ″ (y) = \frac{2}{{(Q - y)}^{3}} \\ E (\frac{1}{Q - y}) ≅ \frac{1}{Q (1 - E (p))} + \frac{Var (y)}{{(Q (1 - E (p)))}^{3}} \end{matrix}

(14)

Besides, the expected value of $(Q - y) / 2$ is

E (\frac{Q - y}{2}) = \frac{Q (1 - E (p))}{2}

(15)

In this article, we consider a storage space restriction, which is dependent on the maximum inventory size. The restriction ensures that even if an item reaches to its maximum inventory position, the maximum available space is still enough for it. Therefore, the random storage space constraint for a partial backlogging policy is as follows

P {(Q - y + r - x + (1 - β) (x - r)^{+}) \leq \frac{F}{f}} \geq γ

(16)

The above constraint forces the probability that total used space is within maximum permissible storage space to be no smaller than $γ$ . By using the chance-constrained programming technique that is originally developed by Charnes and Cooper⁵² and considering Markov inequality, the random storage space constraint is transformed to a crisp constraint, as given below

\begin{matrix} \to γ \leq P {x + (1 - β) {(x - r)}^{+} + y + \frac{F}{f} \geq Q + r} \\ \leq \frac{E (x) - (1 - β) E {(x - r)}^{+} + E (y) + \frac{F}{f}}{Q + r} \\ \to γ \leq \frac{E (x) - (1 - β) E {(x - r)}^{+} + E (y) + \frac{F}{f}}{Q + r} \\ \to γ (Q + r) - \frac{F}{f} - E (x) - E (y) + (1 - β) E (x - r)^{+} \leq 0 \end{matrix}

(17)

In this study, a restriction on the maximum available budget for purchasing is considered. In some cases, the available budget for purchasing is limited, and system managers would like to control it by considering this restriction on the inventory system. In this article, we assume that the purchasing costs are paid when an order arrives. With this assumption, we establish a limitation on the maximum available budget for purchasing. Thus, a form of the budget constraint is as follows

P {C_{p} (Q - y) \leq B} \geq φ

(18)

\begin{matrix} \to P {(Q - y) \leq \frac{B}{C_{p}}} \geq φ \\ \to φ \leq P (y + \frac{B}{C_{p}} \geq Q) \leq \frac{E (y) + \frac{B}{C_{p}}}{Q} \\ \to φ Q - \frac{B}{C_{p}} - E (y) \leq 0 \end{matrix}

(19)

When the lead time demand $X$ follows a normal pdf $f_{X} (x)$ with mean $E (x) = μ_{D} L$ and standard deviation $σ_{x} = σ_{D} \sqrt{L}$ and by giving the reorder point $r = μ_{D} L + k σ_{D} \sqrt{L}$ , the expected shortage quantity $E (x - r)^{+}$ can be expressed by

E (x - r)^{+} = \int_{r}^{\infty} (x - r) f (x) dx, k = \frac{r - E (x)}{σ_{x}}, z = \frac{x - E (x)}{σ_{x}}

(20)

\begin{matrix} \to E (x - r)^{+} = σ_{x} \int_{k}^{\infty} (z - k) f (z) dz \\ \to E (x - r)^{+} = σ_{D} \sqrt{L} [\int_{k}^{\infty} zf (z) dz - \bar{k Φ (k)}], \\ [\int_{k}^{\infty} zf (z) dz - \bar{k Φ (k)}] = [ϕ (k) - \bar{k Φ (k)}] = ψ (k) \\ \to E (x - r)^{+} = σ_{D} \sqrt{L} ψ (k) \geq 0 \end{matrix}

(21)

where $ϕ$ is the standard normal density function and $\bar{Φ}$ is the standard normal complementary cumulative distribution function.

Also, variance of shortages is calculated as follows (see Appendix 3)

Var {(x - r)}^{+} = {σ_{D}}^{2} L ζ (k) \geq 0

(22)

Therefore, considering $r = μ_{D} L + k σ_{D} \sqrt{L}$ , $E (x - r)^{+} = σ_{D} \sqrt{L} ψ (k)$ , $β = ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}$ and respective inspection scenario, the multiobjective constrained is as follows

\begin{array}{l} min E A C (Q, A, k, L) \\ = θ b ln (\frac{A_{0}}{A}) + D E (\frac{1}{Q - y}) [A + c_{z} (L_{z - 1} - L) + \sum_{j = 1}^{z - 1} c_{j} (b_{j} - a_{j})] \\ + h [E (\frac{Q - y}{2}) + k σ_{D} \sqrt{L} + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k)] \\ + D E (\frac{1}{Q - y}) [(π + π_{°} (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)})) σ_{D} \sqrt{L} ψ (k)] + \frac{δ D}{1 - E (p)} + C_{p} D \end{array}

min Var {(x - r)}^{+} = {σ_{D}}^{2} L ζ (k)

Subject to

\begin{matrix} γ (Q + μ_{D} L + k σ_{D} \sqrt{L}) + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) \\ σ_{D} \sqrt{L} ψ (k) - \frac{F}{f} - (μ_{D} L + E (y)) \leq 0 \\ φ Q - \frac{B}{C_{p}} - E (y) \leq 0 Q \geq 0, k \geq 0, A ϵ (0, A_{0}] and L ϵ [L_{z}, L_{z - 1}] \end{matrix}

(23)

where

E (\frac{1}{Q - y}) ≅ \frac{1}{Q (1 - E (p))} + \frac{Var (y)}{{(Q (1 - E (p)))}^{3}}

E (\frac{Q - y}{2}) = \frac{Q (1 - E (p))}{2}

Inspection model with reworking imperfect units

In this section, we consider the assumption of Rosenblatt and Lee⁵³ that imperfect items can be reworked instantaneously at a cost. Thus, the mathematical model can be expressed as follows

\begin{matrix} min EAC (Q, A, k, L) \\ = θ b ln (\frac{A_{0}}{A}) + \frac{D}{Q} [A + c_{z} (L_{z - 1} - L) + \sum_{j = 1}^{z - 1} c_{j} (b_{j} - a_{j})] \\ + h [\frac{Q}{2} + k σ_{D} \sqrt{L} ψ (k) + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k)] \\ + \frac{D}{Q} [(π + π_{\circ} (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)})) σ_{D} \sqrt{L} ψ (k)] \\ + D δ + C_{r} DE (p) + D C_{P} \end{matrix}

min Var {(x - r)}^{+} = {σ_{D}}^{2} L ζ (k)

Subject to

\begin{matrix} γ (Q + μ_{D} L + k σ_{D} \sqrt{L}) - \frac{F}{f} \\ - (μ_{D} L - (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k)) \leq 0 \\ Q - \frac{B}{C_{p}} \leq 0 Q, k \geq 0, A ϵ (0, A_{0}] and L ϵ [L_{z}, L_{z - 1}] \end{matrix}

(24)

Inventory inspection models with inflation

Inspection model without reworking imperfect units

In this study, the average annual cost method with a stochastic inflationary condition is considered. In this article, we consider that the ordering cost will be paid at the time of the replenishment. Hence, $T_{r}$ is the reorder point time and can be expressed by

T_{r} = \frac{I_{max} - r}{D}, I_{max} = Q - y + r - x + (1 - β) (x - r)^{+} \to T_{r} = \frac{Q - y - x + (1 - β) {(x - r)}^{+}}{D}

(25)

Expected ordering cost with inflation (EOCWI) is as follows (refer to Appendix 4)

\begin{array}{l} E O C W I = (A + C (L)) [D E (\frac{1}{Q - y}) (1 + \frac{E (i)}{2}) - \frac{E (i)}{2}] \\ + (A + C (L)) (1 - [(μ_{D} L - (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k)) E (\frac{1}{Q - y})]) E (i) \end{array}

(26)

Total holding costs per year in the average annual cost method is dependent on the average inventory level. Therefore, expected holding cost per year with inflation (EHCWI) can be obtained as follows (refer to Appendix 5)

E H C W I = h (1 + \frac{E (i)}{2}) (\frac{E (Q - y)}{2} + k σ_{D} \sqrt{L} + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k))

(27)

In this study, we assume that shortage cost is calculated at the end of cycle. Thus, expected shortage cost per year with inflation (ESCWI) can be obtained (see Appendix 6 for details)

ESCWI = (π + π_{\circ} (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)})) [DE (\frac{1}{Q - y}) (1 + \frac{E (i)}{2}) + \frac{E (i)}{2}] σ_{D} \sqrt{L} ψ (k)

(28)

In addition, we consider that the purchasing costs are paid at the beginning of the cycle. Hence, total expected purchasing cost with inflation (EPCWI) is as follows (refer to Appendix 7)

EPCWI = [D C_{p} + \frac{D}{1 - E (p)} δ] (1 + \frac{E (i)}{2} (1 - \frac{E (Q - y)}{D}))

(29)

Therefore, the expected annual cost with inflation (EACWI) is the sum of equations (26)–(29). Thus, the multiobjective constrained model can be expressed as follows

min EACWI = θ b ln (\frac{A_{0}}{A}) + EOCWI + EHCWI + ESCWI + EPCWI

min Var {(x - r)}^{+} = {σ_{D}}^{2} L ζ (k)

Subject to

\begin{matrix} γ (Q + μ_{D} L + k σ_{D} \sqrt{L}) + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) \\ σ_{D} \sqrt{L} ψ (k) - \frac{F}{f} - (μ_{D} L + E (y)) \leq 0 \\ φ Q - \frac{B}{C_{p}} - E (y) \leq 0 Q \geq 0, k \geq 0, A ϵ (0, A_{0}] and L ϵ [L_{z}, L_{z - 1}] \end{matrix}

(30)

Note that if we do not have inflation rate (i = 0.00), then the above model (30) is converted to model (23).

Inspection model with reworking imperfect units

In this section, the inventory model’s total EOCWI is calculated as

\begin{matrix} EOCWI = (A + C (L)) [\frac{D}{Q} (1 + \frac{E (i)}{2}) - \frac{E (i)}{2}] \\ + (A + C (L)) (1 - [\frac{(μ_{D} L - (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k))}{Q}]) E (i) \end{matrix}

(31)

The EHCWI can be expressed as follows

EHCWI = h (1 + \frac{E (i)}{2}) (\frac{Q}{2} + k σ_{D} \sqrt{L} + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k))

(32)

The ESCWI is

ESCWI = (π + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) π_{\circ}) [\frac{D}{Q} (1 + \frac{E (i)}{2}) + \frac{E (i)}{2}] σ_{D} \sqrt{L} ψ (k)

(33)

The total EPCWI can be calculated as follows

EPCWI = [D C_{p} + DE (p) C_{r} + D δ] (1 + \frac{E (i)}{2} (1 - \frac{Q}{D}))

(34)

Therefore, our model is reduced to

min EACWI = θ b ln (\frac{A_{0}}{A}) + EOCWI + EHCWI + ESCWI + EPCWI

min Var {(x - r)}^{+} = {σ_{D}}^{2} L ζ (k)

Subject to

\begin{matrix} γ (Q + μ_{D} L + k σ_{D} \sqrt{L}) - \frac{F}{f} \\ - (μ_{D} L - (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k)) \leq 0 \\ Q - \frac{B}{C_{p}} \leq 0 Q, k \geq 0, A ϵ (0, A_{0}] and L ϵ [L_{z}, L_{z - 1}] \end{matrix}

(35)

Note that if we do not have inflation rate (i = 0.00), then the above model (35) is converted to model (24).

Solution method

In this study, a heuristic method is utilized in order to solve multiobjective models. First, EAC function subjects to budget and storage space constraints is minimized as follows

Step 1.

min EACWI = θ b ln (\frac{A_{0}}{A}) + EOCWI + EHCWI + ESCWI + EPCWI

Subject to

\begin{matrix} γ (Q + μ_{D} L + k σ_{D} \sqrt{L}) + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) \\ σ_{D} \sqrt{L} ψ (k) - \frac{F}{f} - (μ_{D} L - E (y)) \leq 0 \\ φ Q - \frac{B}{C_{p}} - E (y) \leq 0 Q \geq 0, k \geq 0, A ϵ (0, A_{0}] and L ϵ [L_{z}, L_{z - 1}] \end{matrix}

(36)

Second, variance of shortages is minimized in a distance of optimum point of first step as follows

Step 2.

min Var {(x - r)}^{+} = {σ_{D}}^{2} L ζ (k)

Subject to

\begin{matrix} EACWI (Q, A, r, L) \approx EACW I^{*} + ξ \\ γ (Q + μ_{D} L + k σ_{D} \sqrt{L}) + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) \\ σ_{D} \sqrt{L} ψ (k) - \frac{F}{f} - (μ_{D} L - E (y)) \leq 0 \\ φ Q - \frac{B}{C_{p}} - E (y) \leq 0 \\ Q \geq 0, k \geq 0, A ϵ (0, A_{0}] and L ϵ [L_{z}, L_{z - 1}] \end{matrix}

(37)

For Step 1, models (30) and (35) can be solved using Lagrange multiplier method with the same solution procedure. For instance, the Lagrangian function of the EAC function of model (30) is as follows

\begin{array}{l} E A C W I (Q, k, L, λ_{1}, λ_{2}) \\ = E A C W I (Q, A, k, L) \\ + λ_{1} [γ f (Q + μ_{D} L + k σ_{D} \sqrt{L}) - F - f μ_{D} L + f (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k)] \\ + λ_{2} [φ Q - \frac{B}{C_{p}} - E (y)] \end{array}

(38)

Notice that for any given $(Q, A, k, λ_{1}, λ_{2})$ , $EACF (Q, A, k, L, λ_{1}, λ_{2})$ is a concave function in $L \in [L_{z}, L_{z - 1}]$ because

\frac{\partial^{2} EACF (Q, k, L, λ_{1}, λ_{2})}{\partial L^{2}} \leq 0

(39)

Hence, for fixed $(Q, A, k, λ_{1}, λ_{2})$ , the minimum EAC will occur at the end of point of the interval $L ϵ [L_{z}, L_{z - 1}]$ . Therefore, for fixed $L ϵ [L_{z}, L_{z - 1}]$ , the Kuhn–Tucker necessary conditions for minimization of function (38) are as follows

\frac{\partial EAC (Q, A, k, L, λ_{1}, λ_{2})}{\partial Q} = 0

(40)

\frac{\partial EAC (Q, A, k, L, λ_{1}, λ_{2})}{\partial k} = 0

(41)

\frac{\partial EAC (Q, A, k, L, λ_{1}, λ_{2})}{\partial A} = 0

(42)

λ_{1} [γ (Q + μ_{D} L + k σ_{D} \sqrt{L}) + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k) - \frac{F}{f} - (μ_{D} L - E (y))] = 0

(43)

λ_{2} [φ Q - \frac{B}{C_{p}} - E (y)] = 0

(44)

Since partial derivatives for these models are very complicated functions; hence, through the extensive testing, we assess the minimum values for the EAC for all models numerically.

Consequently, we can establish the following algorithm to find the optimal ( $Q^{*}, A^{*}, r^{*}, L^{*}$ ).

Step 1. For each $L_{z}$ , $z = 0, 1, 2, \dots, m,$ perform Steps 1.1 and 1.2.

Step 1.1. Input the values of $A, D, h, k, π, π_{0}, B, F, C_{p}, δ, f, i, γ, φ, θ, b, ν, ϑ$ and $ρ$ .

Step 1.2. Compute $EAC (Q^{k}, A^{k}, {λ_{1}}^{k}, {λ_{2}}^{k})$ in terms of different $k$ and $ψ (k)$ and stop when you get to

EACWI (Q^{k_{z}}, A^{k_{z}}, {λ_{1}}^{k_{z}}, {λ_{2}}^{k_{z}}) < \min {EACWI (Q^{k_{z} - ξ}, A^{k_{z} - ξ}, {λ_{1}}^{k_{z} - ξ}, {λ_{2}}^{k_{z} - ξ}), EACWI (Q^{k_{z} + ξ}, A^{k_{z} + ξ}, {λ_{1}}^{k_{z} + ξ}, {λ_{2}}^{k_{z} + ξ})}

| EACWI (Q^{k_{z}}, A^{k_{z}}, {λ_{1}}^{k_{z}}, {λ_{2}}^{k_{z}}) - EACWI (Q^{k_{z} - ξ}, A^{k_{z} - ξ}, {λ_{1}}^{k_{z} - ξ}, {λ_{2}}^{k_{z} - ξ}) | \leq ε

| EACWI (Q^{k_{z}}, A^{k_{z}}, {λ_{1}}^{k_{z}}, {λ_{2}}^{k_{z}}) - EACWI (Q^{k_{z} + ξ}, A^{k_{z} + ξ}, {λ_{1}}^{k_{z} + ξ}, {λ_{2}}^{k_{z} + ξ}) | \leq ε

In this article, we consider $ε = 0.01$

Step 2. Find $mi n_{z = 0, 1, \dots, m} EACWI (Q_{z}, A_{z}, k_{z}, L_{z}, λ_{1_{z}}, λ_{2_{z}}) = EACWI (Q^{*}, A^{*}, k^{*}, L^{*}, λ_{1}, λ_{2})$ , and hence, the optimal reorder point is $r^{*} = μ_{D} L^{*} + k^{*} σ_{D} \sqrt{L^{*}}$ .

In the second optimization level, our goal is to minimize variance of shortages, which is dependent on safety factor and lead time variables. In this case, variance of shortages can be minimized by increasing safety factor or decreasing lead time in a distance ( $ξ$ ) of optimum point of EAC function.

It is obvious that model (35) can be solved with the same solution procedure. Finally, compare the costs obtained for models (30) and (35) and choose the scenario with minimum cost. Moreover, to ensure the convexity of the constrained model of first step, the optimal values of $(Q^{*}, A^{*}, k^{*}, L^{*})$ must satisfy the following sufficient conditions.

For a given value of L, we obtain the Hessian matrix H for objective function as follows

H = [\begin{matrix} \frac{\partial^{2} E A C W I (Q, A, k)}{\partial Q^{2}} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial Q \partial A} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial Q \partial k} \\ \frac{\partial^{2} E A C W I (Q, A, k)}{\partial A \partial Q} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial A^{2}} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial A \partial k} \\ \frac{\partial^{2} E A C W I (Q, A, k)}{\partial k \partial Q} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial k \partial A} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial k^{2}} \end{matrix}]

The first, second, and third principal minor of H must be positive for the optimal values $(Q^{*}, A^{*}, k^{*})$ as follows

| H_{11} | = \frac{\partial^{2} EACWI (Q, A, k)}{\partial Q^{2}} \geq 0

| H_{22} | = \frac{\partial^{2} EACWI (Q, A, k)}{\partial Q^{2}} \times \frac{\partial^{2} EACWI (Q, A, k)}{\partial A^{2}} - {(\frac{\partial^{2} EACWI (Q, A, k)}{\partial k \partial Q})}^{2} \geq 0

And

| H_{33} | = | \begin{matrix} \frac{\partial^{2} E A C W I (Q, A, k)}{\partial Q^{2}} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial Q \partial A} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial Q \partial k} \\ \frac{\partial^{2} E A C W I (Q, A, k)}{\partial A \partial Q} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial A^{2}} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial A \partial k} \\ \frac{\partial^{2} E A C W I (Q, A, k)}{\partial k \partial Q} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial k \partial A} & \frac{\partial^{2} E A C W I (Q, A, k)}{\partial k^{2}} \end{matrix} | \geq 0

Also, for a given value of $L ϵ [L_{z}, L_{z - 1}]$ , the following expression must be positive for the storage space constraint

\frac{\partial^{2} g (Q, k)}{\partial k^{2}} \geq 0

Also, for a fixed $L ϵ [L_{z}, L_{z - 1}]$ , the budget constraint is a linear function. Consequently, the necessary Karush–Kuhn–Tucker (KKT) conditions for the constrained inventory problem are also sufficient if the optimal values ( $Q^{*}, A^{*}, k^{*}$ ) satisfy the above-mentioned expressions. In fact, $(Q^{*}, A^{*}, k^{*})$ is the global minimum.

Numerical example

In this section, we solve some examples using the procedure outlined in the previous section. The purpose is to illustrate the solution procedure, conduct a sensitivity analysis for important model parameters and highlight important features of the developed models. The inventory system parameters are as follows

μ_{D_{1}} = 5 units / week, μ_{D_{2}} = 8 units / week

σ_{D_{1}} = 4 units / week, σ_{D_{2}} = 6 units / week, ρ_{12} = 0.25

Hence, mean and standard deviation of demand are obtained as follows

μ_{D} = μ_{D_{1}} + μ_{D_{2}} = 13 units / week

σ_{D} = \sqrt{{σ_{D_{1}}}^{2} + {σ_{D_{2}}}^{2} + 2 σ_{D_{1}} σ_{D_{2}} ρ_{12}} = 8 units / week

And

D = 700 units / year, A = US$ 200, H = US$ 20, π = US$ 50, π_{0} = US$ 100, f = 1.6 M^{2}, C_{p} = US$ 60, γ = 0.95, φ = 0.95

F = 120 M^{2}, B = US$ 5000, ν = 1, ϑ = 0.7, δ = US$ 1.5, C_{r} = US$ 3

Defective rate $p$ has a beta-distribution function with parameters $α = 1$ and $β = 4$ ; that is, the pdf of $p$ is given by

g (p) = 4 {(1 - p)}^{3}, 0 < p < 1

Therefore, we have

E (p) = \frac{α}{α + β} = 0.2, E (p^{2}) = \frac{α (α + β)}{(α + β) (α + β + 1)} = 0.066 and Var (p) = 0.0266

Besides, for the setup cost reduction, we take $θ = 0.1, b = 5000$ . The lead time has three components with data shown in Table 1. The inflation rate has a normal distribution function. Tables 2 and 3 show the first step of the proposed solution procedure for the unconstrained model without and with reworking imperfect items for $E (i) = 0.00, 0.04, 0.08, and 0.12$ , respectively. A summary of optimal solutions is tabulated in Tables 4 and 5. Considering $ξ = US$ 10$ in the second step of the solution procedure, the effective solutions are tabulated in Tables 6 and 7 for unconstrained inventory inspection models. Also, for instance, optimal EACWI values obtained in Table 4 are tabulated in Table 8 in terms of different safety factor (k). Results show that EACWI is convex in terms of various safety factor amounts (see Figure 1). In addition, the associated results reveal that the larger the inflation rate, the larger the order quantity for inspection inventory models (see Figure 2). Besides, for instance, performance of multiobjective inventory inspection model with discarding defective units for variance of shortages is shown in Figure 3. Figure 3 shows that the larger the amount of safety factor, the smaller the amount of variance of shortages.

Table 1.

Lead time data.

Lead time component z	Normal duration b_z (days)	Minimum duration a_z (days)	Unit crashing cost c_z (US$/day)
1	20	6	0.4
2	20	6	1.2
3	16	9	5.0

Table 2.

Results of first step of solution procedure for unconstrained model without reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$L_{z}$	$C (L_{z})$	$(Q_{z}, A_{z}, r_{z}, k_{z})$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	8	0	(80.33, 44.00, 149.02, 1.99)	(1257.04, 1543.72, 141.04, 43,312.5)	46,254.32
	6	5.6	(85.51, 46.84, 116.40, 1.96)	(1285.45, 1452.74, 122.77, 43,312.5)	46,173.47
	4	22.4	(101.05, 55.38, 82.08, 1.88)	(1344.24, 1410.52, 104.64, 43,312.5)	46,171.91*
	3	57.4	(125.93, 69.04, 63.94, 1.80)	(1447.50, 1506.79, 90.15, 43,312.5)	46,356.94
0.04	8	0	(90.10, 48.59, 148.12, 1.95)	(1207.43, 1636.04, 145.90, 44,089.54)	47,078.92
	6	5.6	(95.17, 51.25, 115.62, 1.92)	(1235.43, 1544.74, 127.75, 44,084.52)	46,992.45
	4	22.4	(110.80, 59.57, 81.60, 1.85)	(1293.53, 1508.54, 106.92, 44,069.05)	46,978.05*
	3	57.4	(136.90, 73.53, 63.52, 1.77)	(1390.51, 1618.08, 92.80, 44,043.21)	47,144.60
0.08	8	0	(101.57, 53.90, 147.44, 1.92)	(1155.50, 1749.49, 145.47, 44,843.88)	47,894.36
	6	5.6	(106.60, 56.41, 115.03, 1.89)	(1182.46, 1657.96, 128.01, 44,833.92)	47,802.36
	4	22.4	(122.76, 64.74, 81.12, 1.82)	(1236.88, 1627.72, 108.11, 44,801.92)	47,774.65*
	3	57.4	(150.27, 79.07, 63.11, 1.74)	(1326.87, 1752.50, 94.60, 44,744.45)	47,921.43
0.12	8	0	(117.43, 61.31, 146.31, 1.87)	(1091.14, 1893.86, 150.72, 45,562.47)	48,698.21
	6	5.6	(121.63, 63.22, 114.25, 1.85)	(1120.06, 1800.85, 129.92, 45,549.99)	48,600.84
	4	22.4	(138.28, 71.52, 80.48, 1.78)	(1170.75, 1777.25, 110.85, 45,500.53)	48,559.38*
	3	57.4	(167.36, 86.24, 62.55, 1.70)	(1253.32, 1919.53, 97.94, 45,414.17)	48,684.97

the minimum value for lead times and respective crash costs have been obtained.

Table 3.

Results of first step of solution procedure for unconstrained model with reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$L_{z}$	$C (L_{z})$	$(Q_{z}, A_{z}, r_{z}, k_{z})$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	8	0	(64.01, 45.72, 148.80, 1.98)	(1237.85, 1536.70, 140.12, 43,470.00)	46,384.68
	6	5.6	(68.29, 48.78, 116.01, 1.94)	(1262.89, 1443.74, 125.52, 43,470.00)	46,302.16
	4	22.4	(80.04, 57.17, 81.92, 1.87)	(1322.05, 1399.28, 104.51, 43,470.00)	46,295.82*
	3	57.4	(99.87, 71.15, 63.66, 1.78)	(1420.04, 1490.05, 92.87, 43,470.00)	46,472.97
0.04	8	0	(71.84, 50.43, 147.89, 1.94)	(1188.84, 1629.01, 144.90, 44,250.17)	47,212.93
	6	5.6	(75.72, 53.14, 115.42, 1.91)	(1215.30, 1536.51, 127.10, 44,245.35)	47,124.27
	4	22.4	(87.83, 61.53, 81.44, 1.84)	(1271.36, 1497.02, 106.72, 44,230.31)	47,105.43*
	3	57.4	(108.18, 75.69, 63.38, 1.76)	(1364.95, 1601.60, 92.89, 44,205.04)	47,264.49
0.08	8	0	(81.51, 56.38, 146.99, 1.90)	(1133.09, 1742.87, 148.22, 45,006.30)	48,030.50
	6	5.6	(85.28, 58.80, 114.64, 1.87)	(1159.65, 1649.94, 130.71, 44,996.94)	47,937.25
	4	22.4	(97.40, 66.92, 80.96, 1.81)	(1214.75, 1616.03, 107.83, 44,966.85)	47,905.47*
	3	57.4	(118.85, 81.44, 62.97, 1.73)	(1301.55, 1735.43, 94.63, 44,913.57)	48,045.19
0.12	8	0	(93.77, 63.78, 146.08, 1.86)	(1071.43, 1887.30, 149.57, 45,728.79)	48,837.10
	6	5.6	(99.42, 65.96, 113.86, 1.83)	(1097.07, 1793.83, 132.56, 45,715.20)	48,738.68
	4	22.4	(109.84, 73.98, 80.32, 1.77)	(1148.60, 1765.59, 110.46, 45,668.91)	48,693.57*
	3	57.4	(132.51, 88.90, 62.41, 1.69)	(1228.21, 1902.01, 97.89, 45,584.45)	48,812.56

the minimum value for lead times and respective crash costs have been obtained.

Table 4.

Summary of results for unconstrained model without reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$(Q^{}, A^{}, L^{}, r^{}, k^{}, β^{})$	$E (\frac{1}{Q - y}), E (Q - y)$	$ζ (k), Var {(x - r)}^{+}$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	(101.05, 55.38, 4, 82.08, 1.88, 0.87)	(0.0129, 80.84)	(0.0080316, 2.05)	(1344.24, 1410.52, 104.64, 43,312.5)	46,171.91
0.04	(110.80, 59.57, 4,81.60, 1.85, 0.86)	(0.0118, 88.64)	(0.0087359, 2.23)	(1293.53, 1508.54, 106.92, 44,069.05)	46,978.05
0.08	(122.76, 64.74, 4,81.12, 1.82, 0.85)	(0.0106, 98.21)	(0.0094932, 2.43)	(1236.88, 1627.72, 108.11, 44801.92)	47,774.65
0.12	(138.28, 71.52, 4,80.48, 1.78, 0.84)	(0.0094, 110.63)	(0.0105946, 2.71)	(1170.75, 1777.25, 110.85, 45,500.53)	48,559.38

Table 5.

Summary of results for unconstrained model with reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$(Q^{}, A^{}, L^{}, r^{}, k^{}, β^{})$	$ζ (k)$	$V a r {(x - r)}^{+}$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	(80.04, 57.17, 4, 81.92, 1.87, 0.87)	0.0082602	2.11	(1322.05, 1399.28, 104.51, 43,470.00)	46,295.82
0.04	(87.83, 61.53, 4, 81.44, 1.84, 0.86)	0.00898164	2.29	(1271.36, 1497.02, 106.72, 44,230.31)	47,105.43
0.08	(97.40, 66.92, 4, 80.96, 1.81, 0.85)	0.0097585	2.49	(1214.75, 1616.03, 107.83, 44,966.85)	47,905.47
0.12	(109.84, 73.98, 4, 80.32, 1.77, 0.84)	0.0108870	2.78	(1148.60, 1765.59, 110.46, 45,668.91)	48,693.57

Table 6.

Results of second step of solution procedure for unconstrained model without reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$(Q^{p}, A^{p}, L^{p}, r^{p}, k^{p}, β^{p})$	$E (\frac{1}{Q - y}), E (Q - y)$	$ζ (k), Var {(x - r)}^{+}$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	(97.28, 53.30, 4,84.64, 2.04, 0.91)	(0.0134, 77.82)	(0.0050666, 1.29)	(1371.19, 1431.25, 66.55, 43,312.50)	46181.51
0.04	(106.48, 57.25, 4, 84.16, 2.01, 0.91)	(0.0122, 85.18)	(0.0055328, 1.41)	(1321.04, 1525.19, 68.16, 44,073.33)	46,987.73
0.08	(117.79, 62.13, 4, 83.68, 1.98, 0.90)	(0.0111, 94.24)	(0.0060373, 1.54)	(1264.72, 1639.31, 69.05, 44,811.76)	47,784.85
0.12	(132.40, 68.50, 4, 83.04, 1.94, 0.89)	(0.0098, 105.92)	(0.0067743, 1.73)	(1199.22, 1781.23, 71.00, 45,517.99)	48,569.45

Table 7.

Results of second step of solution procedure for unconstrained model with reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$(Q^{p}, A^{p}, L^{p}, r^{p}, k^{p}, β^{p})$	$ζ (k)$	$Var {(x - r)}^{+}$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	(77.01, 55.00, 4, 84.48, 2.03, 0.91)	0.0053179	1.33	(1349.01, 1419.93, 66.51, 43,470.00)	46,305.46
0.04	(84.36, 59.11, 4, 84.00, 2.00, 0.91)	0.0056966	1.45	(1298.90, 1513.58, 68.07, 44,234.62)	47,115.18
0.08	(93.42, 64.20, 4, 83.52, 1.97, 0.90)	0.0062145	1.59	(1242.58, 1627.53, 68.91, 44,976.73)	47,915.76
0.12	(105.13, 70.83, 4, 82.88, 1.93, 0.89)	0.0069708	1.78	(1177.05, 1769.47, 70.79, 45,686.46)	48,703.78

Table 8.

Optimal EACWI values in terms of different safety factor (k).

$k$	$ψ (k)$	${EACWI}_{L = 4}^{i = 0.00}$	$k$	${EACWI}_{L = 4}^{i = 0.04}$	$k$	${EACWI}_{L = 4}^{i = 0.08}$	$k$	${EACWI}_{L = 4}^{i = 0.12}$
0.00	0.398994	49,285.10	0.00	49,919.36	0.00	50,528.11	0.00	51,107.02
0.40	0.230439	48,161.78	0.40	48,845.35	0.40	49,508.56	0.40	50,146.02
0.80	0.120207	47,184.42	0.80	47,916.66	0.80	48,633.47	0.80	49,331.22
1.20	0.056102	46,512.26	1.20	47,284.43	1.20	48,044.73	1.20	48,791.74
1.40	0.036668	46,321.84	1.40	47,108.34	1.40	47,884.48	1.40	48,648.93
1.60	0.023242	46,215.47	1.60	47,012.56	1.60	47,800.18	1.60	48,576.33
1.70	0.018288	46,188.87	1.70	46,989.85	1.70	47,781.80	1.70	48,562.67
1.80	0.014276	46,175.10	1.80	46,979.35	1.78	47,775.39	1.74	48,560.22
1.85	0.012575	46,172.38	1.81	46,978.90	1.79	47,775.06	1.75	48,559.87
1.86	0.012257	46,172.13	1.82	46,978.54	1.80	47,774.83	1.76	48,559.61
1.87	0.011946	46,171.98	1.83	46,978.29	1.81	47,774.69	1.77	48,559.45
1.88	0.011642	46,171.91 *	1.85	46,978.05 *	1.82	47,774.65 *	1.78	48,559.38*
1.89	0.011345	46,171.93	1.86	46,978.07	1.83	47,774.70	1.79	48,559.41
1.90	0.011054	46,172.04	1.87	46,978.17	1.84	47,774.83	1.80	48,559.52
1.91	0.010770	46,172.23	1.88	46,978.36	1.85	47,775.05	1.81	48,559.73
1.92	0.010493	46,172.51	1.89	46,978.36	1.86	47,775.35	1.82	48,560.02
1.93	0.010222	46,172.87	1.90	46,978.99	1.90	47,777.89	1.83	48,560.39
2.00	0.008491	46,177.52	1.91	46,979.43	2.00	47,787.64	1.90	48,565.89
2.20	0.004887	46,206.40	2.20	47,018.98	2.20	47,823.69	2.20	48,617.78
2.40	0.002720	46,250.42	2.40	47,065.51	2.40	47,872.48	2.40	48,669.26
3.00	0.000382	46,422.77	3.00	47,242.92	3.00	48,055.12	3.00	48,857.12
3.50	0.000058	46,580.14	3.50	47,403.66	3.50	48,219.25	3.50	49,024.60
4.00	0.000007	46,739.73	4.00	47,566.47	4.00	48,385.44	4.00	49,194.07

minimum values in terms of safety factor.

Figure 1.

Convexity of model in terms of various safety factor (k).

Figure 2.

Optimal order quantity in terms of various lead time and inflation rates.

Figure 3.

Multiobjective model performance for variance of shortages.

The optimal values of the constrained inventory inspection models in the first step are tabulated in Tables 9 and 10 for $E (i) = 0.00, 0.04, 0.08, and 0.12$ . Storage space constraint is binding in this example. Tables 11 and 12 show the summary of optimal values, and the summary of the effective solutions for the second step are tabulated in the Tables 13 and 14. Similar to unconstrained models, the related results show that the larger the inflation rate, the larger the order quantity. In this example, the EAC of without reworking (discarding) inventory system is less than reworking inventory system for both constrained and unconstrained models.

Table 9.

Results of first step of solution procedure for the constrained model without reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$L_{z}$	$C (L_{z})$	$(Q_{z}, A_{z}, r_{z}, k_{z}, λ_{1_{z}}, {λ_{2}}_{z})$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	8	0	(48.43, 26.48, 150.15, 2.04, 8.67, 0)	(1510.80, 1311.09, 199.82, 43,312.5)	46,334.22
	6	5.6	(55.77, 30.51, 116.99, 1.99, 7.64, 0)	(1531.71, 1226.53, 171.55, 43,312.5)	46,242.31*
	4	22.4	(64.73, 35.43, 82.56, 1.91, 9.27, 0)	(1681.37, 1129.44, 149.21, 43,312.5)	46,272.53
	3	57.4	(70.97, 38.86, 63.94, 1.80, 15.86, 0)	(2057.63, 1067.12, 160.15, 43,312.5)	46,597.42
0.04	8	0	(48.72, 26.24, 149.93, 2.03, 9.91, 0)	(1515.36, 1335.04, 209.39, 44,130.51)	47,190.32
	6	5.6	(56.02, 30.14, 116.79, 1.98, 8.87, 0)	(1538.94, 1249.10, 179.97, 44,123.28)	47,091.30*
	4	22.4	(64.93, 34.90, 82.40, 1.90, 10.55, 0)	(1693.79, 1150.42, 156.65, 44,114.47)	47,115.34
	3	57.4	(71.14, 38.21, 63.80, 1.79, 17.07, 0)	(2078.41, 1087.08, 168.17, 44,108.31)	47,441.99
0.08	8	0	(49.00, 26.01, 149.70, 2.02, 11.15, 0)	(1519.83, 1358.91, 219.34, 44,947.96)	48,046.06
	6	5.6	(56.27, 29.79, 116.60, 1.97, 10.11, 0)	(1546.04, 1271.59, 188.72, 4493.58)	47,939.94*
	4	22.4	(65.13, 34.38, 82.24, 1.89, 11.83, 0)	(1706.02, 1171.35, 164.39, 44,916.04)	47,957.81
	3	57.4	(71.32, 37.59, 63.66, 1.78, 18.47, 0)	(2098.96, 1106.99, 176.51, 44,903.78)	48,286.26
0.12	8	0	(49.57, 25.93, 149.25, 2.00, 12.45, 0)	(1523.33, 1380.33, 235.74, 45,764.00)	48,901.42
	6	5.6	(56.51, 29.44, 116.40, 1.96, 11.36, 0)	(1553.01, 1294.01, 197.82, 45,743.39)	48,788.24*
	4	22.4	(65.53, 33.99, 81.92, 1.87, 13.12, 0)	(1715.53, 1190.55, 177.25, 45,716.61)	48,799.95
	3	57.4	(71.66, 37.09, 63.38, 1.76, 19.86, 0)	(2116.21, 1125.41, 190.36, 45,698.40)	49,130.38

the minimum value for lead times and respective crash costs have been obtained.

Table 10.

Results of first step of solution procedure for the constrained model with reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$L_{z}$	$C (L_{z})$	$(Q_{z}, A_{z}, r_{z}, k_{z}, λ_{1_{z}}, {λ_{2}}_{z})$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	8	0	(38.42, 27.44, 149.93, 2.03, 8.62, 0)	(1492.98, 1303.37, 199.06, 43,470.00)	46,465.43
	6	5.6	(44.20, 31.57, 116.79, 1.98, 7.56, 0)	(1511.70, 1218.41, 171.11, 43,470.00)	46,371.23*
	4	22.4	(51.24, 36.60, 82.40, 1.90, 9.09, 0)	(1655.13, 1120.82, 148.94, 43,470.00)	46,394.90
	3	57.4	(56.15, 40.10, 63.80, 1.79, 15.25, 0)	(2018.89, 1058.12, 159.90, 43,470.00)	46,706.92
0.04	8	0	(38.65, 27.19, 149.70, 2.02, 9.84, 0)	(1497.59, 1327.14, 208.62, 44,291.39)	47,324.75
	6	5.6	(44.39, 31.18, 116.60, 1.97, 8.78, 0)	(1518.93, 1240.79, 179.52, 44,284.26)	47,223.51*
	4	22.4	(51.39, 36.04, 82.24, 1.89, 10.34, 0)	(1667.44, 1141.61, 156.38, 44,275.56)	47,241.00
	3	57.4	(56.29, 39.44, 63.66, 1.78, 16.62, 0)	(2039.35, 1077.89, 167.91, 44,269.49)	47,554.65
0.08	8	0	(38.87, 26.95, 149.48, 2.01, 11.08, 0)	(1502.11, 1350.81, 218.55, 45,112.23)	48,183.72
	6	5.6	(44.59, 30.81, 116.40, 1.96, 10.01, 0)	(1526.04, 1263.09, 188.26, 45,098.04)	48,075.44*
	4	22.4	(51.55, 35.51, 82.08, 1.88, 11.60, 0)	(1679.57, 1162.34, 164.12, 45,080.73)	48,086.77
	3	57.4	(56.42, 38.80, 63.52, 1.77, 17.99, 0)	(2059.59, 1097.60, 176.25, 45,068.63)	48,402.08
0.12	8	0	(39.32, 26.86, 149.02, 1.99, 12.37, 0)	(1503.70, 1372.01, 234.93, 45,931.67)	49,042.31
	6	5.6	(44.78, 30.45, 116.21, 1.95, 11.24, 0)	(1533.02, 1285.31, 197.97, 45,911.33)	48,927.03*
	4	22.4	(51.87, 35.10, 81.76, 1.86, 12.88, 0)	(1689.03, 1181.33, 176.97, 45,884.91)	48,932.24
	3	57.4	(56.70, 38.27, 63.24, 1.75, 19.37, 0)	(2076.59, 1115.81, 190.08, 45,866.93)	49,249.40

the minimum value for lead times and respective crash costs have been obtained.

Table 11.

Summary of results for the constrained model without reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$(Q^{}, A^{}, L^{}, r^{}, k^{}, β^{})$	$E (\frac{1}{Q - y}), E (Q - y)$	$ζ (k), Var {(x - r)}^{+}$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	(55.77, 30.51, 6,116.99, 1.99, 0.88)	(0.0234, 44.61)	(0.0058647, 2.25)	(1531.71, 1226.53, 171.55, 43,312.5)	46,242.31
0.04	(56.02, 30.14, 6, 116.79, 1.98, 0.88)	(0.0233, 44.81)	(0.0060373, 2.31)	(1538.94, 1249.10, 179.97, 44,123.28)	47,091.30
0.08	(56.27, 29.79, 6, 116.60, 1.97, 0.88)	(0.0232, 45.02)	(0.0062145, 2.38)	(1546.04, 1271.59, 188.72, 4493.58)	47939.94
0.12	(56.51, 29.44, 6, 116.40, 1.96, 0.87)	(0.0231, 45.21)	(0.0063963, 2.45)	(1553.01, 1294.01, 197.82, 45,743.39)	48,788.24

Table 12.

Summary of results for the constrained model with reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$(Q^{}, A^{}, L^{}, r^{}, k^{}, β^{})$	$ζ (k)$	$Var {(x - r)}^{+}$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	(44.20, 31.57, 6, 116.79, 1.98, 0.88)	0.0060373	2.31	(1511.70, 1218.41, 171.11, 43,470.00)	46,371.23
0.04	(44.39, 31.18, 6, 116.60, 1.97, 0.88)	0.0062145	2.38	(1518.93, 1240.79, 179.52, 44,284.26)	47,223.51
0.08	(44.59, 30.81, 6, 116.40, 1.96, 0.88)	0.0063963	2.45	(1526.04, 1263.09, 188.26, 45,098.04)	48,075.44
0.12	(44.78, 30.45, 6, 116.21, 1.95, 0.87)	0.0065829	2.52	(1533.02, 1285.31, 197.97, 45,911.33)	48,927.03

Table 13.

Results of second step of solution procedure for the constrained model without reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$(Q^{p}, A^{p}, L^{p}, r^{p}, k^{p}, β^{p})$	$E (\frac{1}{Q - y}), E (Q - y)$	$ζ (k), Var {(x - r)}^{+}$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	(52.56, 28.75, 6, 119.54, 2.12, 0.91)	(0.0248, 42.05)	(0.0039918, 1.53)	(1567.13, 1251.58, 121.05, 43,312.50)	46,252.26
0.04	(53.06, 28.54, 6, 119.15, 2.10, 0.91)	(0.0246, 42.44)	(0.0042391, 1.62)	(1571.41, 1272.68, 130.40, 44,126.22)	47,100.72
0.08	(53.30, 28.21, 6, 118.95, 2.09, 0.91)	(0.0245, 42.64)	(0.0043679, 1.67)	(1578.37, 1295.63, 136.72, 44,939.45)	47,950.18
0.12	(53.80, 28.02, 6, 118.56, 2.07, 0.90)	(0.0243, 43.04)	(0.0046361, 1.78)	(1582.45, 1316.46, 147.18, 45,751.46)	48,797.55

Table 14.

Results of second step of solution procedure for the constrained model with reworking imperfect items ( $L_{z}$ in weeks).

$E (i)$	$(Q^{p}, A^{p}, L^{p}, r^{p}, k^{p}, β^{p})$	$ζ (k)$	$Var {(x - r)}^{+}$	$(EOCWI, EHCWI, ESCWI, EPCWI)$	$EACWI$
0.0	(41.66, 29.76, 119.34, 2.11, 0.91)	0.0041137	1.57	(1546.58, 1243.84, 120.71, 43,470.00)	46,381.24
0.04	(42.05, 29.51, 118.95, 2.09, 0.91)	0.0043679	1.67	(1551.57, 1264.72, 130.03, 44,287.16)	47,233.49
0.08	(42.44, 29.27, 118.56, 2.07, 0.90)	0.0046361	1.78	(1556.44, 1285.45, 140.04, 45,103.36)	48,085.30
0.12	(42.64, 28.90, 118.36, 2.06, 0.90)	0.0047755	1.83	(1563.92, 1308.09, 146.78, 45,919.31)	48,938.12

In the discussed inventory system’s data, it is considered that the defective rate follows a beta-distribution function with a small standard deviation. Moreover, in order to show that this assumption is met for the data used in the respective system, a column is added in Tables 4, 6, 11, and 13 for unconstrained and constrained inventory models with discarding defective units, and the values of $E (1 / (Q - y))$ and $E (Q - y)$ are calculated. The results show that the quadratic approximation is satisfied for the used data. For instance, in Table 4, for $E (i) = 0.00$ , the values obtained are as follows

E (\frac{1}{Q - y}) = 0.0129, E (Q - y) = Q - E (y) = 80.84 and E (y) = QE (p)

We know that

\frac{Var (y)}{{(E (Q - y))}^{3}} ≅ E (\frac{1}{Q - y}) - \frac{1}{E (Q - y)} = 0.0129 - 0.0124 = 0.0005

The above calculations show that the effect of variance of defective units is low and the obtained expected value is very close to the situation of fixed defective rate. Also, we know that the expected number of cycle equals $DE (1 / (Q - y))$ . For the above example, the expected number of cycle is 9.03. If we assume that the number of defective item is a fixed constant and equal to $y = Qp$ , then the number of cycle is equal to $D / (Q - y) = 8.68$ . In the case of ordering cost calculation, considering $C (L) = 22.4$ and $A = 55.38$ , the expected ordering cost per year (without considering capital investment) for random defective items is equal to $(A + C (L)) DE (1 / (Q - y)) = 77.78 \times 9.03 = 702.35$ . Besides, the ordering cost for fixed defective items equals $(A + C (L)) D / (Q - y) = 675.13$ . Hence, comparing these two situations, the expected additional cost for considering defective units as a random variable is $702.35 - 675.13 = US$ 27.22$ . Additionally, comparing Tables 4 and 5 for inventory inspection models without and with reworking items, respectively, similar results can be obtained (e.g. $(1 / Q)_{i = 0} = 0.0124, E (1 / (Q - y))_{i = 0} = 0.0129$ ).

The change in the values of system parameters can take place due to uncertainties and dynamic market conditions in any DM situation. In order to examine the implications of these changes in the value of parameters, the sensitivity analysis will be of great help in a DM process. By using the previous numerical example, the sensitivity analysis of various parameters has been carried out in this article. Results of sensitivity analysis are shown in Table 15. Finally, the main conclusions, one can draw from the sensitivity analysis, are as follows:

Table 15.

Effect of changes in various parameters.

	Without reworking model		With reworking model
	$(Q^{}, A^{}, L^{}, r^{}, k^{}, β^{})$	$EACWI$	$(Q^{}, A^{}, L^{}, r^{}, k^{}, β^{})$	$EACWI$
$ϑ$
0.0	(61.76, 32.17, 6, 112.29, 1.75, 1.00)	48,728.57	(48.95, 32.27, 6, 112.09, 1.74, 1.00)	48,865.61
1.0	(55.52, 28.92, 6, 117.19, 2.00, 0.84)	48,802.49	(43.98, 29.91, 6, 116.99, 1.99, 0.84)	48,941.55
5	(51.03, 26.58, 6, 120.71, 2.18, 0.60)	48,886.51	(40.43, 27.50, 6, 120.52, 2.17, 0.60)	49,026.33
100	(49.24, 25.64, 6, 122.09, 2.25, 0.00)	49,000.71	(39.26, 26.95, 6, 121.69, 2.23, 0.00)	49,137.64
$\infty$	(49.24, 25.64, 6, 122.09, 2.25, 0.00)	49,000.71	(39.26, 26.95, 6, 121.69, 2.23, 0.00)	49,137.64
F
100	(39.35, 20.49, 6, 116.79, 1.98, 0.88)	48,993.76	(31.23, 21.26, 6, 116.60, 1.97, 0.88)	49,129.58
150	(82.75, 43.08, 6, 115.42, 1.91, 0.86)	48,651.92	(65.49, 44.45, 6, 115.23, 1.90, 0.86)	48,791.30
200	(138.33, 71.39, 4, 80.48, 1.78, 0.84)	48,560.76	(109.88, 73.85, 4, 80.32, 1.77, 0.84)	48,694.94
B
2000	(44.44, 23.14, 6, 120.52, 2.17, 0.93)	48,887.02	(35.08, 23.80, 6, 120.32, 2.16, 0.93)	49,028.96
2200	(48.88, 25.46, 6, 119.93, 2.14, 0.92)	48,840.23	(38.59, 26.17, 6, 119.73, 2.13, 0.92)	48,982.39
2500	(55.53, 28.92, 6, 117.19, 2.00, 0.89)	48,789.63	(43.85, 29.73, 6, 116.99, 1.99, 0.89)	48,931.83

Keeping all the parameters fixed, if the value of $ϑ$ is increasing, the backorder rate ( $β$ ) is decreasing, then the optimal reorder point ( $r^{*}$ ) and the optimal EAC ( $EACW I^{*}$ ) are increasing simultaneously, yet the optimal order quantity ( $Q^{*}$ ) is decreasing. A simple economic interpretation is as follows. A smaller value of backorder rate indicates a larger shortage cost. Therefore, the optimal reorder point ( $r^{*}$ ) and the optimal EAC ( $EACW I^{*}$ ) are getting larger because a larger backorder rate implies a larger amount of lost sales.

With an augment in the maximum available space (F) and keeping the remaining parameters unchanged, the optimal order quantity ( $Q^{*}$ ) and optimal reorder point ( $A^{*}$ ) increase but optimal EAC and reorder point ( $r^{*}$ ) decrease. A simple economic interpretation is as follows. If the maximum available space (F) increases, the maximum permissible inventory level increases, then the order quantity ( $Q^{*}$ ) should be increased to diminish EAC ( $EACW I^{*}$ ).

With an increase in maximum inventory investment (B) and keeping the remaining parameters fixed, the optimal order quantity ( $Q^{*}$ ) and optimal reorder point ( $A^{*}$ ) increase; however, the optimal reorder point ( $r^{*}$ ) and EAC ( $EACW I^{*}$ ) decrease. From an economic viewpoint, it implies that when maximum permissible inventory investment arises, then the order quantity ( $Q^{*}$ ) should be increased to diminish EAC ( $EACW I^{*}$ ).

Conclusion

In this article, we explored multiobjective multiconstraint stochastic inventory inspection models with and without reworking of imperfect items discovered during inspection. The imperfect items in a lot are assumed to be random variable. The objectives are to minimize EAC and variance of shortages. Lead time and ordering cost are assumed to be controllable. Backorder rate is dependent on the length of lead time through the amount of shortages. Hence, with mixture of backorder and lost sales, this study assumes the order quantity, reorder point, lead time, ordering cost, and backorder rate as decision variables. A random storage space constraint is considered since the inventory level is random when an order arrives. So, in this case, a chance-constrained programming technique is used to make it crisp. A budget constraint is also added to the model. The piece-wise linear crashing cost function that is widely used in project management is considered in this article. This article assumes that the lead time demand follows a normal distribution. A numerical example is also given to illustrate model and its solution procedure. The proposed model can be extended in several directions. For instance, we may consider sampling inspection in the models or optimizing model for periodic review policy instead of continuous review policy.

Footnotes

Appendix 1

Appendix 2

It is considered that defective items in an order lot to be a random variable (y), which has a beta-binomial distribution with parameters ( $Q, α, β$ ), as shown below

(45)

P (y | p) = (\begin{matrix} Q \\ y \end{matrix}) p^{y} {(1 - p)}^{Q - y}, y = 0, 1, 2, \dots, Q

(46)

f (p) = \frac{Γ (α + β)}{Γ (α) Γ (β)} p^{(α - 1)} {(1 - p)}^{β - 1}, 0 \leq p \leq 1

(47)

\begin{matrix} f (y) = \int_{0}^{1} f (p) P (y | p) dp = (\begin{matrix} Q \\ y \end{matrix}) \\ \frac{Γ (α + β)}{Γ (α) Γ (β)} \int_{0}^{1} p^{y + α - 1} {(1 - p)}^{Q - y + β - 1} dp \\ \to f (y) = (\begin{matrix} Q \\ y \end{matrix}) \frac{Γ (α + β) Γ (α + y) Γ (Q + β - y)}{Γ (α) Γ (β) Γ (Q + α + β)}, y = 0, 1, 2, \dots, Q \end{matrix}

In this case, we have

(48)

E (y | p) = Qp

(49)

Var (y | p) = Qp (1 - p)

Hence, unconditioning on p, the expected value of $Y$ is computed by

(50)

E (y) = E (E (y | p)) = QE (p)

The variance of $Y$ is calculated by

(51)

Var (y) = E (Var (y | p)) + Var (E (y | p))

(52)

E (Var (y | p)) = E (Q (p (1 - p))) = QE (p (1 - p))

(53)

Var (E (y | p)) = E (Q^{2} p^{2}) - {(E (p))}^{2} = Q^{2} E (p^{2}) - Q^{2} E^{2} (p) = Q^{2} Var (p)

(54)

\to Var (y) = QE (p (1 - p)) + Q^{2} Var (p)

Appendix 3

Variance of shortages is calculated as follows

(55)

Var {(x - r)}^{+} = E [{(Max (x - r, 0))}^{2}] - {[E (Max (x - r, 0))]}^{2}

E [{(Max (x - r, 0))}^{2}] = \int_{r}^{\infty} {(x - r)}^{2} f (x) dx, k = \frac{r - E (x)}{σ_{x}}, z = \frac{x - E (x)}{σ_{x}}, P (x \leq r) = p (z \leq k)

(56)

\to \int_{r}^{\infty} {(x - r)}^{2} f (x) dx = \int_{k}^{\infty} {(z σ_{x} - k σ_{x})}^{2} f (z) dz = {σ_{x}}^{2} \int_{k}^{\infty} {(z - k)}^{2} f (z) dz

And

(57)

{[E (Max (x - r, 0))]}^{2} = {σ_{x}}^{2} {[\int_{k}^{\infty} zf (z) dz - \bar{k Φ (k)}]}^{2}

Hence, variance of shortages is as follows

(58)

Var {(x - r)}^{+} = {σ_{x}}^{2} {\int_{k}^{\infty} {(z - k)}^{2} f (z) dz - {[\int_{k}^{\infty} zf (z) dz - \bar{k Φ (k)}]}^{2}}

(59)

ζ (k) = {\int_{k}^{\infty} {(z - k)}^{2} f (z) dz - {[\int_{k}^{\infty} zf (z) dz - \bar{k Φ (k)}]}^{2}}

(60)

\to Var {(x - r)}^{+} = {σ_{D}}^{2} L ζ (k) \geq 0

Appendix 4

The total ordering cost per year paid at the time of order placing is computed as follows (( $n = 1 / T = D / (Q - y)$ ), $(C (L) = c_{z} (L_{z - 1} - L) + \sum_{j = 1}^{z - 1} c_{j} (b_{j} - a_{j}))$ )

(61)

\begin{matrix} [(A + C (L)) (1 + T_{r} i) + (A + C (L)) (1 + T_{r} i + \frac{Q - y}{D} i) + (A + C (L)) (1 + T_{r} i + 2 \times \frac{Q - y}{D} i) + \dots + (A + C (L)) (1 + T_{r} i + (n - 1) \times \frac{Q - y}{D} i)] \end{matrix}

So, the total ordering cost paid at the time of order placing can be expressed by

(62)

[(A + C (L)) \sum_{j = 0}^{n - 1} (1 + T_{r} i + \frac{Q - y}{D} \times i \times j)]

Hence, equation (62) can be written as

(63)

[(A + C (L)) (n + n T_{r} i + \frac{Q - y}{D} \times i \times \frac{n (n - 1)}{2})]

By considering $(I, X, Y)$ as independent, the expected value of equation (63) can be expressed by

(64)

\begin{array}{l} E O C W I = (A + C (L)) {E [\frac{D}{Q - y} + (\frac{D}{Q - y} \times \frac{i}{2}) - \frac{i}{2}] + E [(1 - \frac{x - (1 - β) {(x - r)}^{+}}{Q - y}) i]} \\ \to E O C W I = (A + C (L)) {D E (\frac{1}{Q - y}) (1 + \frac{E (i)}{2}) - \frac{E (i)}{2} \\ + (1 - [(E (x) - (1 - β) E {(x - r)}^{+}) E (\frac{1}{Q - y})]) E (i)} \end{array}

Therefore, considering $E (x - r)^{+} = σ_{D} \sqrt{L} ψ (k)$ and $β = ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}$ , the expected ordering cost with inflation (EOCWI) is transformed as

(65)

\begin{array}{l} E O C W I = (A + C (L)) [D E (\frac{1}{Q - y}) (1 + \frac{E (i)}{2}) - \frac{E (i)}{2}] \\ + (A + C (L)) (1 - [(μ_{D} L - (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k)) E (\frac{1}{Q - y})]) E (i) \end{array}

Appendix 5

Total holding cost per year is a random variable and can be computed as follows $(n = 1 / T = D / (Q - y))$

(66)

\begin{matrix} [\int_{0}^{nT} h (\frac{Q - y}{2} + r - x + (1 - β) {(x - r)}^{+}) (1 + it) dt] \\ = h (1 + \frac{i}{2}) (\frac{Q - y}{2} + r - x + (1 - β) {(x - r)}^{+}) \end{matrix}

Hence, considering $(I, X, Y)$ as independent, the expected value of the holding cost per year with inflation (EHCWI) is computed by

(67)

EHCWI = h (1 + \frac{E (i)}{2}) (\frac{E (Q - y)}{2} + r - E (x) + (1 - β) E {(x - r)}^{+})

Therefore, considering $r = μ_{D} L + k σ_{D} \sqrt{L}$ , $E (x - r)^{+} = σ_{D} \sqrt{L} ψ (k)$ , and $β = ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}$ , EHCWI is transformed as

(68)

EHCWI = h (1 + \frac{E (i)}{2}) (\frac{E (Q - y)}{2} + k σ_{D} \sqrt{L} + (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}) σ_{D} \sqrt{L} ψ (k))

Appendix 6

The average shortage cost per year per unit with inflation is calculated by ( $n = 1 / T = D / (Q - y)$ )

(69)

\begin{array}{l} \frac{((π + π_{°} (1 - β)) (1 + \frac{1}{n} i) + (π + π_{°} (1 - β)) (1 + \frac{2}{n} i) + \dots + (π + π_{°} (1 - β)) (1 + \frac{n}{n} i))}{n} \\ = \frac{(π + π_{°} (1 - β))}{n} \sum_{j = 1}^{n} (1 + \frac{j}{n} i) = (π + π_{°} (1 - β)) (1 + \frac{i}{2} (1 + \frac{1}{n})) \end{array}

By multiplying the average stock out cost in the number of cycle (n), the total shortage cost per year is obtained by ( $n = 1 / T = D / (Q - y)$ )

(70)

(π + π_{\circ} (1 - β)) {(x - r)}^{+} {[(1 + \frac{i}{2} (1 + \frac{Q - y}{D}))] \frac{D}{Q - y}}

Hence, by considering $(I, X, Y)$ as independent, the expected shortage cost per year with inflation (ESCWI) is obtained by

(71)

\begin{matrix} ESCWI = (π + π_{\circ} (1 - β)) E {(x - r)}^{+} \\ E [\frac{D}{Q - y} + (\frac{D}{Q - y} \times \frac{i}{2}) + \frac{i}{2}] \\ = (π + π_{\circ} (1 - β)) [DE (\frac{1}{Q - y}) (1 + \frac{E (i)}{2}) + \frac{E (i)}{2}] E {(x - r)}^{+} \end{matrix}

Therefore, considering $E (x - r)^{+} = σ_{D} \sqrt{L} ψ (k)$ and $β = ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)}$ , ESCWI is transformed as

(72)

ESCWI = (π + π_{\circ} (1 - ν e^{- ϑ σ_{D} \sqrt{L} ψ (k)})) [DE (\frac{1}{Q - y}) (1 + \frac{E (i)}{2}) + \frac{E (i)}{2}] σ_{D} \sqrt{L} ψ (k)

Appendix 7

The average purchasing cost per unit is obtained as follows $(n = 1 / T = D / (Q - y))$

(73)

\begin{matrix} \frac{(C_{p} + C_{p} (1 + \frac{1}{n} i) + C_{p} (1 + \frac{2}{n} i) + \dots + C_{p} (1 + \frac{n - 1}{n} i))}{n} \\ = \frac{C_{p} \sum_{j = 0}^{n - 1} (1 + \frac{j}{n} i)}{n} = C_{p} (1 + \frac{i}{2} (1 - \frac{1}{n})) = C_{p} (1 + \frac{i}{2} (1 - \frac{Q - y}{D})) \end{matrix}

Hence, considering (I, X, Y) as independent, the expected purchasing cost is obtained by

(74)

D C_{p} (1 + \frac{E (i)}{2} (1 - \frac{E (Q - y)}{D}))

Also, the expected inspecting cost is as follows

(75)

\frac{D}{1 - E (p)} δ (1 + \frac{E (i)}{2} (1 - \frac{E (Q - y)}{D}))

Therefore, the expected total purchasing cost with inflation (EPCWI) is

(76)

EPCWI = [D C_{p} + \frac{D}{1 - E (p)} δ] (1 + \frac{E (i)}{2} (1 - \frac{E (Q - y)}{D}))

Acknowledgements

The authors gratefully acknowledge the constructive comments made by the anonymous referees on an earlier version of this article.

Conflict of interest

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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