A collaborative vendor-buyer production-inventory systems with imperfect quality items, inspection errors, and stochastic demand under budget capacity constraint: a Karush-Kuhn-Tucker conditions approach

In this paper, we develop an integrated inventory model considering the imperfect quality items, inspection error, controllable lead time, and budget capacity constraint. The imperfect items were uniformly distributed and detected on the screening process. However there are two types of possibilities. The first is type I of inspection error (when a non-defective item classified as defective) and the second is type II of inspection error (when a defective item classified as non-defective). The demand during the lead time is unknown, and it follows the normal distribution. The lead time can be controlled by adding the crashing cost. Furthermore, the existence of the budget capacity constraint is caused by the limited purchasing cost. The purposes of this research are: to modify the integrated vendor and buyer inventory model, to establish the optimal solution using Kuhn-Tucker’s conditions, and to apply the models. Based on the result of application and the sensitivity analysis, it can be obtained minimum integrated inventory total cost rather than separated inventory.


Introduction
Nowadays, integrated inventory management has enjoyed a great deal of attention. The integration model exists because the inventory replenishment decisions are treated separately from the viewpoints of buyer and vendor based on economic order quantity (EOQ) and economic production quantity (EPQ). The results shows that the optimal EPQ solution for the vendor is not acceptable to the buyer, and vice versa. To identify the efficient solutions, both buyer and vendor should coordinate and collaborate to achieve good inventory management. The supply chain collaborative advantages from a firm's perspective have been showed by Chao and Zhang [5], such as process efficiency, offering flexibility, business synergy, quality and innovation. Goyal [7] was a pioneer in the study of the integrated inventory model. He developed a joint economic lot-sized model for single buyer and single vendor with infinite production rate. Banerjee [3] developed the model by incorporating a finite production rate and following a lot-for-lot policy for the vendor. After that, more researchers considered different cases in integrated inventory model about permissible delay in payments [6], inflation [4], service level constraint [18], price-dependent demand [22], and fuzzy random framework [23]. However, a common unrealistic assumption of the above integrated inventory models is that all items received by a buyer are of good quality. In a real system, due to the imperfect production process of the vendor, damage in transit, or other unforeseeable circumstance, goods that were received by the buyer often contain some defective items. Further, Goyal et al. [8] and Huang [12] developed integrated vendor-buyer inventory models for items with imperfect quality. They assumed that items of poor quality detected in the screening process of a lot are sold at a discounted price. Several researchers, including Hsu and Hsu [10], Lin [20], Bag and Chakraborty [2], Kurdhi et al. [16], Jindal and Solanki [13] proposed integrated vendor-buyer production-inventory model under imperfect quality items in various conditions. However, the above mentioned papers assumed that there is no error in the inspection process. Khan [14] stated that there are still wastes generated by errors in screening, although there have been advancements in technology. This means that the screening errors may occur with imperfect quality in practice. Moreover, Khan [14] asserted that the inspection accuracy is influenced by some factors. More specifically, these factors can be categorized into three groups: inspector related factors, task related factors and environmental factors. Lin [19] proposed an integrated vendor-buyer model for items with imperfect quality and inspection errors. He suggested two types errors usually could be found in inspection process. These are Type I error, in which the inspector may incorrectly classify a non-defective item as defective; and Type II error, in which a defective item is classified as non-defective. The Lin's [19] model assumed that both the Type I and Type II inspection errors are known constants. Hsu and Hsu [11] then developed a production-inventory model with defective items and inspection errors, where the both type of errors are viewed as random variable. There are more papers related to inspection errors on integrated model such as Widianto et al. [25], Khan et al. [15], Priyan and Uthayakumar [21].
In the integrated inventory management system with both imperfect quality items and inspection errors, the integrated inventory model under deterministic demand to be widely discussed. However, it is noticed that the inventory literature on the integrated inventory model under stochastic demand discussing screening errors is quite sparse. Nearly all integrated models that consider the inspection errors assume no uncertainty in demand and shortages are not allowed. On the other hand, Al-Salamah [1] proposed two EOQ models with stochastic demand, imperfect quality, and inspections errors. In the models, shortages are backordered and demand during lead time has a certain probability density function. However, the Al-Salamah's [1] models focused on determining optimal policy for the buyer only (one-echelon model). These models neglect the opportunity that buyer and vendor can cooperate and negotiate with each other to obtain a better integrated policy. We also notice that in the inventory models under inspection errors, shortages, reorder point, lead time reduction, and budget constraint were not considered. Separately, Priyan and Uthayakumar [5] considered budget capacity constraint on an integrated inventory model. On the model, the buyer has limited capacity to purchase products. On the other hand, there is an upper bound on the purchase of products. According to Kurdhi et al. [16,17], in many practical situations, lead time can be reduced, by an additional crashing cost, customer service level improved, inventory in safety stocks reduced, and the competitive edge in business increased; in other words, it is controllable.
Summarizing the above description, we seek to investigate a two-echelon production-inventory model by considering shortage backlogging, imperfect quality items, inspection errors, lead time, budget capacity constraint, and stochastic demand. In this study, the percentage of defective item and lead time demand are assumed to follow a uniform distribution and a normal distribution, respectively. It is also assumed that the partial backorder situation is considered. This means that the shortages are partial backordered and partial lost sale with a certain backorder rate. The aim is to minimize the integrated inventory system total cost by optimizing the order quantity, reorder point, lead time, and the number of deliveries per batch production run. This paper continuous with notations and assumptions in Section 2. Section 3 and 4 present model formulation and solution methodology, respectively. A numerical example and sensitivity analysis are discussed in Section 5. Finally, conclusions are given in Section 6.

Notations and Assumptions
To develop the proposed model, the following notations and assumptions are introduced.

Notations
ܳ : size of production batch of items at the vendor ܳ : size of the deliveries from the vendor to the buyer (decision variable) ‫ݎ‬ : reorder point ݇ : safety factor (decision variable) ݊ : number of deliveries per batch production run (decision variable) ‫ܦ‬ : annual demand of the buyer ܲ : annual production rate at the vendor ‫ݕ‬ : inspection rate ܵ ௩ : setup cost per production run for the vendor ܵ : ordering cost per order for the buyer ߛ : probability that an item produced is defective ݂(ߛ) : probability density function of ߛ ݁ ଵ : probability of a Type I error (classifying a non-defective item as defective) ݁ ଶ : probability of a Type II error (classifying a defective item as non-defective) ܿ ௦ : buyer's inspection cost per unit ܿ ௪ : vendor's unit cost for producing a defective item (warranty) ܿ : buyer's cost of a post-sale defective item ܿ ௩ : vendor's cost of a post-sale defective item ܿ : cost of accepting a defective item (ܿ = ܿ + ܿ ௩ ) ܿ : cost of rejecting a non-defective item ℎ ௩ : : lead time demand with finite mean ‫ܮܦ‬ and standart deviation ‫ܮ√ߪ‬ ≥ 0 ‫)•(ܧ‬ : mathematical expectation ‫ݔ‬ ା : maximum value of ‫ݔ‬ and 0, i.e., ‫ݔ‬ ା = maxሼ‫,ݔ‬ 0ሽ * : superscript representing optimal value.

Assumptions
1. There are a single vendor and a single buyer with a single product.
2. The buyer's inventory is continuously reviewed and replenishments are made whenever the inventory level falls to the reorder point ‫.ݎ‬ The reorder point ‫ݎ‬ = expected demand during lead time + safety stock (ܵܵ), and ܵܵ = ݇ × (standard deviation of lead time demand), i.e., ‫ݎ‬ = ‫ܮܦ‬ + ‫,ܮ√ߪ݇‬ where ݇ is the safety factor. 3. The production processes are imperfect and may produces defective items. The defective percentage (ߛ) is variable random uniformly distributed with probability density function ݂(ߛ). The inspection process is also imperfect. There are two types of possibilities. The first is type I of inspection error (when a non-defective item classified as defective) and the second is type II of inspection error (when a defective item classified as non-defective). 4. The lead time ‫ܮ‬ has ݉ mutually independent components. The ݅th component has a normal duration ܾ , a minimum duration ܽ , and a crashing cost per unit time ܿ . Furthermore, for convenience, we rearrange ܿ such that ܿ ଵ ≤ ܿ ଶ ≤ ⋯ ≤ ܿ . Then, it is clear that the reduction of lead time should first occur on component 1 (because it has the minimum unit crashing cost), and then component 2, etc. If we let ‫ܮ‬ = ∑ ܾ ୀଵ and ‫ܮ‬ be the length of components 1, 2, … , ݅ crashed to their minimum duration, then ‫ܮ‬ can be expressed as , ݅ = 1,2, … , ݊; and the lead time crashing cost ‫)ܮ(ܥ‬ per cycle for a given ‫ܮ‬ ∈ ‫ܮ[‬ , ‫ܮ‬ ିଵ ], is given by Shortage is allowed for the buyer and partially backordered. 6. The purchasing cost for all products is limited, mathematically, ‫ܳ‬ ≤ ‫.ܤ‬

Model Formulation
In this paper, an integrated single-vendor and single-buyer inventory model involving defective items, inspection errors, stochastic demand, controllable lead time, and partial backorder under budget capacity constraint. The coordination mechanism is such that the vendor receives the buyer's demand and produces the single product at a finite production rate P. The vendor replenishes the order in a number of equal-sized shipments. It is assumed that the vendor's production processes are imperfect and devective items may be produced. Thus, once the buyer receives the lot-size ܳ, a 100% screening process is conducted. The length of the screening process is ‫ݐ‬ ଵ = ‫ݕ/ܳ‬ year. The screening process is also imperfect in that an inspector may incorrectly classify a non-defective product as defective (Type I error), or a defective product as non-defective (Type II error). It is assumed that the customers who buy the defective items will detect the quality problem and return them to the buyer and receive a good item from the buyer. Later, the buyer returns all items classified as defective and those returned from the customers to the vendor, and receives a full price refund from the vendor. Both the vendor and the buyer incur a post-sale failure cost (e.g., loss of good will) for the items returned from the market. The goal of this study is to simultaneously optimize the order quantity, safety factor, lead time and the number of lots delivered from vendor to buyer with the objective of minimizing the total supply chain integrated cost and the constraint is satisfied.

The buyer's cost formulation
The buyer's total inventory cost consists of ordering cost, transportation cost, screening cost, post-sale failure cost for the buyer, crashing lead time cost, holding cost, and shortage cost. The behaviour of the inventory level over time for the buyer is described in Fig.1. The buyer's inventory system with random lead time demands, the inventory level is continuous reviewed and new order is triggered until the inventory level declines to ‫ݎ‬ units. If the lead time demand exceeds the reorder point, the inventory system will experience shortages. The shortages are partial lost sale and partial backordered with backorder rate ߚ. Let , and ߶ and Φ are standard normal distribution and cumulative distribution function, respectively. Hence, the shortage cost for the buyer per production cycle will be ݊[ߙ + ߙ (1 − ߚ)]ߪ√‫ܮ‬Ψ(݇), where ߙ and ߙ represent the buyer's penalty cost and marginal profit.

Figure 1. Behavior of the buyer's inventory level over time
By definition, the number of items that are classified as defective include those that are non-defective, ܳ(1 − ߛ), and incorrectly classified as defective (with probability ݁ ଵ ), and those that are defective, ܳߛ, and classified as defective (with probability 1 − ݁ ଶ ); thus, we get ‫ܤ‬ ଵ = ܳ(1 − ߛ)݁ ଵ + ܳߛ(1 − ݁ ଶ ). The number of defective items returned from the market are those that are defective, ܳߛ, and incorrectly classified as non-defective (with probability ݁ ଶ ); thus, we have ‫ܤ‬ ଶ = ܳߛ݁ ଶ . Since we assume that the defective items returned from the market are replaced with good items, the inventory will be depleted at a rate of ‫ܦ‬ ᇱ = ‫ܦ‬ + మ ் . By definition, the cycle length . Hence, the holding cost per production cycle is The ordering cost, transportation cost, screening cost, and crashing lead time cost per delivery cycle are ܵ , ‫,ܨ‬ ܿ ܳ, and ‫,)ܮ(ܥ‬ respectively. There are ݊ deliveries in one production cycle, then the ordering cost, transportation cost, screening cost, and crashing cost per production cycle are ݊ܵ , ‫,ܨ݊‬ ݊ܿ ௦ ܳ, and ‫,)ܮ(ܥ݊‬ respectively. Further, the buyer incur a post-sale failure cost ܿ for each item being returned from the market. Then, the post-sale failure cost for the buyer per production cycle is ݊ܿ ܳߛ݁ ଶ .
Then, the buyer's total cost in a production cycle is the sum of ordering, transportation, screening, post-sale failure, crashing, holding, and the shortage costs

The vendor's cost formulation
The vendor prepares for the repeating flow of orders size ܳ = ݊ܳ from the buyer by producing items in in batches of size ܳ and by planning to have each batch delivered to the buyers in ݊ deliveries, each with a lot of ܳ units. Fig. 2 subject to ‫ܳ‬ ≤ ‫.ܤ‬

Solution Technique
Taha [24] discussed how to solve the optimum solution of nonlinear programming problem subject to inequality constraints by using the Kuhn-Tucker conditions. The development of the Kuhn-Tucker conditions is based on the Lagrangian method. Suppose that the problem (1) can be written as follow. with The explicit general solution for Q and k cannot be obtained by solving equations (5) and (6) because the evaluation each of the expressions requires knowledge of the value of the other. The value of (ܳ, ݇) can be obtained by adopting a similar graphical technique used in Hadley and Within [9]. The same numerical search technique also has been used in Lin [19,20] and Kurdhi et al. [16,17]. Therefore, the following iterative algorithm is established to determine the solution of (݊, ܳ, ݇, ‫)ܮ‬ for ߣ = 0. ii. For ߣ ≠ 0, from (3) and (4), we have and With the same argument as case ߣ = 0, the following iterative algorithm is established to determine the solution of (݊, ܳ, ݇, ‫)ܮ‬ for ߣ ≠ 0. , then go to step (e), otherwise go to step (g). g. Set ‫݊(ܥܶܧ‬ * , ܳ * , ݇ * , ‫ܮ‬ * , ߣ * )= ‫݊(ܥܶܧ‬ − 1, ܳ ିଵ * , ݇ ିଵ * , ‫ܮ‬ ିଵ * , ߣ ିଵ * ), then ‫݊(ܥܶܧ‬ * , ܳ * , ݇ * , ‫ܮ‬ * , ߣ * ) is the solution.

A numerical example
In this section, an example is solved using the proposed solution in the previous section. The purpose is to illustrate the solution procedure and conduct some sensitivity analysis for important model parameters. Consider an integrated vendor and buyer inventory model with the following parameters: Production rate (ܲ) = $75/unit Buyer's purchasing price ‫)(‬ = $10/unit Buyer's maximum available budget to purchase products ‫)ܤ(‬ = $30,000/year. The lead time has three components with data shown in Table 1. Moreover, we assume that the defective percentage is uniformly distributed with p.d.f. as By the method of Kuhn-Tucker conditions, consider the two cases 1) ߣ = 0 and 2) ߣ ≠ 0. Table 2 shows the solutions of ݊, ܳ, ݇, ‫,ܮ‬ ‫,ݎ‬ and the minimum integrated inventory total cost ‫ܥܶܧ‬ for ߣ = 0. From Table 2, we obtain ݊ * = 3 times, ‫ܮ‬ * = 8 weeks, ܳ * = 1710.55 units, ݇ * = 2.65040, ‫ݎ‬ = 7702.70 units, and the minimum total expected annual cost is $323432.15. We also have ‫ܳ‬ − ‫ܤ‬ = −12894.50 < 0, then the lot size satisfy the budget constraint. The Kuhn-Tucker conditions are satisfied, so the solution is optimal and feasible. Table 3 shows the solutions of ݊, ܳ, ݇, ‫,ܮ‬ ‫,ݎ‬ and the minimum integrated inventory total cost ‫ܥܶܧ‬ for ߣ ≠ 0. From the table, we get ߣ * = 0.190555, ݊ * = 1 times, ‫ܮ‬ * = 8 weeks, ܳ * = 5119.92 units, ݇ * = 2.25670, ‫ݎ‬ = 7701.16 units, and the minimum total expected annual cost is $323992.96. However, the lot size does not satisfy the budget constraint, that is ‫ܳ‬ − ‫ܤ‬ = 21199.22 > 0. Hence, the solution is not feasible. In order to study how the parameters affect the integrated optimal solution, the sensitivity analyses for transportation cost ‫,)ܨ(‬ vendor's holding cost (ℎ ௩ ), buyer's holding cost (ℎ ), upper limit of defective percentage (ߤ), probability of Type I error (݁ ଵ ), and probability of Type II error (݁ ଶ ) are performed. Table 4 shows the optimal solutions for different transportation costs. When the transportation cost increases, the joint total cost also increase. The percentage increase in joint total cost is 0.05%. We may also observe that increasing in transportation cost, will increase the order quantity and decrease the reorder point. The smaller the transportation cost up to $10, the larger the cost reduction of the integrated model in comparison to independent decision is.  Table 5 shows the optimal solutions for different vendor's holding cost per unit. From the table, one can see that the higher the vendor's holding cost is, the greater the joint total cost is. The percentage increase in joint total cost is 0.40%. One can also see that when the vendor's holding cost decreases, it is more profitable to decrease the order quantity and to deliver the items more frequently from the vendor to the buyer. The greater the vendor's holding cost up to $8, the larger the cost reduction of the integrated model in comparison to independent decision is.  Table 6 shows the optimal solutions for different buyer's holding cost per unit. One can see that the smaller the buyer's holding cost is, the greater the advantages to deliver the items more rarely from the vendor to the buyer. Moreover, when the buyer's holding cost increases, the joint total cost also increase. The percentage increase in joint total cost is 0.82%.  Table 7 shows the optimal solutions for different probability of Type I error. One can see that when the probability of Type I error increases, both the buyer and the vendor incur a higher joint total annual cost. The percentage increase in joint total cost is 82.16%. The smaller the probability of Type I error up to 0.08, the smaller the cost reduction of the integrated model in comparison to independent decision is. Table 7. Optimal solution for different probability type I error.  Table 8 shows the optimal solutions for different probability of Type II error. From the table, we may see that the probability of Type II error has a similar impact as the probability of Type I error. When the probability of Type II error increases, the joint total cost also increases. The percentage increase in joint total cost is 6.57%. The smaller the probability of Type II error up to 0.12, the smaller the cost reduction of the integrated model in comparison to independent decision is.  Table 9 shows the optimal solutions for different upper limit of defective percentage. From the table, we may see that when the upper limit of defective percentage increases, the joint total cost and the order quantity tend to increase. The percentage increase in joint total cost is 25.78%. The smaller the upper limit of defective percentage up to 0.08, the smaller the cost reduction of the integrated model in comparison to independent decision is.

Conclusions
In this paper, we propose the two-echelon supply chain inventory model by considering shortage backlogging, imperfect quality items, inspection errors, lead time, budget capacity constraint, and stochastic demand. An analytic solution procedure is developed to determine the optimal number of shipments, the size of each shipment, the reorder point, and the lead time. An investigation of the effects of six important parameters (transportation cost, vendor's holding cost, buyer's holding cost, upper limit of defective percentage, Type I error, and Type II error) on the optimal solution is also made. Numerical results show that (1) the joint total annual cost is more sensitive to the variation of the upper limit of percentage defective, Type I error, and Type II error; (2) the cost reduction of the integrated model in comparison to independent decision can be increased by decreasing the transportation cost; or by increasing the upper limit of defective percentage, the probability of Type I and II errors, and the vendor's holding cost. Possible extensions for future research could be by considering learning in production. In this case, the vendor experiences learning in the production process while some of the units are defective. The effect of learning in buyer's inspection errors on supply chain cost also can be investigated. Furthermore, the other possibility is considering the alternative inventory models such as periodic review.