Retailer Integrated Inventory Model in Supply Chain - Essay Example

INTEGRATED INVENTORY CONTROL Yahya Elahibakhsh Department of Engineering St. Mary’s San Antonio, Texas, 78228, U.S. Maximizing profit is the major aim of most of the organizations. Two methods that are preferred widely by organizations to increase the profit are to increase the revenue or reduce the cost. Most of the companies become outdated due to the fact that they fail to maintain the cost down. Over the past few years, researchers have become more focused on improving the supply chain system by different methods. Supply chain management is an important activity of every organization. This paper aims to minimize the total costs by considering various methods and taking elements that are important in the supply chain into consideration. Contents Abstract 1 Contents 2 Introduction 3 Literature Review 5 Methodology 8 Problem Assumptions and Description 8 Mathematical Model of the Problem 9 A solution algorithm 9 Determining the optimal values 10 Numerical example 19 Conclusion and Discussion 24 References 25 Introduction Business has become very competitive in the present world. Customers are now exposed to more information regarding the product and this allows them to select the best product in the market. Globalization has made business to be competitive outside the region of operation. It is very important that every organization determines suitable method to make the product and service to the people at lowest price. Company will be able to provide products under reduced cost only if the operating cost of the organization is reduced. In the business world every amount that is spent on the product is countable, by reducing the cost of operation the firm will be able to reduce the price of the product and this will increase the competitive advantage of the organization. Supply chain management is now being identified as an important factor for the effective functioning of an organization. Supply chain management provides large opportunity for reducing the cost of operation and to ensure that the firm functions effectively. Supply chain management covers all the process in the production of a product or service starting from the collection of raw materials to the delivery of the product. Effective supply chain management depends on several factors. With the increase in sequences and participants in the supply chain the complexity of the chain increases. Inventory management is an important aspect of the supply chain management. The main focus of inventory management is to determine the activities of inventory like determining the amount of inventory that a company would require to meet the demand of the customer. Proper inventory control enhances the firm to have the optimum amount of inventory to meet the demands of the customer. Low inventories may result in the organization having deficient inventories and will have to face a shortage of inventories. Most of the companies in the past used the push-inventory system under which the organizations would keep producing the goods based on the forecast of demand. This method became not suitable under the situation where the actual demand and the demand forecasted did not match. This lead to a situation in which the inventory generated was high and ultimately resulted in increase in the cost of production. Many researchers have focused on developing models that would help in allocating the optimum amount of inventory for any organization. Most of the researches have been focused on reducing the cost of the total operation. Literature Review Most of the researches that have been done in the past are based on the two-echelon supply chains. The first and foremost research regarding the integrated inventory problems was done by Goyal (1977). In this work the single vendor single buyer integrated inventory system was introduced. The system was meant to determine the joint replenishment policy in economical terms for the vendor and the buyer. The model focused to provide equal benefits to both the parties. The demand was assumed to be finite under this case. In 1986, based on the deterministic condition a model was developed by Banerjee. In this model products are produced by the vendors to meet the demands that are placed by the customers. This model aim was to determine the ordering quantity that is beneficial for both the vendor and the buyer. The model was further extended by Goyal in 1988 by allowing the production lot to be produced in small shipments. According to Goyal, the system reduced the system cost. In 1995, an optimal model was developed by Lu for the single vendor-single buyer problem with the assumption that all the shipments were of the same size. In 1995, Goyal proved that all the shipments do not have the same size and that they are different. The ordering policies for the system was developed by Sarker and Parija in 1994 and 1996 that consisted of fixing the intervals for manufacturing and supplying the raw materials. The optimal policy was developed in 2002 by Wee and Yang algebraically without the aid of differential calculus. It was claimed by the authors that the model is more profitable than the one that was proposed by Banerjee (1986) and Goyal (1988). Even though these researches seem to have played an important role in the development of various inventory control models, it is not applicable practically. This is mainly due to the fact that in the real world system it is not possible to have only one vendor and one buyer always. Clark and Scarf were the initiators in identifying the optimal model for multi vendors and buyers in 1960 to increase the quality of production. Khouja (2003) studied supply chain network that consisted of more than one echelon inventory problems. Khouja (2003) identified three mechanisms namely the equal cycle time mechanism, integer multipliers mechanism, and the integer’s power of two multipliers mechanism. In both, the researches that were done based on single buyer and multi buyer, shortage that could occur in the process was not taken into account. In 1999, Grubbstrom and Erdem extended their work by considering the EOQ with shortage and Cardenas- Barron (2001) further extended the Grubbstrom and Erdem model in 2001 to determine the EPQ by allowing the shortage. Under this model it was assumed that there occurred one shortage cost per unit in unit time. AN inventory model with single vendor-single buyer system was developed by Hill and Omar in 2006 that had a higher holding cost and considered unequal shipment sizes. Methodology Problem Assumptions and Description Various variables were important in minimizing the combined average annual cost by applying the optimal integrated inventory policy. These variables include: (i) retailer (ii) distributor (iii) deterministic demand rate for a non-finite horizon at the retail location (iv) fixed set-up cost for the retailer (v) fixed set-up cost for the distributor (vi) proportional or variable per-unit cost paid by the retailer, (vii) proportional or variable per-unit cost paid by the distributor, (viii) inventory holding cost per unit per year for the retailer, (ix) Inventory holding cost per unit per year for the distributor. The assumption made was that the procurement lead-time is negligible for both the retailer and the distributor, and there are no stock outs. In order to minimize the total costs, the following parameters were defined: In short, the specifications of the supply chain in which the supplier and the retailer interact with each other are defined as follows: a) There is Womack and CRC Manufacturing b) There is a sub-assembly c) There is deterministic demand rate for a non-finite horizon at the retail location d) There is fixed set-up cost for the CRC Manufacturing e) There is fixed set-up cost for the Womack f) The proportional or variable per-unit cost paid by the CRC Manufacturing and Womack g) There is an inventory holding cost per unit per year for the CRC Manufacturing and Womack h) There is not lead-time for both the CRC Manufacturing and the Womack, and there are no stock outs Mathematical Model of the Problem A solution algorithm Exploiting this problem characteristic as in the EOQ model, it was assumed that a stationary policy with inventory levels for the distributor and the retailer as shown in Figure 1 below. It should be observed that the distributor procured a lot size that is an integer multiple of the retailers order quantity. Unlike the DDP model in which the EOQ model is applied to compute the retailers order quantity, in this model, the integer multiple and the retailers order quantity are determined simultaneously. Further, in Figure 1, the maximum inventory level for the distributor is less than the distributors procurement size because, whenever the distributor replenishes inventory, there is an instantaneous shipment to the retailer. Moreover, in every replenishment cycle of the distributor, the inventory level remains zero during the retailers final shipment cycle. Figure1: Inventory Levels for the Stationary Policy for Distributor (top) and Retailer (bottom) Determining the optimal values The average annual cost functions was used to determine the required optimal values and the following and the following decisions variables were defined by taking Figure 1 into consideration: : CRC’s manufacturing order quantity (number of units) per purchase order : CRC’s manufacturing replenishment time : Womack’s order quantity (number of units) per purchase order : Womack’s replenishment time  : Integer multiple factor that determines the Womack’s inventory policy in terms of CRC Manufacturing inventory policy. That is, Womack’s order quantity =, and Womacks replenishment time =. To establish the benefits of the integrated inventory policy, three deferent scenarios are analyzed: (i) the retailer enforces the order quantity  as per his EOQ with no consideration of costs incurred by the distributor, and the distributor determines  given the retailers EOQ (i.e., the independent inventory policies are determined sequentially), (ii) the optimal inventory policies (and ) are determined simultaneously using the combined average annual cost function, and (iii) the distributor dictates the order quantity  (and the other policy variable () that defines the retailers inventory policy, without any consideration of costs incurred by the retailer. i. Sequentially Determined Retailer Enforced Independent Inventory Policies Retailers independent policy: When the retailer enforces the order quantity, the EOQ model can be applied to compute the economic order quantity  and the replenishment time , which will cost the retailer an average annual cost  per year. Distributors independent policy: When deciding independently, the distributor should work with the retailers order quantity and determine the distributors order quantity  =  The sequential decision framework, while being identical to the DDP framework, considers distributors cost function that includes the inventory holding cost component. In other words, the distributor operates on the average annual cost function using the retailer enforced  to determine the policy variable  that is optimal for this situation. The distributors average annual cost is   It should be noted that the total inventory per cycle is the area under the inventory-level curve for the distributors cycle (Figure 1). It is expressed as the sum of the areas of the rectangles associated with each replenishment cycle of the retailer, which is (K-1) +(K-2)+…..+(1)+(0) =[(K- 1)+(K-2)+….+1].  Distributors average annual cost Zd(K|) = =  This is the cost function that will be used to find K, which is an integer multiple, and the following two optimality criteria are applied. 1.  That is,   (1) 2. Similarly,  (2) The two inequalities (1) and (2) can be used to determine and the associated optimal. The overall cost of the sequentially determined retailer enforced independent inventory policy is+. ii. Integrated Inventory Policy As in the DDP model, in the sequentially determined retailer enforced independent inventory Policy, the decisions were made in two stages. In the first stage, the retailers inventory policy was determined based on the EOQ model. Following this, in the second stage, the best inventory policy for the distributor was determined given the retailers order quantity. The inventory policies determined in this manner is sub-optimal in comparison to the simultaneous optimization of the underlying decision variables. In the integrated inventory policy method, simultaneous optimization of the decisions pertaining to the distributor and the retailer will be performed by minimizing the combined average annual costs, which is The combined cycle length is The combined total average annual cost expressed as a function of decision variables and  is  It should be noted that  is a non-negative integer and  is a non-negative real variable. The following steps are used to find both and : Step 1: First, determine the optimal order quantity  in terms of the decision variable  and denote the function using (). Recall that, For notational convenience, we define two functions as follows. Exploiting the similarities between  and the EOQ cost function, we have  (3) Step 2: Substituting equation (3),  is obtained in terms of the decision variable.  (4) This cost function is then used to find. Since  is an integer multiple, the following two optimality criteria are applied. K can be founded in cycle cost and cycle length, that means K should be integer multiple. 1.  (5) 2. Similarly, (6) Using the two inequalities (5) and (6) to determine, following which,  and the associated optimal average annual cost Z*combined () can be computed, which is the overall cost of the integrated inventory policy. Alternatively, the value of  can be determined using the solution methods applicable for single-variable non-linear integer optimization problem. One such heuristics enumerates K (=1, 2,...,M, where M is a large integer) and chooses the value that results in minimum .  can then be determined using equation (4) to establish the integrated inventory policy. (iii). Distributor Regulated Inventory Policy Based on the power structure, if the distributor regulates the inventory policy, then the replenishment decisions will be based entirely on the distributors average annual cost function. In such case, the retailer will implement the distributor-imposed. The distributors average annual cost, as a function of the decision variables  and K, is expressed as Zd(K, ) = cd  + + (hd(K - 1))  . It should be noted that the average cost function will be minimal for the distributor if K = 1 and when the retailers order quantity  is arbitrarily large. Obviously, such a policy is infeasible and will violate supply, transportation, or storage capacity restrictions. However, if the distributor dictates the order size () such that the resulting policy is feasible for implementation, then the distributors inventory policy can be determined by setting= 1.For example, setting = 3 and = 1, the distributor will procure once every three years and will not hold any inventory. The retailer will have to procure once every three years as well, and may incur substantially higher inventory holding cost. Clearly, the inventory policy thus determined is sub-optimal in comparison to the integrated inventory policy. Numerical example Womack Machine Supply is an industrial distributor of hydraulic, pneumatic and automation equipment with corporate office located in Farmers Branch, Texas. Womack represents one of the worlds leading manufacturers of fluid power and industrial control products, and maintains one of the largest inventories in the South-West region in conveniently located regional service centers. CRC Manufacturing Inc., one of Womacks customers, requires a sub-assembly supplied by an international supplier. The estimated demand for the sub-assembly is 25,000 per year and is expected to be fairly steady over the next several years. Womack pays $25,000 per shipment for importing the sub-assembly from its supplier located in China. The inventory holding cost, inventory tax, and inventory maintenance cost for Womack is $48 per sub-assembly.CRC Manufacturing buys the sub-assembly from Womack and uses a third party logistics service provider, and pays a fixed cost of $5000 per shipment to transport the sub-assemblies to CRCs warehouse. CRC estimates $60 per sub-assembly for inventory holding, maintenance, and taxes. Assume the cost of sub-assembly to be $0.10 per sub-assembly for both Womack and CRC Manufacturing (as per a long-term contract between the international supplier, Womack and CRC). Case (a): When the strategic power structure allows CRC Manufacturing to dictate the ordering policy. Determine the sequentially determined retailer enforced independent inventory policies for CRC Manufacturing and Womack. CRCs EOQ, = 2,041 sub-assemblies per procurement cycle. CRCs replenishment time  = = 0.0816 year (or 20.42 days assuming 250 days per year). CRCs average annual cost () = $124,974,49 per year. Given the CRC enforced, optimality conditions (1) and (2) are used to determine the inventory policy parameter K for Womack as follows:  ( - 1)  6.25 ( + 1) >> = 3. Womacks order size= 6,124 sub-assemblies per procurement cycle. Womacks average annual cost (|) = $202,541,66 per year. The total average annual cost for the CRC-dictated sequentially determined independent inventory policies is  () + (|) = $327,516,15 per year. Case (b): When Womack and CRC are strategic partners and can coordinate inventory decisions. Determine the decision variables and the combined total average annual cost associated with the integrated inventory policy. Optimality conditions (5) and (6) can be used and Womacks optimal inventory parameter determined As follows:  ( - 1)  1.25  (+ 1) >>= 1. A() = $30,000 per procurement and B() = $60 per sub-assembly per year can then be computed. Condition (3) can be used to determine the optimal order quantity for CRC,  = 5,000 sub- assemblies per procurement cycle. CRC’s replenishment time  Womack’s order size  CRC’s average annual cost  Womack’s average annual cost  The combined total average annual cost associated with the integrated inventory policy Case (c): Assuming that the strategic power-structure allows Womack to dictate the inventory policy, determine the associated policy decisions. For Womack dictated policy, = 1. It is assumed that = = 25,000 sub-assemblies are feasible for implementation. Womacks order size = 25, 000 sub-assemblies per order. Womack’s average annual cost  CRCs order size = 25, 000 sub-assemblies per order. CRCs replacement time  CRC’s average annual cost  The total annual for Womack dictated inventory policy  Conclusion and Discussion There have been certain limitations and shortcomings in the researches that have been done before. Some of the limitations were that no shortage was allowed; complexity of the supply chain network, and the improving the service was not done in the research. This research to an extent has been able to overcome these shortcomings and limitations. By implementing the process that has been discussed in the paper, companies will be able to replenish the inventories and consequently reduce the cost of operation in the firm. This will allow the firms to determine the supply chain network that is to be ordered and the number of units that is to be ordered over the time. The end customer demands will be satisfied and of course the operating price of the firm will also be reduced. This method can be adopted in almost every industry and will be successful. References Bertazzi, L., Paletta, G., & Speranza, M. G. (2005).Minimizing the total cost in an integrated vendor-managed inventory system. JournalofHeuristics, 11: 393–419 Biskup, D., Simons, D. & Jahnke, H. (2003).The effect of capital lock up and customer trade credits on the optimal lot size–a confirmation of the EPQ. Computers and Operation Research, 30: 1509–1524 Cardenas-Barron, L. E. 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