International Journal for Management Science And Technology (IJMST)

Transcription

1 Volume 3; Issue 2 Manuscript- 3 ISSN: (Online) ISSN: (Print) International Journal for Management Science And Technology (IJMST) VALIDATION OF A MATHEMATICAL MODEL IN A TWO ECHELON SUPPLY CHAIN CASE USING MONTE CARLO SIMULATION \ M. B. Parjane Loni, India Dr. B. M. Dabade Loni, India M. B. Gulve Loni, India March 2015

2 1. Introduction The main aim of this paper is to consider a supply chain case, so as to develop a model which will help to reduce the total cost across echelons in a multiple product system. Every product here has got Poisson demand which is independent. The costs that are considered at retailers end are holding cost and back order cost. For manufacturer two things are important i.e. setup cost and setup time. Every product whenever taken up for production takes time. Basic aim of this paper is to find the optimal batch size and lowest expected total cost with finding the recorder level or sequencing number. Here necessary is to find the production batch size (q) for product. Model will be developed when the demands are known for product. When the inventory level reduces below the reorder level, then the product is queued for production. Here we have considered products as identical, so total cost per unit time of the supply chain would be total cost per unit time for single product multiplied by number of products. Expected total cost = Setup cost + Holding cost + Back order cost Notations M Number of products ETC Expected total supply chain cost Retailer D- Demand per unit time S- Order up to level h- Per unit holding cost per unit time b- Back order cost k- Safety factor Manufacturer q- Batch size T- Changeover (Setup) time t- Per unit processing time A- Setup cost ρ- Plant utilization including production and changeover ISSN: (O.)/ (P.) Page 2 March, 2015

3 If the demands for the products are deterministically known over the planning horizon, we can model it as a capacitated multi-product lot-sizing problem. The objective would be to determine the production / product lot for multiple products over a finite or infinite planning horizon so as to minimize total costs, while known demands are satisfied and capacity restrictions are respected. The total relevant cost generally consists of setup costs, inventory holding costs, production/procurement costs, and backorder costs. The limited availability of production resources introduces some interdependency among products, which leads to complex coordination problems where decisions can no longer be made for each product separately. Even set-up times are considered. The literature consists of a whole lot of heuristic approaches to obtain good solution to this problem. Our problem differs from the capacitated lot sizing problem in the sense that the demands are probabilistic and we need not fix any time horizon in our model. [1] Over the last three decades, there has been a lot of work going on in the area of multi-echelon inventory systems. The focus of our research in inventory control literature is on a single product. The multi-product production scenario at the manufacturer has also been discussed but they do not consider the cost incurred at the retailer. The situation also assumes that the demand is deterministic proper lot size and schedule lead to smooth running and provide a cost efficient production process. A model for analyzing the lead time in a multi-product, two echelon system is developed models the decision processes of supply chain to minimize the customer s dissatisfaction by making a tradeoff between price and delivery lead time. Our work differs in the sense that we have dealt with multi-retailer scenario, considering both cost and the time. [2] We have considered the integrated retailer manufacture scenario handling multiple products under capacity constraints, taking into account the time and cost for both setup and processing.[3] The demand s met with existing capacity by appropriate choice of production batch size. We explore how the costs of the various echelons balance against each other, and how the existence to multiple products affects the inventories at the retailer. It gives us an insight into how to incorporate the setup and processing time interactions in the cost calculations. For similar products after rigorous calculated experiments, we have built an approximate analytical mode. It will help us know the intricacies of handling multiple times and in formulating simple rules for inventory control in an integrated supply chain environment. ISSN: (O.)/ (P.) Page 3 March, 2015

4 In traditional supply chain situations, each player made a decision that was best suited to meet his needs. Each player worked towards optimizing individual objective functions. But with the rise in competition, the players are now willing to coordinate in order to reduce the total cost of the supply chain, and thus, stay in business. The motivation for our work seems form the fact that even in the reduced echelon system we need to address the problems arising due to the interactions of demand variability, limited production capacity, information sharing and various cost components. [4][9] The companies are on the lockout for cost reduction strategies and better information sharing technologies so as to attain profits.here we have considered a closely coupled supply chain where a manufacture produces several products on a shared resource and on completion of each batch ships it to the retailer. We have assumed the products to be similar, that is the set of parameters for each product is same. It should be noted that in today s factory that focuses on a group of products, the parameters are relatively similar. The customer demand for the different products is assumed to be Poisson. Though the model explicitly mentions only one retailer, it can be considered as an aggregation of retailers facing similar demand for the product. Shortages are completely back ordered. The retailer and the manufacturer are seamlessly integrated and try to optimize the overall goals. Thus the goal there is not of individual cost minimization but the one that leads to minimal supply chain cost. Lesser setups are of interest to the manufacture whereas lesser inventory is the preference of the retailer. This calls for information sharing, and centralized decisions have to be taken so as to minimize the cost across the supply chain. The supply chain policy is such that the retailer shares his inventory information at the end of every transaction. This is more like a vendor managed inventory where the retailer transfers the point of sale information. The policy is (S-1, S) where the inventory position is always brought back to the order up to level. The manufacturer produces all the M products on a shared resource with limited capacity. Each product takes T + qt time and incurs a setup cost of A to produce a batch of size q units. The manufacturer has to meet the demand of all products. The customer demands at the retailer for each product can reasonably be assumed to follow Poisson distribution with parameter D. the retailer incurs a holding cost for the inventory held and a backorder cost for not meeting the customer requirements. The retailer holds safety stock, as a precaution to uncertainties in demand. The point of sale information at the retailer is passed on to the manufacturer. If the retailer does not hold buffer stock, then the chance of incurring ISSN: (O.)/ (P.) Page 4 March, 2015

5 backorder increases. The retailer will incur holding cost if he holds buffer stock. At the manufacturer a larger batch will reduce the number of setups and hence the total setup cost. [5][10] But this will mean long time for the production of a batch and the other products need to wait in the queue for longer duration before getting processed and retailer has to hold more inventories. So a trade-off among all these costs is essential. [6][13] The manufacturer should dynamically assess the situation and determine the product that needs to be taken for production. This is diagrammatically shown in Fig. 1. Fig. 1 Manufacturer Retailer Relation Given the scenario and the cost components, the objective is to build a quantitative supply chain model that can then be used to optimize the cost. Optimization is done by deciding (i) the production batch size, and (ii) the production queue point. Batch size is the quantity that will be produced at the manufacturer once the processor has been set up for a product. When the inventory level falls below the queuing point at the retailer, that particular product will be queued for production at the manufacturer. It will be directly taken up for production if the manufacturing facility is free; else it waits for its turn to be produced in the queue which is processed on a first come first served basis. The goal is to minimize the total supply chain which is essentially the setup cost at the manufacturer and the holding and backorder costs at the retailer. We assume that the backorder cost in much greater than the holding cost. The Expected Total Cost represents the cost of the entire supply chain per time period obtained by averaging the total cost over a long period of time. [7][11] ISSN: (O.)/ (P.) Page 5 March, 2015

6 2. An Approximate Model Here C language is used to calculate the ETC, since the products are identical, the total cost per unit time of the supply chain would be the total cost per unit time for a single product (ETC) multiplied by the number of products. So it is enough to analyze the total cost of a single product by considering interactions with the other products. The total expected supply chain cost per product per unit time could be written as ETC = Setup Cost + Holding + Backorder Cost. The expected setup cost per unit time will be the set up cost apportioned to each unit product multiplied by the demand per unit time and is given by (A* D/q) The second part is the expected holding which consists of three terms. Since the production batch size is q, the average inventory held during replenishment is q/2, which is the first term. The second term is the buffer inventory to be held at the retailer so as to account for the interactions among the various products. When a product reaches the production queue point it is queued for production. If this queue is empty then the product is immediately taken up for production, else it has to wait till the manufacturer has produced all other products that are in front of it. There are (M - 1) potential products to be in the queue and it takes (T + qt) time produce each batch of products. Therefore the average demand during this period is (M - 1)(T + qt)d. But the waiting time is dependent of the plant utilization. If ρ is low then there is high chance that the queue is empty and a large ρ value implies that the product should wait for its turn to get processed. If the plant is 100% utilized then on an average the waiting time for a particular product is (M - 1) (T + qt)/2 and the average demand during this period is (M - 1) ( T + qt) D/2. Considering the plant utilization, we can assume that the average inventory to be held at the retailers to account for product interactions is (ρ/2)* (M-1) (T+qt) D. We call this term to be the interaction term. A proper estimate of the interactions among the various products plays a crucial role in balancing the various cost components. To optimize the cost we need to make a tradeoff between the holding and backorder cost and hence multiply this with a safety factor of k. The third part is the expected backorder cost. Backorders can occur only before the replenishment points and hence is multiplied by (D/q). The standard deviation during this period due to demand variations is given by the following equation ISSN: (O.)/ (P.) Page 6 March, 2015

7 .. (1) where D is the probability of this happening is [Ø(k) k + k Ø (k)]which is obtained assuming the error of variation to be standard normal having the above standard deviation. While most of the other expressions are obvious, the expression for the expected stock out per replenishment cycle can be found. Therefore the expected total cost per product can be written as ETC product = ρ 1.. (2) Note that for a single product (M = 1), the above equation reduces to.. (3) Where is the standard deviation of demand during lead time. This is the standard expression of batch size for a single product. Queuing point can be given as.. (4) ISSN: (O.)/ (P.) Page 7 March, 2015

8 The policy followed by the supply chain is to meet the customer demand (backordered in some cases by paying a penalty cost) in all cases. The lower bound for the system is obtained from the plant utilization, if the batch size q obtained by solving the unconstrained ETC equation is less than the lower bound mentioned above, the lower bound on q will be used as the production batch size.[8][12]. We now look into the framework for obtaining the solution for the proposed model. It is not easy to obtain a closed form expression of q and k that will optimize the cost. We have adopted a spreadsheet based approach to determine q and k simultaneously by enumeration of the total supply chain cost. Here ABC Company is considered and the data of the same is used for evaluating the total cost of the supply chain. ISSN: (O.)/ (P.) Page 8 March, 2015

9 Fig. 2 Effect of Batch size on Expected Total Cost As the batch size is below certain level the expected total cost is also high and even when we vary the batch size it also affects the expected total cost. In the graph, we see by increasing or decreasing the batch size does not help to reduce the expected total cost. At certain point, the ETC is minimum and the batch size is near 1257 units, with a reorder level of 341 units. Fig. 3 shows that with the increase in the number of products the interactions among the products increases more inventories to be held. If the inventory held, is the same then the chances of back order increases. Thus, the Total supply chain cost is directly proportional to the no of products M. Fig. 3 Variations with No of Products ISSN: (O.)/ (P.) Page 9 March, 2015

10 Fig. 4 Variation with demand As the Demand increases, the inventory that needs to be held increases to meet the customer demand. This calls for bigger production batches so as to save on the set up cost. The steep rise in the graph is obtained when the plant utilization reaches unity. This calls for increasing the batch size so that we can still meet the demand 3. Validation by Monte Carlo Simulation In using the Monte Carlo simulation, a given problem is solved by simulating the original data with random number generators. Basically, its use requires two things. First, we must have a model that represents an image of the reality of the situation. The distribution may be obtained by direct observation or from past records. Here we have considered the information related to the sales of the product for 200 days of ABC Sinter Metals Pvt. ltd.. From the past record, the data is obtained for demand. Table No. 1 Demand distribution Demand Total No. of days Table No. 2 Probability distribution Demand Probability ISSN: (O.)/ (P.) Page 10 March, 2015

11 Table No. 3 Random Number Interval Demand Probability Cumulative Probability Random no. interval Table No. 4 Probability Distribution for Number of Products M Probability Cumulative Probability Random no. interval Table No. 5 Probability Distribution for Unit Processing Time (t) T Probability Cumulative Probability Random no. Interval Table No. 6 Simulation in Inventory Control for Case 1 Batch size q 1257 Units Reorder Level s 341 Units Setup Cost T Rs. ISSN: (O.)/ (P.) Page 11 March, 2015

12 Total Processing Time t 1080 min. Holding Cost h 0.02 Rs./ Unit Back Order Cost b 500 Rs./Batch Stock at Retailer Day R.No. Dem and R. No Units Table No. 7 Simulation Worksheet (q=1257, s=341) M Setup Time T No.of Set up R.N o. Units Proce ssing Time t Total Mfg. Time Produ cts Mfg. in a day Bal. at retaile r Setup Cost Holding Total Cost Back Order Cost Expected total cost (ETC) = Setup Cost (A) + Holding Cost (h) + Back Order Cost (b) ETC = 12,50, ETC = Rs.12,53, /- Similarly, the results were obtained for case 2 and case 3 ISSN: (O.)/ (P.) Page 12 March, 2015

13 Table No.8 Result Table for simulation Sr. No. Batch Size q Reorder level sequencing Expected Total cost Case Rs.12,53,330.74/- Case Rs. 12,53,830.64/- Case Rs. 12,53,840.68/- 4. Conclusions The mathematical model and Monte Carlo simulation were used to obtain the recommended optimal policies and the various costs of the supply chain. The cost measures of interest are the expected setup cost at the manufacturer, expected holding and back order costs at the retailer. These costs can be used to derive the performance measures of supply chain like retailer inventory level, expected number of backorders at the retailer, fill rate, delay and so on. Completing a 20 day simulation, we find that the backorder cost for case 1 = Rs.3000/-, the holding cost = Rs /- and the setup cost = Rs.12,50,000/-, the three adding up to Rs.12,53,330.74/-. In the case 2, the addition goes up to Rs.12,53,830.64/- where as in case 3 the total is Rs.12,53,840.68/-. Therefore, the optimum batch size that can be considered is 1257 units with a reorder level of 341 units, as the expected total supply chain cost is minimum for the same. ISSN: (O.)/ (P.) Page 13 March, 2015

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