Simulation of surge effect in supply chain for demand uncertainty at the end customer.Introduction
The main objective of supply chain management is to provide a high velocity flow of high quality relevant in formation that will enable suppliers and manufacturers to provide an uninterrupted and precisely time flow of materials to customers.
The complex and dynamic interactions between supply chain entities lead to considerable uncertainty in planning. Uncertainty tends to propagate up and down the supply chain and this affects performance of the supply chain(Ref-1).. The uncertainty demand or inaccurate forecasts are causes up and down the supply chain and undergoes with surge or bullwhip effect.
The Bullwhip Effect is problematic: order variability increases as orders propagate along the supply chain. The Bullwhip Effect has been documented as a significant problem in an experimental, managerial, as well as in a wide variety of companies and industries. Many proposed strategies for mitigating the Bullwhip Effect have a history of successful application (Ref-2).
A surge in demand depletes inventory, quality problems, higher raw material costs, overtime expenses and shipping costs. In the worst-case scenario, customer service goes down, lead times lengthen, sales are lost, costs go up and capacity is adjusted. An important element to operating a smooth flowing supply chain is to mitigate and preferably eliminate the bullwhip effect.
Companies can effectively counteract the bullwhip effect by thoroughly understanding its underlying causes. Industry leaders are implementing innovative strategies that pose new challenges such as integrating new information systems, defining new organizational relationships, and implementing new incentive and measurement systems (Ref-3).
In this paper the demand is assumed as a random variable and the distribution of the demand assumed as normal distribution. Monte Carlo simulation method and Excel software is used to simulate the surge effect. The surge effect is simulated for five stages of supply chain by assuming the demand is sudden rising and sudden falling to the limits of the normal distribution. The mean value plus 0.67 std or 1 std or 2 std or 3 std is taken as sudden rise to the demand and mean value minus 0.67 std or 1 std or 2 std or 3 std is taken as sudden fall to the demand. From the results it can be found that for lower fluctuation of the demand, the variability of the demand is small that will not give any significant effect on the performance of the supply chain. But for larger demand fluctuations the variability of the demand is very large (identified at the limit of 3 std) that will significantly effect on the performance of the supply chain.
Surge Effect or Bullwhip Effect
The surge effect or bullwhip effect describes the phenomenon that the variation of demand increases up the supply chain from end customer to supplier. This effect leads to inefficiencies in supply chains, since it increases the cost for logistics and lowers its competitive ability. Particularly, the bullwhip effect negatively affects a supply chain in dimensioning of capacities, variation of demand and high level of safety stock.
Normal Distribution of the Demand:
The normal distribution describes many random phenomena that occur in every day life, including demand fluctuations, test scores, weights, heights, and many others. The probability density function of the normal distribution is defined as
F(x) = 1/[square root of (2[[PI][[sigma].sup.2])] e -[alpha]< x < [alpha]
Where [mu] is the mean of demand and [sigma] is standard deviation
In fig-1 the graph shows the normal F(x) and is symmetrical around the mean value [mu]. The cumulative distribution of the random variable cannot be determined in a closed form. So that normal tables have been prepared for this purpose. These tables apply to the standard normal for which the mean is zero and variance is 1. Any random variable x with mean [mu] and [sigma] standard deviation can be converted to a standard normal z by using the transformation z =(x-[mu])/[sigma].The limits under normal distribution curve are taken for 50% of the spread of the demand as [+ or -] 0.67[sigma], for 68.3% of the spread of the demand as [+ or -] [sigma], for 95.5% of the spread of the demand as [+ or -]2[sigma] and for 99.7% of the spread of the demand as [+ or -]3[sigma].
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
The dynamics of inventory management can be under stand by considering effect of demand fluctuations on the desirable level of stock in all stages in supply chain. A typical supply chain as shown in fig-2, which consist of five stages such as suppliers, factory, distributors, wholesalers and retailers. The retailers receive the demand of end customers. At all the stores of different stages must hold some stock of items to compensate the variation of lead times, demand fluctuations and for other disturbances for smooth running of the supply chain. If the mean demand at the customers is D and if the policy for reserve stock specifies that a portion r should be held, then rD is the safety stock for finished products at the retailer's stores. If there is a sudden change in demand, it may be sudden rise or sudden fall of the demand at customers, so that D (1+x) quantities of products are needed at the retailers, the reserve stock now should be rD(1+x). where x is % of sudden rise or sudden fall of the demand. Here x is considered as x = (maximum or minimum limit of normal distribution of the demand--mean demand) X 100/ mean demand. The following results are derived for all stages from retailers to suppliers through supply chain (Ref4).
Mean Demand at End Customers = D Stage 1 Retailer: Output required = D(1+x) Safety stock required = rD(1+x)
Stage 2 Wholesalers: Output required = D (1+x) for the final stage + (rD(1+x)--rD) for the Safety stock at the final stage = D (1+x(1+r)) Safety stock required = rD(1+x(1+r))
Stage 3 Distributors: Output required = D(1+x(1+r))for the 2 stage + (rD(1+x(1+r)) - rD) for the safety stock at the 2 stage = D(1+x[(1+r).sup.2]) Safety stock required = rD(1+x[(1+r).sup.2])
Stage 4 Factories: Output required = D(1+x[(1+r).sup.2])for the 3 stage + (rD(1+x[(1+r).sup.2])- rD) for the safety stock at the 3 stage = D(1+x[(1+r).sup.3]) Safety stock required = rD(1+x[(1+r).sup.3])
Stage 5 Suppliers: Output required = D(1+x[(1+r).sup.3])for the 4 stage + (rD(1+x[(1+r).sup.3])- rD) for the safety stock at the 4 stage = D(1+x[(1+r).sup.4]) Safety stock required = rD(1+x[(1+r).sup.4])
And so on. It can be shown in generally that the output required at the nth stage is [D.sub.n] then
Dn/D = 1+x[(1+r).sup.n-1]
From which it is evident that [D.sub.n] will increase with policy r, the impulse x and with the number of stages n. the x and n are dictated by outside circumstances, where as r is an expression of inventory policy. Obviously, the smaller the reserve stock, the smaller the effects of fluctuations caused by this chain reaction along the supply chain.
Simulation of Surge Effect in Supply Chain
The simulation is defined as the process of creating representative model (usually by using computers) of an existing system or proposed system (in this case, surge effect in supply chain)) in order to identify and understand the factors that controlling the system. Any system that quantitatively described using equations and rules can be simulated. Here a simulation model has been prepared for variability of the demand through the supply chain by Monte Carlo simulation method and Ms-Excel software for uncertainty demand.
The demand at the end customer is assumed as random variable and distributed as continuous normal distributed pattern. In simulation process mean of demand is assumed as 500 units and standard deviation [sigma] is assumed as 25 units. The demand is calculated for the normal distribution limits such as [+ or -] 3[sigma], [+ or -] 2[sigma][+ or -], 1[sigma][+ or -], and 0.67[sigma] and cumulative probabilities are taken from normal distribution tables to the Z values such as -3, -2, -1, 0, 1, 2 and 3 (Ref-3). All these values are tabulated in the table -1. A graph has been plotted for calculated values of demand and cumulative values of probability distribution values as shown in fig-3.
The Monte Carlo simulation method is used to simulate the demand for various random numbers. The 20 random numbers are generated by using Excel with command Rand(). These 20 random numbers are considered as the probability of occurrence of the demand for 20 months. The value of demands for 20 months are taken from graph (From fig-3) for corresponding random numbers and tabulated in table--2. The mean and standard deviation are calculated for demands of 20.
[FIGURE 3 OMITTED]
Five-stage supply chain is considered as shown in fog-2 to simulate the surge effect in supply chain for impulse of the demand. The five stages are suppliers, factory, distributors, wholesalers and retailers. The end customers received the products from the retailers. The suppliers supply raw materials to the factory; the factory produces the products and supply to the distributors. The distributors act as reservoirs to the products and distribute to the wholesalers, when they ordered and wholesalers distributes the retailers according to their requirements.
Here the demand is considered as two different cases for the limits of o.67std, 1std, 2std, 3std of the normal distribution curve. In the first case sudden raise of the demand is considered from mean to upper limit and x is calculated in terms of percentage. For example the x is the % sudden rise of the demand for 1 std upper limit is calculated as x = 1 std/ mean demand. Similarly x in terms of percentage is calculated for all limits of the demand. In the second case the demand is assumed as sudden falling and x in terms of percentage is calculated in the same way to the lower limit. Here the policy decision r in terms percentage is also is used to determine quantity. The policy decisions depends upon the supply chain management decisions. By using this x in terms percentage and policy decision r in terms of percentage the quantity required at various stages through supply chain from downstream to upstream is calculated. The results are tabulated in table-3 and a graph is drawn as shown in fig-4 for the policy decision r=20%.and sudden rise of the x. And for sudden fall of x and for r=20% the results are tabulated in table-4 and a graph is drawn as shown in fig-5.
Besides using the x in terms percentage and policy decision r in terms of percentage the safety stock required at various stages through supply chain from downstream to upstream is also calculated. The results for policy decisions r=20% and for sudden rise and sudden fall to x are tabulated in table-5, table-6 and graphs are shown in fig-6 and fig-7. Comparison of quantity required and safety stock at stages through supply chain from downstream to upstream for 3 std limit of rising of the demand and inventory policy r=60% is given in table-7 and a graph shown in fig-8. From this, for higher inventory policy and for higher rising of demand safety stock required more and it also crossed from mean demand.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
[FIGURE 8 OMITTED]
The surge effect in supply chain is simulated for uncertainty demand. The demand of the end customer is assumed as random demand of continuous normal distribution. From the simulation results it is found that the quantity required at the stages through the supply chain from down stream to upstream is increased for the sudden rise of the demand of the end customers and decreased for the sudden fall of the demand of the end customers.
Safety stock also at the stages through the supply chain from downstream to upstream is increased for sudden rise of the demand and decreased for sudden fall of the demand.
At higher policy decision r and high sudden rise of the demand (3 Std) the safety stock required at supplier is more than the mean demand of the end customer.
 Rohit Bhatnagar, Amrik S. Sohal, 2003, "Supply chain competitiveness: measuring the impact of location factors, uncertainty and manufacturing practices", Technovation, Elsevier (Available online at www.elsevier.com/locate/technovation).
 Warburton Roger D. H., 2004, "An analytical investigation of the Bullwhip Effect", Production and Operations Management, 13(2), pp. 150-160,
 Hau L Lee, V Padmanabhan, and Seungjin Whang,1997, "The Bullwhip Effect In Supply Chains" Sloan Management Review, 38(3), pp. 93-102.
 Samuel & Ellon., 1994, "Elements of production planning and control", Navaneethan Prakashan Ltd., Bombay, India.
B. Chandra Mohana Reddy (1), K. Hemachandra Reddy (2), C. Nadha Muni Reddy (3), K. Vijaya Kumar Reddy (4) and B. Durga Prasad (5)
(1) Asst. Prof. in Mechanical Engineering Department., J.N.T.U. College of Engineering, Anantapur--515002, AP. E-mail: email@example.com
(2) Professor in Mechanical Engineering Department, J.N.T.U. College of Engineering, Anantapur--515002, AP.
(3) Professor & Principal, S.V.P.C.E.T, Puttur, Chittur (dist), A.P.
(4) Professor in Mechanical Engineering Department, Controller of Examinations, J.N.T. University., Hyderabad, A.P.
(5) Associate Professor in Mechanical Engineering Department, J.N.T.U. College of Engineering, Anantapur--515002, AP.
Table-1 Mean ([mu]) = 500 Standard deviation ([sigma])=25 Z Demand Cumulative probability 0 0 -3 425 0.0013 -2 450 0.0228 -1 475 0.1587 0 500 0.5 1 525 0.8413 2 550 0.9772 3 575 0.9987 Table-2 Simulated Months Random numbers demand (X) [mu] X [([mu] X).sup.2] 1 0.30236 485 16.2 262.44 2 0.99278 572 -70.8 5012.64 3 0.047325 455 46.2 2134.44 4 0.662034 510 -8.8 77.44 5 0.293601 480 21.2 449.44 6 0.901139 530 -28.8 829.44 7 0.125719 468 33.2 1102.24 8 0.749202 515 -13.8 190.44 9 0.694485 510 -8.8 77.44 10 0.184959 472 29.2 852.64 11 0.901331 530 -28.8 829.44 12 0.367548 490 11.2 125.44 13 0.459338 492 9.2 84.64 14 0.696288 512 -10.8 116.64 15 0.406118 490 11.2 125.44 16 0.619101 508 -6.8 46.24 17 0.566073 504 -2.8 7.84 18 0.188001 477 24.2 585.64 19 0.323692 490 11.2 125.44 20 0.263223 482 19.2 368.64 9972 52 13404 Mean (D)=498.6 Standard deviation (Std) = [square root of [(D- X).sup.2]]/n = 25.88822 Table-3 Quantity Required at various stages in supply chain for sudden rise of demand and Inventory policy r=20% Upper Standard limits 0.67std 1std 2std 3std X of sudden rising 0.03478762 0.051922 0.103844 0.155765 Retailer Stage 1 515.945297 524.4883 550.3766 576.2644 Wholesalers Stage 2 519.414356 529.666 560.7319 591.7973 Distributors Stage 3 523.577227 535.8792 573.1583 610.4368 Manufacturers Stage 4 528.572673 543.335 588.07 632.8041 Suppliers Stage 5 534.567207 552.282 605.964 659.645 Table 4 Quantity Required at various stages in supply chain for sudden fall of demand and Inventory policy r=20% Lower Standard limits 0.67std 1std 2std 3std X of sudden falling 0.03478762 0.051922 0.103844 0.155765 Retailer Stage 1 481.254703 472.7117 446.8234 420.9356 Wholesalers Stage 2 477.785644 467.534 436.4681 405.4027 Distributors Stage 3 473.622773 461.3208 424.0417 386.7632 Manufacturers Stage 4 468.627327 453.865 409.13 364.3959 Suppliers Stage 5 462.632793 444.918 391.236 337.555 Table 5 Safety stock required at various stages in supply chain for sudden rise of demand and Inventory policy r = 20% Upper Standard limits 0.67std 1std 2std X of sudden rising 0.03478762 0.051922 0.103844 Retailer Stage 1 103.189059 104.8977 110.0753 Wholesalers Stage 2 103.882871 105.9332 112.1464 Distributors Stage 3 104.715445 107.1758 114.6317 Manufacturers Stage 4 105.714535 108.667 117.614 Suppliers Stage 5 106.913441 110.4564 121.1928 Upper Standard limits 3std rD X of sudden rising 0.155765 Retailer Stage 1 115.2529 99.72 Wholesalers Stage 2 118.3595 99.72 Distributors Stage 3 122.0874 99.72 Manufacturers Stage 4 126.5608 99.72 Suppliers Stage 5 131.929 99.72 Table 6 Safety stock required at various stages in supply chain for sudden fall of Demand and Inventory policy r=20% Lower Standard limits 0.67std 1std 2std X of sudden fall 0.03478762 0.05192182 0.103844 Retailers Stage 1 96.2509406 94.5423382 89.36468 Wholesalers Stage 2 95.5571288 93.5068058 87.29361 Distributors Stage 3 94.7245545 92.264167 84.80833 Manufacturers Stage 4 93.7254654 90.7730003 81.826 Suppliers Stage 5 92.5265585 88.9836004 78.2472 Lower Standard limits 3std rD X of sudden fall 0.155765 Retailers Stage 1 84.18711 99.72 Wholesalers Stage 2 81.08054 99.72 Distributors Stage 3 77.35264 99.72 Manufacturers Stage 4 72.87917 99.72 Suppliers Stage 5 67.51101 99.72 Table 7 Comparison of demand rising and safety stock for 3std limit of demand and inventory Policy r = 60% Safety stock Demand rising Mean demand Retailers Stage 1 345.758657 576.2644 498.6 Wholesalers Stage 2 373.717852 622.8631 498.6 Distributors Stage 3 418.452563 697.4209 498.6 Manufacturers Stage 4 490.028101 816.7135 498.6 Suppliers Stage 5 604.548961 1007.582 498.6