Simulation of surge effect in supply chain for demand uncertainty at the end customer.

Reddy, B. Chandra Mohana ; Reddy, K. Hemachandra ; Reddy, C. Nadha Muni 等

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]

Conclusions

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.

References

[1] 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).

[2] Warburton Roger D. H., 2004, "An analytical investigation of the Bullwhip Effect", Production and Operations Management, 13(2), pp. 150-160,

[3] Hau L Lee, V Padmanabhan, and Seungjin Whang,1997, "The Bullwhip Effect In Supply Chains" Sloan Management Review, 38(3), pp. 93-102.

[4] 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: cmr_b@yahoo.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