CHAPTER IX OPTIMIZATION OF INTEGRATED VENDOR-BUYER PRODUCTION INVENTORY MODEL WITH BACKORDERING WITH TRANSPORTATION COST IN AN UNCERTAIN ENVIRONMENT This chapter develops an integrated single-vendor, single-uyer production inventory model y including transportation cost where ackordering is allowed. In this we extend the inventory model proposed y Chun et al [17] y incorporating transportation cost. We applied cost comparison method and differential calculus method to find optimal ackorder ratio, lotsize and the numer of deliveries. We also extended the model in fuzzy environment y taking vendor s unit production cost and uyer s unit purchasing cost as triangular fuzzy numers. Graded Mean Integration Representation Method is used for defuzzification. 175
9.1. Introduction Inventory management and supply chain management are the ackone of any usiness operations. Supply Chain management involves procurement of raw materials, manufacturing and distriution management of finished goods. The inventory together with the transport, the location of production centers represent an important factor that influences the performance of the supply chain. Vendors, retailers, etc. ie. a supply chain are generally interested in minimizing their own cost rather than that of a chain as a whole. This is not suitale for competitive environment. Coordination of vendors and uyers definitely result in reduced cost and lead time without sacrificing customer service. Hence coordination has received great deal of attention. Yang et al [68] developed an integrated vendor-uyer inventory system for deriving optimal policy using calculus technique. Also two approaches namely cost comparison and arithmetic geometric mean inequality was used in [44] and [57]. Backordering policy provides a etter cost control in inventory management and several papers involving time varying demand, constant deterioration rate, partial and imperfect quality items. Pentico et al [49] 176
considering inventory with partial ackordering under inflation. Cardenas Barron [11] derived EPQ model with two ackorders using analytic geometry and algera. However two stage inventory issues has een neglected in the aove studies. In the inventory model of [10] two decision variales are taken namely numer of deliveries and atch size Q. Our study is ased on [17] which contain 3 variales namely numer of deliveries, atchsize Q and ackordering rate K in the development of two stage sample vendor-uyer integrated model. We have included transportation cost in the total cost and study the influence of it. Also in real life situation, the cost parameters are uncertain and change continuously. Hence we have developed a fuzzy inventory model y taking vendor s unit production cost and uyer s unit purchasing cost as triangular fuzzy numers. We applied graded mean integration representation method for defuzzification and derived optimal values of n, Q and K y using differential calculus approach. In crisp mathematical model we applied differential calculus, arithmetic-geometric mean inequality and costdifference rate comparison methods. 177
9.. Assumptions a) Both the production and demand rates are constant and the production rate is greater than the demand rate. ) Single-vendor and single uyer is considered. c) Vendor and uyer name complete knowledge of each other. d) The uyer s shortage is allowed. e) System does not consider waiting-in-process inventory (WIP). f) The lead times of the integrated system is known. 9.3. Notations The following notations are used Q : Buyer s lot size per delivery. n : Numer of deliveries from the vendor to the uyer per vendor s replenishment interval. nq : Vendor s lot size per delivery. S : Vendor s setup cost per setup. A : Buyer s ordering cost per order. C v : Vendor s unit production cost. C : Unit purchase cost paid y the uyer. B : Unit ackordering cost of the uyer. 178
r : Annual inventory carrying cost per dollar invested in stocks. P : Production rate per year, P > D. D : Demand rate per year. K : The ackordering ratio. F : Fixed transportation cost incurred y vendor per delivery. C v : Fuzzy Vendor s unit production cost. C : Fuzzy Unit purchase cost paid y the uyer. 1 K : The non ackordering ratio. TC(Q, n, K) : The average integrated total cost of vendor-uyer model ~ ~ TC(Q, n, K) considering ackordering overtime period of T years. : Fuzzy average integrated cost P TC( Q, n, K) : Fuzzy average integrated cost 9.4. Model Description Crisp Mathematical Method A finite planning horizon with length T is taken and demand is replenished y n equal atch sizes, DT Q =, n 1. In this study we extend n the model proposed y Chun-Jen Chung [] y incorporating transportation cost incurred, y the vendor. 179
Fig. 9.1 Assuming that the ackordering quantity is KQ (see fig.9.1), the total cost is given y TC(Q, n, K) = (The vendor s setup cost + carrying cost + transportation cost) + (the uyer s ordering cost + carrying cost) + the uyer s ackordering cost = DS rqcv D D DF DA r(1 - K) QC K QB + (n-1) 1 + nq P P nq Q... (9.1) Equation (1) can e rearranged in Q as follows: TC(Q, n, K) = 1 y + Q C Q... (9.) 180
Where y(n) = DA + DS + DF n... (9.3) and D P C(n, K) = r(1 - K) C + K B + rc v (n - 1) 1 - + D P... (9.4) 9.4.1. Optimal ratio K y cost difference rate comparison method To determine the optimal ackorder for a given horizon T y cost difference rate comparison method we have the following conditions TC(Q, n, K + x) TC(Q, n, K) 0 TC(Q, n, K - x) TC(Q, n, K) 0... (9.5a)... (9.5) where 0 < x < 1. We know that n - 1 n + 1 lt = lt = 1 n T,n n T,n DT DT DT lt = lt = lt = Q n + 1 n n - 1 T,n T,n T,n (9.5a) and (9.5) give x rc TC(Q, n, K + x) TC(Q, n, K) = K + B + rc K x rc B + rc = +... (9.6a) rc K x TC(Q, n, K + x) TC(Q, n, K) = -... (9.6) B + rc 181
The optimal values of K and x can e otained as K x rc K x + -... (9.7) B + rc Using Sandwich Theorem in limit when x 0 we have K = rc B + rc... (9.8) Sustituting (8) into the total cost function TC(Q, n) = 1 Y(n) Q W(n) Q... (9.9a) where Y(n) = DA + DS + DF n and B D D W(n) = rc + rc v (n - 1) 1 - + rc + B P P... (9.9) 9.4.. Optimization of Q y using cost difference rate comparison method To derive optimal lotsize Q for given n, T and K we have the inequalities TC(Q -, n) TC(Q, n)... (9.10a) and TC(Q +, n) TC(Q, n) where 0 < < 1... (9.10) Applying the aove conditions 18
TC(Q +, n) - TC(Q, n) = Y W + 0... (9.11a) Q(Q + ) and TC(Q -, n) - TC(Q, n) = Y W - 0... (9.11) Q(Q - ) Rewriting (9.11a) and (9.11) Q(Q - ) Y W Q(Q + )... (9.1) when T, n and Q approach infinity and 0 lt Q - = 1 = lt Q 0 Q Q Q + Q which implies Q - = Q = Q + From equation (9.1) the optimal conditions of Q can e written as y w lt Q(Q - ) Q and 0 y w lt Q(Q + ) Q... (9.13) 0 By using sandwich theorem in limit we have y Q = w For which Q 0 = y D(A + (s + F)/n) = w D D r C [B/(rC + B)] + C v (n - 1) 1 - +... (9.14) 183
TC(n) = Dr(A + (s + F)/n) + C [B/(rC + B)] D D + C v(n - 1) 1 - + P P... (9.15) (9.15) can also e written as TC(n) = DrG(n) + H(n)... (9.16) G(n) is independent of H D G = A C [B/(rC + B)] + C v[(n - 1) 1 - + P D P U H(n) = nu + n... (9.17) When D U = ACv 1 - p B D V = (S + F) C Cv 1 rc +B P V must e positive.if V is negative then TC(n) attains minimum when n * =1. 9.4.3. Optimization of Q with arithmetic geometric mean inequality For any two real positive numers a and, the arithmetic geometric mean inequality states that a + a and equality is valid when a =. 184
Here we have two functions Q D D r(1 k) C + k B + rc v (n 1) 1 + and 1 DA + Q DS + DF n In order to apply the inequality the following three conditions must e satisfied (i) the functions must e non-negative, (ii) the product of functions must e constant, (iii) when the functions are equalized the system can e solved applying condition (iii) (since the system has a solution). Hence we can write Q D D r(1 k) C + k B + rc v (n 1) 1 + = 1 DS + DF DA + Q n Solving for Q we get... (9.18) * Q = D(A + (s + F)/n) D r C [B/(rC + B)] + C v (n - 1) 1 - + P D P... (9.19) which is same as (9.14) We conclude that cost difference rate comparison method and arithmetic geometric mean inequality yield the same expression. 185
9.4.4. Optimization of n y using cost difference rate comparison method As given in [9.16] the expressions of U and V ecome (after including the transportation cost) D U = ACv 1 - p B D V = (S + F) C Cv 1 rc +B P and V n(n - 1) n(n 1)... (9.0) U Allowing n tend to infinity and applying sandwich theorem in limit We have n = V U... (9.1) and optimal value of n is B D n * = V (S + F) C Cv 1 U = rc +B P D ACv1 P... (9.) The positive value of n satisfies n * 1 1 V * + + (n + 1)... (9.3) 4 U If 1 1 V + + 4 U is not an integer then optimal solution is 186
n * = 1 1 V + + 4 U 9.5. Solution Procedure Step 1 : Determine K = rc C + B U = D ACv 1 P and V = B D (S + F) C Cv 1 rc +B P If V > 0 then go to step. Otherwise n * = 1 Step : Determine n * If 1 1 V + + 4 U is not integer then n * = n = 1 + 1 + V 4 U Otherwise if 1 1 V + + 4 U is an integer then n * = 1 + 1 + V 4 U and n * = 1 + 1 + V - 1 4 U Step 3 : Determine Q using (9.14) Step 4 : Determine the total inventory cost with (9.15) 187
9.6. Numerical Example Example 1 : Let D = 000 units/year P = 4000/year r = 0./unit/year C v = Rs.0/unit C = Rs.30/unit B = Rs.45/unit A = Rs.30 per order S = Rs.400/setup F = Rs.100 per shipment Now K = rc rc B = 0.117647 U = D ACv 1 P = 300 B D V = (S + F) C Cv 1 rc +B P = 11383.5 > 0. n = 1 + 1 + V 4 U = 19.98 which not integer. Then n* = n = 19 deliveries Q * = 163.88 TC (Q * = 163.88, n = 19, K = 0.117647) = Rs.666.41 188
Example : Let D = 1980 units/year P = 3960/year r = 0./unit/year C v = Rs.0/unit C = Rs.30/unit B = Rs.45/unit S = Rs.400/setup A = Rs.5 per order F = Rs.100 per shipment Now K = rc rc B = 0.117647 n * = [6.3] = 6 9.7. Fuzzy Mathematical Model Let numers. C = C, C, v v1 v v3 C, C = C, C, C 1 3 e triangular fuzzy The total cost is given y ~ TC= DS rqcv D D DF DA r(1 - K) QC K QB + (n - 1) 1 + nq nq Q...(9.4) = 1 Y(n) + Q C(n, K) Q where D Y(n) = DA + (S + F) n... (9.5) 189
C(n, D D K) = (1 - K) rc +K B rc v (n - 1) 1... (9.6) D D (1 - K) rc 1 + rc v1 (n - 1) 1 + K B, C(n, K) = d C(n, K) D D (1 - K) rc + rc v (n - 1) 1 + K B, D D (1 - K) rc 3 + rc v3 (n - 1) 1 + K B 1 D D (1 - K) rc 1 + rc v1 (n - 1) 1 6 = D D 4 (1 - K) rc + rc v (n - 1) 1 D D (1 - K) rc 3 + rc v3 (n - 1) 1 6K B... (9.7) 9.7.1. Optimization of K d C(n, K) K = 0 gives (1 K)(-1)(rC 1 + 4rC + rc 3 ) + 1KB = 0 which gives K = r C 4C + C 1 3 6B + r C 4C + C 1 3... (9.8) Let X = C 1 4C + C 3 190
So that K = Xr 6B + Xr Sustituting the value of K in PTC we have PTC = 1 Y(n) + Q W(n) Q Where Y(n) = 6D A + S + F n... (9.9) W(n) = 6B D D r X + Z (n1) 1 6B + Xr... (9.30) Where Z = (C v1 + 4C v + C v3 ) 9.7.. To find the optimal value of Q Equating ~ P TC Q to 0 we have 1 1 Y(n) = W(n) Q Q = Y(n) W(n) ; Q* = Y(n) W(n) Q * = S + F 1DA + n 6B D D rx + Z (n1) 1 6B + Xr P P... (9.31) 191
9.7.3. To find the optimal value of n Sustituting the values of Q *, n *, K in the expression for the total cost, the terms which depend on the value of n after algeraic calculations are written as V G(n) = nu + n... (9.3) where U = D 6AZ1 P... (9.33) V = 6B D 6(S + F)X Z 1 6B + Xr P... (9.34) Differentiating G(n) with respect to n and equating to zero we have D 6AZ1 = 1 6B D X + Z 1 P n 6B + Xr P n = 6B D S + FX + Z 1 6B + Xr P D AZ1 P n * = 6B D S + FX + Z 1 6B + Xr P D AZ1 P... (9.35) The total cost can e calculated. 19
9.8. Numerical Example Let D = 1980 units/year P = 3960/year r = 0./unit/year C V = (19, 0, 1) C = (7, 8, 9) B = Rs.45/unit A = Rs.5 per order S = Rs.400/setup F = Rs.100 per shipment X = C 1 4C + C 3 = 168 Z = 10 Step 1 : K = U = V= Xr 6B + Xr =.1106 D 6AZ1 = 9000 P 6B D 6SX Z 1 rx + 6B P = 3765059.8 > 0 Step : n * = V U = 0.45 Take n = 0 as n should e integer Step 3 : Q * = 145.36 TC = Rs.6538.68 193
9.9. Sensitivity Analysis D P r CV CB B A S F K N NOPT Q TC 1980 3960 0. 0 8 45 5 400 100 0.11 1.0 1 139.16 6537.99 1980 3960 0. 0 8 45 5 400 10 0.11 1.5 1 139.45 6551.53 1980 3960 0. 0 8 45 5 400 15 0.11 1.6 1 139.5 6554.91 1980 3960 0. 0 8 45 5 400 140 0.11 1.30 1 139.74 6565.03 1980 3960 0. 5 8 45 5 400 100 0.11 19.0 19 137.71 76.96 1980 3960 0.3 0 8 45 5 400 100 0.16 0.65 0 119.03 7984.46 1980 3960 0. 0 30 45 5 400 100 0.1 1.84 1 138.70 6559.79 1980 3960 0. 0 8 50 5 400 100 0.10 1.3 1 139.08 6541.87 1980 3960 0. 0 8 45 30 400 100 0.11 19.40 19 153.67 6604.95 1980 3960 0. 0 8 45 5 4300 100 0.11 1.44 1 140.60 6605.39 1980 3960 0.3 5 8 45 5 400 100 0.16 18.53 18 118.37 888.06 000 3980 0. 0 8 45 5 400 100 0.11.39 134.45 6558.54 For oth crisp and fuzzy models, If we increase one parameter, keeping the other parameters fixed, the total cost increases. The optimal values of Q, n, Total cost remain the same for oth the models. 194