An Integrated Inventory Model with Geometric Shipment Policy and Trade-Credit Financing under Stochastic Lead Time

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1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu An Integrated Inventory Model with Geometric Shipment Policy and Trade-Credit Financing under Stochastic Lead Time S. Hemapriya 1 and R. Uthayakumar 1, The Gandhigram Rural Institute - Deemed University, Gandhigram. 1 hemapriya9194@gmail.com uthayagri@gmail.com Abstract This article explores a single-vendor and a single-buyer integrated production inventory system with main concern is on the trade-credit policy and lead time reduction. A geometric shipment policy is adopted to deliver the items to the buyer. In particular, the supplier offers a credit period that is less than the average duration of the inventory model. The objective of this study is to minimize the Joint Economic Total Cost (JETC) by simultaneously optimizing the order quantity, lead time and safety factor. An iterative algorithm of finding the optimal solution is developed and numerical examples are given to illustrate the results. AMS Subject Classification: 90B05. Key Words and Phrases: Integrated inventory model, Geometric shipment policy, Lead time reduction. 1 Introduction To succeed in the competitive business environment and to optimize performance, integrated policy is a new paradigm adopted by business organizations. Many organizations have begun to embrace the integrated policy as it enables shorter lead time and lower inventory cost. Thus, the integrated policy has drawn the attention of several researchers to study the effect of integration between a vendor and a buyer. [1 studied a model in which the shipment size increases geometrically. Hill developed the [1 model where the geometric growth rate in shipments size is a decision variable. Later, several researchers [3, [4 and [5 have developed integrated 01

2 production-distribution inventory models by extending the idea of [ and incorporating various realistic assumptions. But it is not always suitable in a realistic environment. In real life business via share marketing, trade credit financing becomes a influential tool to improve sales and reduce on-hand stock. This permissible delay in payments reduces the cost of holding stock since it reduces the amount of capital invested in stock for the duration of the permissible period. During the delay period the buyer can accumulate revenue on sales and earn interest on that revenue via share market investment or banking business. It makes economic sense for the distributor to delay period allowed by the producer. Wu established the continuous review inventory model with permissible delay in payments. [10 developed supply chain model with stochastic lead time, trade-credit financing and transportation discounts. [6 revealed that lead time could be reduced by investing an additional amount on hi-tech equipment, information technology, order expedite and logistics. The first attempt to formulate a probabilistic inventory model with lead time as decision variable was made by [7. By treating the lead time as a decision variable, [9 generalized [1 model and found that the joint total expected cost and lead time are less compared to that of [1 model. Researchers [8 have worked on lead time reduction by formulating integrated production-inventory models in a single-vendor and a single-buyer supply chain environment. In the aforesaid literature it is not clear that most of the integrated inventory models were developed under the assumption that the products are delivered in equal sized shipments. Therefore, the efforts has been taken by the authors to adopt a geometric shipment policy so as to fill this remarkable gap in the inventory literature. Here an integrated inventory model with geometric shipment policy and trade-credit financing under stochastic lead time is discussed. Notations and assumptions.1 Notations We need the following assumptions and notations, to develop the mathematical model of this model. The following terminology is used: Q Vendor s production batch size q 1 Size of the first shipment from the vendor to the buyer q i Size of the i th shipment D Average demand per year P Production rate of the buyer P > D A The ordering cost of the buyer per order S Vendor s setup cost per setup λ Geometric growth factor h b The holding cost rate of the buyer per unit per unit time h v The holding cost rate of the vendor per unit per unit time n Number of lots in which the item are delivered from the vendor to the buyer L Length of lead time β Fraction of the shortage that will be backordered at the buyer s end, 0 β < 1 π 0 Marginal profit per unit 0

3 π x Price discount offered on backorder by the vendor per unit, 0 π x π 0 R Reorder point of the buyer k Safety factor a Fixed component of the transportation cost, ($/shipment) b Variable component of the transportation cost, ($/unit item/shipment) R(L) Lead time crashing cost i b Buyer s interest or opportunity cost in annual percentage i s Vendor s interest or opportunity cost in annual percentage N Length of credit-period per unit time pc Purchasing cost per unit time. Assumptions The fundamental assumptions used in developing the model are as follows: 1. The system deals with a single-vendor and a single-buyer.. The reorder point R equals the sum of the expected demand during lead time and safety stock (SS) and SS=k standard deviation of lead time demand. (i.e)., R = µl + kσ L where k is the safety factor. 3. Replenishments are made when the on hand inventory reaches the reorder point R. 4. The buyer orders a lot size of Q units and the vendor produces them with a finite production rate P in units per unit time in one setup. Produced items are supplied to the buyer in n unequal sized shipments. 5. The trade-credit financing is used to make it a cost-reduced supply chain. 6. The transportation cost is a + bq where a is the fixed cost per shipment, and b is the unit transportation cost per item. 3 Model Development Production and delivery are arranged as follows: The vendor produces Q units at one set-up with a finite production rate P (P > D). Consider the production batch of size Q which is made up of n unequal-sized shipments which are delivered to the buyer and the size of the i t h shipment within a batch is q i = βq i 1 = β i 1 q 1, i =, 3,..., n and represents the lot size of the first shipment. Therefore, the total batch production lot size is, Q = n q i = i=1 n i=1 λ i 1 q 1 = q 1(λ n 1) (λ 1) and the cycle length of the inventory is T = Q D = q 1(λ n 1) D(λ 1) 03

4 3.1 Buyer s Perspective The buyer receives a lot of q i units in each shipment which could be utilized during the time period q i. So the average inventory of the buyer during the time period q i D D is q i which gives the time-weighted inventory during a complete production cycle as, n i=1 q i D = n i=1 Therefore, the average inventory of the buyer is, (λ i 1 q 1 ) D = q 1(λ n 1) D(λ 1) q1(λ n 1) D(λ 1) /q 1(λ n 1) D(λ 1) = q 1(λ n + 1) (λ + 1) The buyer s expected annual total cost per unit time is the sum of the ordering cost, holding cost, transportation cost, stock-out cost and the lead time crashing cost and is given by, ET C b (q 1, k, L, n) = [ A + (π x β + π 0 (1 β))σ nd(λ 1) Lψ(k) + F + R(L) q 1 (λ n 1) [ q1 (λ n + 1) h b (λ + 1) + kσ L + (1 β)σ Lψ(k) (1) 3. Vendor s Perspective The buyer order a lot size of Q units and the vendor produces them at the rate of P > D and supplies them in n unequal-sized shipments. When the production process is about to start, the vendor s inventory level is zero and there are Dq 1 units P in the buyer s inventory which is just first shipment arrives. Then the vendor s total inventory level increases at a rate of P D units and it reaches the maximum height of, Dq 1 + (P D) q 1(λ n 1). P P (λ 1) The vendor s expected annual total cost per unit time is the sum of setup cost and the holding cost is given by, ET C v (q 1, n) = [ SD(λ 1) q 1 (λ n 1) + h Dq1 v P + q 1(P D)(λ n 1) q 1(λ n + 1) () P (λ 1) (λ + 1) 4 Integrated Approach The objective of this section is to determine the optimal values of the decision variables by minimizing the expected total cost per unit time and the buyer s expected annual total cost per unit time. Therefore, the joint expected total cost per unit 04

5 time of the integrated system is, JET C(q 1, k, L, n) = ET C b (q 1, k, L, n) + ET C v (q 1, n) [( = A + S ) + (π x β + (1 β)π 0 )σ Lψ(k) + F + R(L) n [ nd(λ 1) q 1 (λ n 1) + h q1 (λ n + 1) b (λ + 1) + kσ L + (1 β)e(x R) + [ Dq1 +h v P + q 1(P D)(λ n 1) q 1(λ n + 1) P (λ 1) (λ + 1) (3) To make the profitable supply chain an attempt of trade-credit policy is used. During the credit period, the buyer saves his/her total interest by using the trade-credit policy which is similar to [10. Thus the trade-credit cost is for the buyer which is offered by the vendor is defined as, pc(q DN) i s D D pcn i b D pcni b(1 β) Dβ Therefore, the trade-credit cost for the buyer per unit time is, pc( q 1(λ n 1) DN) λ 1 D pcn i b (λ 1) pcni b(1 β)(λ 1) q 1 (λ n 1) q 1 (λ n 1) βq 1 (λ n 1) Therefore, the above problem reduces to, JET C(q 1, k, L, n) = ET C b (q 1, k, L, n) + ET C v (q 1, n) [( = A + S ) + (π x β + (1 β)π 0 )σ Lψ(k) + F + R(L) n [ nd(λ 1) q 1 (λ n 1) + h q1 (λ n + 1) b (λ + 1) + kσ L + (1 β)e(x R) + [ Dq1 +h v P + q 1(P D)(λ n 1) q 1(λ n + 1) P (λ 1) (λ + 1) D pcn i b (λ 1) q 1 (λ n 1) pcni b(1 β)(λ 1) βq 1 (λ n 1) + pc( q 1(λ n 1) λ 1 DN) q 1 (λ n 1) At first for fixed (q 1, k, n), JET C(q 1, k, L, n) is a strictly concave in L due to the fact that, [ L JET C(q 1, k, S, θ, n) = (π x β + π 0 (1 β )) σψ(k) nd(β 1) L c i (β n 1) + h b L [kσ + (1 β )σψ(k) L JET C(q 1, k, S, θ, n) = 1 [(π L 3 x β + π 0 (1 β ))σψ(k) h b [kσ + (1 β )σψ(k) < 0 nd(β 1) (β n 1) (4) 05

6 Thus, JET C(q 1, k, L, n) is concave in L [L i, L i 1. Therefore, for fixed (q 1, k, n), the minimum value of JET C(q 1, k, L, n) on L lies at the end point of the interval [L i, L i 1. Secondly, if we relax the integer constraint on n, it is possible to note that JET C(q 1, k, L, n) is strictly convex in n, for fixed (q 1, k, L), L [L i.l i 1 and noticing that, n JET C(q 1, k, L, n) = ( ) [ [ Y D(λ 1) (λ n 1) n log λe n log λ SD(λ 1) log λe n log λ q 1 (λ 1) q 1 (λ n 1) [ q 1 log λ +(h B h v ) log λen + h vq 1 (P D) log λe n log λ (λ + 1) P (λ 1) + pci sq 1 log λe n log λ (λ 1) + pcd N i s (λ 1) q 1 (n log λ n log λ + (λ n 1)) n JET C(q 1, k, L, n) = where Y = M log λλn (λ n 1) 3 {λn nλ n SD(λ 1)(log λ) log λ} + (λ n + 1) q 1 (λ n 1) 3 + (h b h v )q 1 (log λ) λ n + h vq 1 (P D)(log λ) λ n (λ + 1) P (λ 1) pci s q 1 (log λ) λ n pcd N i s (λ 1) + (λ 1) q 1 ( log λλ n + n(log λ) λ n ) > 0 and M = [ (A + S n) + (πx β + (1 β)π 0 )σ Lψ(k) + F + R(L) ( ) Y D(λ 1) q 1 The above equation proposes that the Joint expected total cost is strictly convex in n for fixed (q 1, k, L),[L i, L i 1 and with the aid of taking account of equation 4. So we want to able to take the derivatives of JET C(q 1, k, L, n) with respect to q 1, and to attain, q 1 JET C(q 1, k, L, n) = 1 q 1 q 1 = [ ( A + S ) n + (π x β + (1 β)π 0 )σ Lψ(k) + a + R(L) nd(λ 1) + pcdn n (i s i b ) pcni b(1 β) βnd +h v [ D P + (P D)(λn 1) P (λ 1) (λn + 1) (λ + 1) Solving the above equation 5, we bear the value of q 1 as, [ [ ( ) A + S n + (πx β + (1 β)π 0 )σ Lψ(k) + a + R(L) + pcdn n (h b h v ) ( (λ n +1) (λ+1) ) [ D + h v + (P D)(λn 1) P P (λ 1) (λ n 1) + h b [ (λ n + 1) (λ + 1) + pci s(λ n 1) (λ 1) (i s i b ) pcni b(1 β) βnd + pcis(λn 1) (λ 1) Similarly for fixed (q 1, k, L),L [L i, L i 1 and with the aid of taking account of equation 4, so we want to able to take the derivatives of JET C(q 1, k, L, n) with respect to k and to attain, k JET C(q 1, k, L, n) = σ L [ ( ) nd(λ 1) (π x β + π 0 (1 β)) q 1 (λ n 1) + (1 β)h b (Φ(k) 1) + h b (7) (5) nd(λ 1) (λ n 1) 1 (6 06

7 Solving the above equation 7, we get the value of k and to attain, Φ(k) = 1 h b q 1 (λ n 1) (π x β + π 0 (1 β))nd(λ 1) + (1 β)h b q 1 (λ n 1) Moreover, it can be shown that the SOSC are satisfied since the Hessian matrix is positive definite at point (q 1, k) (see the appendix for the proof). Algorithm Step 1: Set n = 1. Step : Find Q and k for fixed n. Step 3: For every L i, i = 1,,..., n perform steps (3.1) to (3.3). 3.1 Set k (1) i = 0[implies ψ(k i ) (1) = , Φ(k i ) (1) = Substitute k (1) i = 0 and in equation 6 and evaluate q (1) i Utilizing q (1) i1 to determine the values of k () i from equation Repeat (3.1) to (3.3) until no change occurs in the values of q 1i and k i. Denote the solutions by (q i1, k i ). Step 4: Use equation 4 find the corresponding joint expected total cost JET C(q i1, k i, L, n) for i = 1,,..., n. Step 5: Find MinJET C(qi1, ki, L, n) for i = 1,,..., n and denote it by JET C(q(n)1, k (n), L, n)=min JET C(qi1, ki, L, n) and (q(n)1, k (n), L, n) are the optimal solutions for the given n. Step 6: Replace n by n+1 and repeat steps () to (5) to get JET C(q(n)1, k (n), L, n). Step 7: If JET C(q(n)1, k (n), L, n) JET C(q (n 1)1, k (n 1), L, n) then go to step 6 otherwise go to step 8. Step 8: Set (q 1, k, L, n) = (q (n 1)1, k (n 1), L, n) and JET C(q 1, k, L, n) is the optimal solution. 5 Numerical Results In this section, to illustrate the above solution procedure we consider the adaptive parameters similar to that proposed by Lin. The optimal values of the decision variables are obtained by using the solution procedure and the MATLAB software. D =00 units/year, P =1000 units/year, A =$00/order, S =$1500/setup, a =$5/unit, b =$10/unit, h b =$5/unit, h v =$0/unit, =7, pi x =$100/unit, pi 0 =$300/unit, s =$75/unit, α =0.5, N =3, pc =$/unit, i b =0.06, i s =0.0. We solve the cases for which the geometric growth factor λ = 1.5,, and.5. For the given 3 values of λ, we use the proposed algorithm to find the optimal solution of the model and the optimal solutions are summarized in??.?? shows that when L =4 weeks and λ =.5 the ordered units could be supplied in third shipment with an initial lot size of 35 units and the joint expected total cost reduced to $4363$. (8) 07

8 Table 1: Comparison of the previous model vs. proposed model λ L n q 1 k F JET C Conclusion In this paper, we presented an integrated inventory model with geometric shipment policy and trade-credit financing under stochastic lead time. We then developed an exact algorithm that authorizes the optimization of ordering quantity, safety factor, transportation cost and lead time. Numerical example conferred that this optimization approach achieves a high level of efficiency, which may offer promising in practice. The model can be extended to embody price discount, various reduction factors etc. Acknowledgment The first author research work is supported by DST INSPIRE Fellowship, Ministry of Science and Technology, Government of India under the grant no. DST/INSPIRE/03/016/00457 and UGC-SAP, Department of Mathematics, The Gandhigram Rural Institute Deemed University, Gandhigram 6430, Tamilnadu, India. Appendix For a given value of L [L i, L i 1, we first obtain the Hessian matrix H as follows: H = [ JET C(q 1,k,L,n) q1 JET C(q 1,k,L,n) kq 1 For the first minor, one can easily obtain as, JET C(q 1,k,L,n) q 1 k JET C(q 1,k,L,n) q1 [ H 11 = det JET C(q q1 1, k, L, n) = q 3 1 [ ( A + S ) + (π x β + (1 β)π 0 )σ Lψ(k) n +a + R(L) + pcdn n (i s i b ) pcni b(1 β) βnd nd(β 1) (β n 1) Therefore, H 11 > 0 For the second minor, one can easily obtain easily as, H = det [ JET C(q 1,k,S,θ,L,n) q1 JET C(q 1,k,S,θ,L,n) kq 1 JET C(q 1,k,S,θ,L,n) q 1 k JET C(q 1,k,S,θ,L,n) k 08

9 ( [ ( = A + S ) + (π q1 3 x β + (1 β)π 0 )σ Lψ(k) + R(L) + a + pcdn n )( nd(β 1) σ [ ( L (β n 1) n (i s i b ) pcni b(1 β) βnd ) ) nd(β 1) (π x β + π 0 (1 β)) q 1 (β n 1) + (1 β)h b φ(k) ( [ nd(β 1) + (π q (β n x β + (1 β )π 0 )σ L(Φ(k) 1)) > 0 1) Since φ(k) > 0, ψ(k) > 0 and φ(k)ψ(k) [Φ(k) 1 > 0 for all k > 0. Therefore, H > 0. From the above derivations, all the principal minors of the Hessian matrix is positive. Hence the given Hessian Matrix H is positive definite at (q 1, k). References [1 S.K. Goyal, A one-vendor multi-buyer integrated inventory model: A comment. European Journal of Operations Research, 8, (1995), [ S.K. Goyal, A joint economic-lot-size model for purchaser and vendor: A comment. Decision Sciences, 19, (1988), [3 M.A. Hoque, S.K. Goyal, A heuristic solution procedure for an integrated inventory system under controllable lead-time with equal or unequal sized batch shipments between a vendor and a buyer, International Journal of Production Economics, 10, (006), [4 A. Pandey, M. Masin, V. Prabhu, Adaptive logistic controller for integrated design of distributed supply chains, Journal of Manufacturing Systems, 6, (007), [5 J.T. Teng, L.E. Cardenas-Barron, K.R. Lou, The economic lot size of the integrated vendor-buyer inventory system derived without derivatives: a simple derivation, Applied Mathematics and Computation, 1, (011), [6 S.L. Hsu, C.C. Lee, Replenishment and lead time decisions in manufacturerretailer chains, Transportation Research Part E: Logistics and Transportation Review, 45, (009), [7 C.J. Liao, C.H. Shyu, An analytical determination of lead time with normal demand, International Journal of Operations and Production Management, 11, (1991), [8 L.Y. Ouyang, K.S. Wu and C.H. Ho, An integrated vendorbuyer inventory model with quality improvement and lead time reduction, International Journal of Production Economics 108, (007), [9 J.C.H. Pan, J.S. Yang, A study of an integrated inventory with controllable lead time, 40, (00),

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