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30220130403002

5. 5. International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383 (Print), ISSN 0976 – 6391 (Online) Volume 4, Issue 3, September - December (2013), © IAEME 3.1 Supplier’s total profit per unit time The supplier manufactures nQ units in batches whereQ is defined in Appendix B and incurs a batch set-up cost As . The supplier’s set-up cost per unit time is As / ( nT ). FollowingJoglekar (1988), the supplier’s inventory holding cost per unit time is T T 1 {Cs ( I s + I sp ) [(n − 1)(1 − γ ) + γ ]} ∫ I (t ) dt. (See Appendix C for computation of ∫ I( t )dt ). T 0 0 ( ) The purchase cost of an item for the buyer is 1 − K j β v, whenaccount is settled at time M j ; where j = 1, 2; K1 = 1, K 2 = 0. Hence, for the permissible delay period, the opportunity 1 cost per unit time is (1 − K j β )vI sp M j Q; where j = 1, 2; K1 = 1, K 2 = 0. When the buyer T pays at time M1 , the supplier can use the revenue (1 − β ) v to shrinka cash flow crisis during time M 2 − M1 . This timely payment acquires gain at the cash flexibility rate per unit time and is given by 1 ( 1 − β )vf sc ( M 2 − M1 )Q. Hence, the supplier’s total profit per unit time is, T sales revenueplus the interest earned on the timely payment, minus total cost which is sum of the manufacturing cost, set-up cost, inventory holding cost and opportunity cost,is given by TSPj ( n ) = ( 1 − K j β )vQ − T T C Q A 1 − s − s − Cs ( I s + I sp )[( n − 1 )( 1 − γ ) + γ ] ∫ I( t )dt T nT T 0 ( 1 − K j β )vI sp M j Q T { + } ( 1 − β )vf sc ( M 2 − M1 )Q T j = 1, 2; K1 = 1, K 2 = 0 (1) 3.2Buyer’s total profit per unit time A The ordering cost per unit time is b for each transfer of Q units. The buyer’s purchase cost per unit time is T ( 1 − K j β )vQ T and inventory holding cost per unit time is T (1 − K j β )vI b ∫ I (t ) dt 0 ; where j = 1, 2; K1 = 1, K 2 = 0. T On the basis of choice of payment time of the buyer two cases may arise. 1. T < M j 2. T ≥ M j ; j = 1, 2 . Case 1: T < M j (Fig.2) 18
6. 6. International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383 (IJPTM), (Print), ISSN 0976 – 6391 (Online) Volume 4, Issue 3, September - December (2013), © IAEME Fig. 2: Interest earned when T < M j ; j =1,2 In this case, the buyer’s stock level depletes to zero before the permissible delay period. So, the opportunity cost for the buyer is zero. The interest earned on the generated revenue per T  PIbe  ∫ t ⋅ R( P, t ) dt + Q ( M j − T )  0  ; j = 1, 2. (See Appendix D for unit time is given by T T ∫ t ⋅ R( P,t )dt ). Hence, buyer’s total profit per unit timeis 0 T ( 1 − K j β )vI b ∫ I( t )dt PQ ( 1 − K j β )vQ Ab − − − T T T T  PIbe  ∫ t ⋅ R( P,t )dt + Q ( M j − T ) Q( 0  + T TBPj1 ( P,T ) = j = 1, 2; K1 = 1, K 2 = 0 (2) Case 2: T ≥ M j ; j = 1,2 (Fig.3) 19 0 T
7. 7. International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383 (Print), ISSN 0976 – 6391 (Online) Volume 4, Issue 3, September - December (2013), © IAEME Fig. 3 Interestearned and charged when T ≥ M j ; j = 1, 2 In this case, the buyer’s permissible payment time offered by the supplierovers on or before the cycle time. The interest earned per unit time by the buyer at the rate Ibe during 0, M j  ; j = 1,2 is    Mj 1 PIbe ∫ t ⋅ R1 ( P, t ) dt ; 0 ≤ M j ≤ u1 T 0    Mj Mj   1 1  u1  PIbe ∫ t ⋅ R( P, t ) dt =  PIbe  ∫ t ⋅ R1 ( P, t ) dt + ∫ t ⋅ R2 ( P, t ) dt  ; u1 ≤ M j ≤ u2 T u1 0 T 0       Mj u u    1 PI  1 t ⋅ R ( P, t ) dt + 2 t ⋅ R ( P, t ) dt + ∫ ∫ t ⋅ R3 ( P, t ) dt  ; u2 ≤ M j ≤ T ∫ 1 2  T be u1 u2    0   ;where j = 1, 2 and interest paid per unit time at the rate Ibc during  M j , T  ; j = 1,2 is   20
8. 8. International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383 (Print), ISSN 0976 – 6391 (Online) Volume 4, Issue 3, September - December (2013), © IAEME   u1  u2 T  1 (1 − K β )vI  ∫ I (t ) dt + ∫ I (t ) dt + ∫ I (t ) dt  ; M ≤ u ≤ T  j bc  j 1 2 3 1 T u1 u2 M j        T T 1 1   u2  (1 − K j β )vI bc ∫ I (t ) dt =  (1 − K j β )vI bc  ∫ I 2 (t ) dt + ∫ I3 (t ) dt  ; M j ≤ u2 ≤ T T Mj u2 T M j     T 1 ; u2 ≤ M j ≤ T  (1 − K j β )vIbc ∫ I 3 (t ) dt T Mj   j = 1, 2; K1 = 1, K 2 = 0 Therefore, total profit of buyer per unit time is ( ( ( ) ) ) TBPj 2 ( P, T ) : 0 ≤ M j ≤ u1   TBPj 2 ( P, T ) = TBPj 2 ( P, T ) : u1 ≤ M j ≤ u2  TBPj 2 ( P, T ) : u2 ≤ M j ≤ T ; j =1,2  (See Appendix ( ( Efor TBPj 2 ( P, T ) : 0 ≤ M j ≤ u1 (3) ) , ( TBPj 2 ( P,T ): u1 ≤ M j ≤ u2 ) , ) TBPj 2 ( P, T ): u2 ≤ M j ≤ T ; j = 1,2 ). The buyer’s total profit per unit time is TBPj1 ( P, T ) ; T < M j  TBPj ( P, T ) =  TBPj 2 ( P, T ) ; T ≥ M j  (4) 3.3 Joint total profit per unit time The joint profit per unit time of integrated system is given by π j1 ( n, P, T ) ; T < M j  π j ( n, P, T ) =  π j 2 ( n, P, T ) ; T ≥ M j ; j = 1, 2  ;where (5) π j1 (n, P, T ) = TSPj (n) + TBPj1 ( P, T ) π j 2 ( n, P, T ) = TSPj ( n) + TBPj 2 ( P, T ); j = 1, 2 The objective is to decide optimal values of discrete variable n and continuous variables P and T , which maximize π j ( n,P,T ) , j = 1, 2 .We use following steps to maximize the joint profit of the supply chain. 21
9. 9. International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383 (Print), ISSN 0976 – 6391 (Online) Volume 4, Issue 3, September - December (2013), © IAEME 4. COMPUTATIONAL PROCEDURE To maximize joint profit, execute following steps: Step 1: Assign parametric values in proper units to all model parameters. Step 2: Set n = 1 . Step 3: Solve ∂π j ∂P = 0 and ∂π j = 0 , j = 1, 2 simultaneously for P and T . ∂T Step 4: Increment n by 1. Step 5: Continue steps 3 and 4 until π j ( n − 1, P ( n − 1) , T ( n − 1) ) ≤ π j ( n, P, T ) ≥ π j ( n + 1, P ( n + 1) , T ( n + 1) ) ; j = 1, 2 is satisfied. Step 6: Stop. The optimal value of ( n, P , T ) determines the optimal purchase quantity Q (Appendix B) pertransfer for the buyer. 5. NUMERICAL EXAMPLE Let us illustrate the developed model with the following numerical values to model parameters. a = 1,00,000, b1 = 7%, b2 = 5%, η = 1.5, u1 = 15 days, u2 = 45 days, γ = 0.9, C s = \$ 2/unit, v = \$ 4.5/unit, As = \$ 1000/set-up, Ab = \$ 300/order, Is = 5% /unit/year, Ib = 8% /unit/year, f sc = 17% /\$/year and I sp = 9% /\$/year, I bc = 16%/\$/year, I be = 12% /\$/year and θ = 0.12. The supplier offers buyerthe credit term ‘3/10 net 30’ means if buyer pays by 10 days then he will be offered 3% discount in the unit purchase price otherwise the buyer has to settle the account due against purchases in 30 days. From Table 1, we see that for 10-shipments, the buyer’s selling price is \$ 6.59/unit and cycle time is122 days maximizing joint total profit of \$ 25319 of the integrated system. The corresponding profit of the supplier is \$ 13507 and that of buyer is \$ 11812.Each transfer is of 2018 units. Optimal payment time is 10 days in ‘3/10net 30’credit terms. The concavity of joint total profit with respect to number of transfers, n and retail sale price, P are shown in figures 3and 4 respectively. 3-D plot given in figure 5 for n = 10establishes the convavity of the total joint profit. The variations in permissible delay periods; M 1 and M 2 are worked out to study the changes in decisionvariable and total joint profit in Table 1. The profit gain is compared with benchmark of no credit period. 22
10. 10. International Journal of Production Technology and Management (IJPTM), ISSN 0976 – 6383 (Print), ISSN 0976 – 6391 (Online) Volume 4, Issue 3, September - December (2013), © IAEME Fig.4: Concavity of Joint Profit w.r.t. no. of Shipments (n) Fig. 5: Concavity of Joint Profit w.r.t. Retail Price (P) Fig. 6 Concavity of joint profit w.r.t. cycle time and retail price 23