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- 1. 1 KING FAHD UNIVERSITY OF PETROLEUM & MINERALS INDUSTRIAL & SYSTEMS ENGINEERING DEPARTMENT Report on: CONSIGNMENT AND VENDOR MANAGED INVENTORY IN SINGLE-VENDOR MULTIPLE BUYERS SUPPLY CHAINS Mohamed Ben-Daya, Elkafi Hassini, Moncer Hariga, and Mohammad M. AlDurgam International Journal of Production Research 51(2013), 1347-1365. By Mohammed Al-Marhoun Term 131 1. Objective
- 2. 2 Study the values of using a hybrid Vendor Managed Inventory and Consignment (VMI&CS) partnership inventory program. Through finding the number of shipments and replenishment cycle time of each batch such that the joint manufacturer and retailer cost is minimized 2. Introduction Vendor-buyer partnerships: i. The vendor and the buyers act independently. ii. The vendor enters in a VMI&CS partnership with the buyers. iii. The vendor and the buyer belong to a vertically integrated firm where a single decision maker decides about the ordering policy. Consignment (CS): The process of a supplier placing goods at a customer location without receiving payment until after the goods are used or sold. Vendor Managed Inventory (VMI): The vendor is responsible for managing the inventory for the buyer, including initiating orders on behalf of the buyer. The vendor in return gets more visibility about the product’s demand. Share of cost and decisions in a supply chain under VMI, CS and VMI&CS inventory management programs
- 3. 3 Decision Cost Order quantity Number of shipments Ordering Holding VMI Vendor Vendor Shared Buyer CS Buyer Buyer Buyer Shared VMI&CS Vendor Vendor Shared Shared 3. Notation A bpi : the cost of placing an order by the i th buyer ($/order) A bri : the cost of receiving a shipment by the i th buyer ($/order) A bi : i th buyer’s total ordering cost composed of the cost of placing an order and the cost of receiving a shipment (A bi = A bpi + A bri ) ($/order) h boi : i th buyer’s opportunity cost of holding one unit in stock for one unit of time ($/unit/unit time) H bsi : i th buyer’s physical storage cost for one unit of stock held for one unit of time ($/unit/unit time) h bi : i th buyer’s total holding cost per unit of stock per unit of time (h bi = h boi + h bsi ) A vs : Vendor’s setup cost ($/order) A vri : Vendor’s shipment release cost to the i th buyer ($/order) h v : Vendor’s total cost of holding one unit in stock for one unit of time ($/unit/unit time)
- 4. 4 c : Unit purchase price paid by the buyers ($/unit). n : equal number of shipment that is sent to buyers during a cycle N : Number of buyers di : demand from buyer i (units) D : total demand of buyers = ∑ di N i=1 (units) P : vendor’s production rate (units/unit time) Decision and consequence variables q i : shipment size for buyer i Q : Total shipments sizes to all buyers = ∑ 𝑞𝑖 𝑁 𝑖=1 T : replenishment cycle length TC k s : Total cost for supply chain party k, where k = v (vendor) and k = bi (buyer i) under system s=1 (no partnership), s = 2 (Vendor managed inventory and consignment) and s = 3 (centralized) TC s : Total cost for system s=1 (no partnership), s = 2 (Vendor-managed inventory and consignment) and s = 3 (centralized). 4. Assumptions
- 5. 5 1. Share of ordering and holding costs in the different supply chain scenarios Supply Chain Structure Supply Chain Partner Independent parties VMI&CS Centralized Costs Costs Costs Ordering Holding Ordering Holding Ordering Holding Vendor Avs Avri hv Abpi Avs Avri hboi hv Abi Avs Avri hbi hv Buyer Abi hbi Abri hbsi 2. The shipments to the buyers are time-phased and their sizes are not proportional to the buyer’s demand (equal shipments). 3. A cyclic delivery policy where the shipment is sent to each buyer and then repeat this cycle until all shipments are delivered.
- 6. 6 Figure 1. Cyclic delivery policy 5. Models 1.1 No partnership No coordination between the vendor and buyers and all parties act independently and attempt to optimize their own cost without taking into consideration the decision of the other parties. Buyer: 𝑇𝐶 𝑏𝑖 1 = 𝐴 𝑏𝑖 𝑇𝑏𝑖 + ℎ 𝑏𝑖 𝑞𝑖 1 2 = 𝐴 𝑏𝑖 𝑑𝑖 𝑞𝑖 1 + ℎ 𝑏𝑖 𝑞𝑖 1 2
- 7. 7 Figure 2. Buyer inventory cycle in no partnership policy 𝜕 𝜕𝑞𝑖 1 𝑇𝐶 𝑏𝑖 1 = 0 𝑞𝑖 1 = √ 2𝐴 𝑏𝑖 𝑑𝑖 ℎ 𝑏𝑖 , 𝑖 = 1,2, … , 𝑁 𝜕 𝜕𝑇𝑏𝑖 1 𝑇𝐶 𝑏𝑖 1 = 0 𝑇𝑏𝑖 1 = √ 2𝐴 𝑏𝑖 𝑑𝑖ℎ 𝑏𝑖 , 𝑖 = 1, 2, …, 𝑁 Substituting by the value of qi 1 and Tbi 1 into TCbi 1 𝑇𝐶 𝑏𝑖 1 = √2𝐴 𝑏𝑖ℎ 𝑏𝑖 𝑑𝑖 , 𝑖 = 1,2, … , 𝑁 Vendor: 𝑇𝐶 𝑣 1 = 𝐴 𝑣𝑠 𝑇𝑣 1 + ℎ 𝑣 𝐷𝑇𝑣 1 2 (1 − 𝐷 𝑃 ) + ∑ 𝐴 𝑣𝑟𝑖 𝑇𝑏𝑖 1 𝑁 𝑖=1 + ℎ 𝑣 ∑ 𝑞𝑖 1 𝑁 𝑖=1 Figure 3. Vendor inventory cycle in no partnership policy 𝐷 = ∑ 𝑑𝑖 𝑁 𝑖=1
- 8. 8 𝜕 𝜕𝑄𝑣 1 𝑇𝐶 𝑣 1 = 0 𝑄𝑣 1 = √ 2𝐴 𝑣𝑠 𝐷 ℎ 𝑣 (1 − 𝐷 𝑃 ) 𝜕 𝜕𝑇𝑣 1 𝑇𝐶 𝑣 1 = 0 𝑇𝑣 1 = √ 2𝐴 𝑣𝑠 ℎ 𝑣 𝐷 (1 − 𝐷 𝑃 ) Substituting by the value of Qv 1, Tv 1and Tbi 1 into TCv 1 𝑇𝐶 𝑣 1 = √2𝐴 𝑣𝑠 𝐷ℎ 𝑣 (1 − 𝐷 𝑃 ) + ℎ 𝑣 ∑ 𝑞𝑖 1 𝑁 𝑖=1 + ∑ (𝐴 𝑣𝑟𝑖 √ ℎ 𝑏𝑖 𝑑𝑖 2𝐴 𝑏𝑖 ) 𝑁 𝑖=1 𝑇𝐶1 = 𝑇𝐶 𝑣 1 + ∑ 𝑇𝐶 𝑏𝑖 1 𝑁 𝑖=1 𝑇𝐶1 = √2𝐴 𝑣𝑠 𝐷ℎ 𝑣 (1 − 𝐷 𝑃 ) + ℎ 𝑣 ∑ 𝑞𝑖 1 𝑁 𝑖=1 + ∑ (𝐴 𝑣𝑟𝑖 √ ℎ 𝑏𝑖 𝑑𝑖 2𝐴 𝑏𝑖 + √2𝐴 𝑏𝑖 𝑑𝑖ℎ𝑖) 𝑁 𝑖=1 Example 1: P = 3200 items/year D = 1500 items/year d1 = 500 items/year d2 = 1000 items/year Avs = $400 per setup Avri = 0 per shipment Ab1 = $25 per order Ab2 = $75 per order hb1 = $5 per item per year hb2 = $5 per item per year hv = $4 per item per year Tv= 0.501 T1= 0.141 T2= 0.173 q1= 70.71 q2= 173.20 TCv= 2572.53 TC1= 353.55 TC2= 866.03 TC= 3792.11
- 9. 9 1.2 VMI&CS Partnership Figure 4. Inventory cycle in VMI&CS partnership policy Buyer average inventory: Figure 5. Triangle of type I and rectangle for average inventory 𝑅1 = ( 𝑄 𝑃 ) (𝑞𝑖 − 𝑄 𝑃 𝑑𝑖) = 𝑄 𝑃 𝑞𝑖 + 𝑄2 𝑃2 𝑑𝑖
- 10. 10 𝑇11 = 1 2 ( 𝑄 𝑃 )( 𝑄 𝑃 𝑑𝑖) = 1 2 𝑄2 𝑃2 𝑑𝑖 Figure 6. Triangle type II for average inventory 𝑇2 = 1 2 (𝑞𝑖 + ( 𝑛 − 1)(𝑞𝑖 − 𝑄 𝑃 𝑑𝑖)) ( 1 𝑑𝑖 (𝑞𝑖 + ( 𝑛 − 1)(𝑞𝑖 − 𝑄 𝑃 𝑑𝑖))) = 1 2 (𝑞𝑖 + ( 𝑛 − 1) 𝑞𝑖 − ( 𝑛 − 1) 𝑄 𝑃 𝑑𝑖 )( 𝑞𝑖 𝑑𝑖 + ( 𝑛 − 1) 𝑞𝑖 𝑑𝑖 − ( 𝑛 − 1) 𝑄 𝑃 𝑞𝑖 𝑑𝑖 ) = 1 2 (𝑛𝑞𝑖 − ( 𝑛 − 1) 𝑄 𝑃 𝑑𝑖) ( 𝑛𝑞𝑖 𝑑𝑖 − ( 𝑛 − 1) 𝑄 𝑃 ) = 1 2 ( 𝑛2 𝑞𝑖 2 𝑑𝑖 − 𝑛( 𝑛 − 1) 𝑞𝑖 𝑄 𝑃 − 𝑛( 𝑛 − 1) 𝑞𝑖 𝑄 𝑃 + ( 𝑛 − 1)2 𝑄2 𝑃2 𝑑𝑖) = 𝑛2 𝑞𝑖 2 2𝑑𝑖 − 𝑛( 𝑛 − 1) 𝑞𝑖 𝑄 𝑃 + ( 𝑛 − 1)2 2 𝑄2 𝑃2 𝑑𝑖 𝑇 = ( 𝑛 − 1) 𝑄 𝑃 + (𝑛 𝑞𝑖 𝑑𝑖 − ( 𝑛 − 1) 𝑄 𝑃 ) = 𝑛 𝑞𝑖 𝑑𝑖
- 11. 11 𝐼̅ = 1 𝑇 (( 𝑛 − 1) 𝑇11 + 𝑇12 + 𝑛( 𝑛 − 1) 2 𝑅1) = 1 𝑇 ( 𝑛 − 1 2 𝑄2 𝑃2 𝑑𝑖 + 𝑛2 𝑞𝑖 2 2𝑑𝑖 − 𝑛( 𝑛 − 1) 𝑄 𝑃 𝑞𝑖 + ( 𝑛 − 1)2 2 𝑄2 𝑃2 𝑑𝑖 + 𝑛( 𝑛 − 1) 2 𝑄 𝑃 𝑞𝑖 − 𝑛( 𝑛 − 1) 2 𝑄2 𝑃2 𝑑𝑖) = 1 𝑇 ( 𝑛2 𝑞𝑖 2 2𝑑𝑖 − 𝑛( 𝑛 − 1) 2 𝑄 𝑃 𝑞𝑖) Substitute by 𝑇 = 𝑛𝑞 𝑖 𝑑𝑖 = 𝑛𝑞𝑖 2 − 𝑛𝑄 2𝑃 𝑑𝑖 + 𝑄 𝑃 𝑑𝑖 = 𝑇𝑑𝑖 2 − 𝑛𝑇𝐷 2𝑃𝑛 𝑑𝑖 + 𝑇𝐷 𝑛𝑃 𝑑𝑖 = 𝑇 2 𝑑𝑖 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 ) Vendor : 𝑇𝐶 𝑣 2 = 𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴 𝑣𝑟𝑖 𝑁 𝑖=1 𝑇 + ℎ 𝑣 2𝑛 𝑇 ∑ 𝑑𝑖 2 𝑃 𝑁 𝑖=1 + 𝑛 ∑ 𝐴 𝑏𝑝𝑖 𝑁 𝑖=1 𝑇 + 𝑇 2 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 ) 𝑁 𝑖=1 𝜕 𝜕𝑇 𝑇𝐶 = 0
- 12. 12 − 𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴 𝑣𝑟𝑖 𝑁 𝑖=1 𝑇2 + ℎ 𝑣 2𝑛 ∑ 𝑑𝑖 2 𝑃 𝑁 𝑖=1 − 𝑛 ∑ 𝐴 𝑏𝑝𝑖 𝑁 𝑖=1 𝑇2 + 1 2 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 ) 𝑁 𝑖=1 = 0 1 𝑇2 (𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴 𝑣𝑟𝑖 𝑁 𝑖=1 + 𝑛 ∑ 𝐴 𝑏𝑝𝑖 𝑁 𝑖=1 ) = ℎ 𝑣 2𝑛 ∑ 𝑑𝑖 2 𝑃 𝑁 𝑖=1 + 1 2 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 ) 𝑁 𝑖=1 𝑇∗ = √ 2(𝐴 𝑣𝑠 + 𝑛 ∑ (𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖)𝑁 𝑖=1 ) ∑ ( ℎ 𝑣 𝑛𝑃 𝑑𝑖 2 + ℎ 𝑏𝑜𝑖 𝑑𝑖 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 ))𝑁 𝑖=1 𝑇𝐶 𝑣 2 (𝑛) = √2 (𝐴 𝑣𝑠 + 𝑛 ∑(𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖) 𝑁 𝑖=1 )∑ ( ℎ 𝑣 𝑛𝑃 𝑑𝑖 2 + ℎ 𝑏𝑜𝑖 𝑑𝑖 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 )) 𝑁 𝑖=1 Minimizing 𝑇𝐶 𝑣 2 (𝑛) Means minimizing (𝐴 𝑣𝑠 + 𝑛 ∑(𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖) 𝑁 𝑖=1 ) ∑ ( ℎ 𝑣 𝑛𝑃 𝑑𝑖 2 + ℎ 𝑏𝑜𝑖 𝑑𝑖 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 )) 𝑁 𝑖=1 Which equivalent to minimizing 𝐴 𝑣𝑠 𝑛𝑃 ∑ (ℎ 𝑣 𝑑𝑖 2 + ℎ 𝑏𝑜𝑖 𝑑𝑖 𝐷) 𝑁 𝑖=1 + 𝑛 (1 − 𝐷 𝑃 )∑(𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖) 𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1 Applying the first difference approach 𝑇𝐶 𝑣 2( 𝑛) < 𝑇𝐶 𝑣 2( 𝑛 + 1) 𝐴 𝑣𝑠 𝑛𝑃 ∑ (ℎ 𝑣 𝑑𝑖 2 + ℎ 𝑏𝑜𝑖 𝑑𝑖 𝐷) 𝑁 𝑖=1 + 𝑛 (1 − 𝐷 𝑃 )∑(𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖) 𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1
- 13. 13 < 𝐴 𝑣𝑠 (𝑛 + 1)𝑃 ∑ (ℎ 𝑣 𝑑𝑖 2 + ℎ 𝑏𝑜𝑖 𝑑𝑖 𝐷) 𝑁 𝑖=1 + (𝑛 + 1) (1 − 𝐷 𝑃 )∑( 𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖) 𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1 𝑛( 𝑛 + 1) > 𝐴 𝑣𝑠 (ℎ 𝑣 ∑ 𝑑𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1 ) ( 𝑃 − 𝐷)∑ (𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖)𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1 Let 𝛽 = 𝐴 𝑣𝑠 (ℎ 𝑣 ∑ 𝑑 𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑜𝑖 𝑑 𝑖 𝑁 𝑖=1 ) ( 𝑃−𝐷) ∑ ( 𝐴 𝑣𝑟𝑖+ 𝐴 𝑏𝑝𝑖)𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑 𝑖 𝑁 𝑖=1 𝑛2 + 𝑛 − 𝛽 > 0 𝑛 = −1 ± √1 + 4𝛽 2 Since n can’t be negative 𝑛 = −1 + √1 + 4𝛽 2 𝑇𝐶 𝑣 2( 𝑛) < 𝑇𝐶 𝑣 2( 𝑛 − 1) 𝐴 𝑣𝑠 𝑛𝑃 ∑ (ℎ 𝑣 𝑑𝑖 2 + ℎ 𝑏𝑜𝑖 𝑑𝑖 𝐷) 𝑁 𝑖=1 + 𝑛 (1 − 𝐷 𝑃 )∑(𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖) 𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1 < 𝐴 𝑣𝑠 (𝑛 − 1)𝑃 ∑ (ℎ 𝑣 𝑑𝑖 2 + ℎ 𝑏𝑜𝑖 𝑑𝑖 𝐷) 𝑁 𝑖=1 + (𝑛 − 1) (1 − 𝐷 𝑃 )∑( 𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖) 𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1 𝑛( 𝑛 − 1) < 𝐴 𝑣𝑠 (ℎ 𝑣 ∑ 𝑑𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1 ) ( 𝑃 − 𝐷)∑ (𝐴 𝑣𝑟𝑖 + 𝐴 𝑏𝑝𝑖)𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑𝑖 𝑁 𝑖=1 Let 𝛽 = 𝐴 𝑣𝑠 (ℎ 𝑣 ∑ 𝑑 𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑜𝑖 𝑑 𝑖 𝑁 𝑖=1 ) ( 𝑃−𝐷) ∑ ( 𝐴 𝑣𝑟𝑖+ 𝐴 𝑏𝑝𝑖)𝑁 𝑖=1 ∑ ℎ 𝑏𝑜𝑖 𝑑 𝑖 𝑁 𝑖=1 𝑛2 − 𝑛 − 𝛽 < 0 𝑛 = 1 ± √1 + 4𝛽 2 Since n can’t be negative 𝑛 = 1 + √1 + 4𝛽 2
- 14. 14 −1 + √1 + 4𝛽 2 < 𝑛 < 1 + √1 + 4𝛽 2 𝑛2 ∗ = ⌈ −1 + √1 + 4𝛽 2 ⌉ 𝑇𝐶𝑏𝑖 2 = 𝑛𝐴 𝑏𝑟𝑖 𝑇 + 𝑇 2 𝑑𝑖 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 ) Example 2: Consider the same data as in example 1 Abp1= 15 Abp2= 50 hbo1= 2.5 hb02= 2 n2 * = 3 T2* = 0.657 TCv 2= 1810.72 < TCv 1 TCb1 2= 328.04 < TCb1 1 TCb2 2= 791.86 < TCb2 1 TC2= 2930.61 < TC1 The partnership is efficient since all members realized cost savings “Efficient partnership” Example 3: Consider the same data as in example 1 Abp1= 20 Abp2= 65 hbo1= 4.5 hbo2=4.5 n2 * = 3 T2 * = 0.504 TCv 2= 2600.29 > TCv 1 TCb1 2= 73.07 < TCb1 1 TCb2 2= 146.14 <TCb2 1
- 15. 15 TC2= 2819.50 < TC1 The supply chain cost is smaller than the one with no partnership, but the vendor is now worse off. “potentially efficient partnership” Partnership coordination through side payments • If vendor is worse off, some of the buyers’ savings can be transferred to the vendor through a unit price increase. • The maximum price increase is the one that makes at least one buyer indifferent to go for the partnership. • The minimum price increase is the one that makes the vendor no worse off without partnership. • Similarly, when the partnership achieves system-wide savings and some buyers (or all of them) are worse off, the vendor can offer a price discount to these buyers as an incentive to accept the partnership. Proposition 1. To achieve coordination the vendor can vary the unit price c in the range [Cmin, Cmax] where 𝑐 𝑚𝑖𝑛 = 𝑐 + 𝑇𝐶 𝑣 2 − 𝑇𝐶 𝑣 1 ∑ 𝑑𝑖 𝑁 𝑖=1 𝑐 𝑚𝑎𝑥 = 𝑐 + 𝑚𝑖𝑛1≤𝑖≤𝑁 [ 𝑇𝐶 𝑏 2 − 𝑇𝐶 𝑏 1 𝑑𝑖 ] Proof. The minimum price increase is the one that makes the vendor no worse off without partnership.
- 16. 16 𝑐 𝑚𝑖𝑛 − 𝑐 = 𝑇𝐶 𝑣 2 − 𝑇𝐶 𝑣 1 ∑ 𝑑𝑖 𝑁 𝑖=1 By definition the maximum price increase is obtained by finding the maximum price that satisfies all the following inequalities: 𝑐𝑑𝑖 + 𝑇𝐶 𝑏𝑖 1 ≥ 𝑐 𝑚𝑎𝑥 𝑑𝑖 + 𝑇𝐶 𝑏𝑖 2 For i= 1, 2, … , N Example 4: Abp2= 40 TCv 2= 1759.99 TCb1 2= 342.31 TCb2 2= 883.99 TC2= 2986.29 Only Buyer 1 is better off 1.3 Centralized supply chain The vendor and buyers are part of a vertically integrated supply chain under a common control. 𝑇𝐶3 = 𝑇𝐶 𝑣 3 + ∑ 𝑇𝐶 𝑏𝑖 3 𝑁 𝑖=1 𝑇𝐶3 = 𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴 𝑣𝑟𝑖 𝑁 𝑖=1 𝑇 + 𝑇 2 ( ℎ 𝑣 𝑛𝑃 ∑ 𝑑𝑖 2 𝑁 𝑖=1 )
- 17. 17 + ∑( 𝑛𝐴 𝑏𝑖 𝑇 + 𝑇 2 (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 )ℎ 𝑏𝑖 𝑑𝑖) 𝑁 𝑖=1 Let Ai = Avri + Abi be the total ordering cost 𝑇𝐶3 = 𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴𝑖 𝑁 𝑖=1 𝑇 + 𝑇 2 ( ℎ 𝑣 𝑛𝑃 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 )∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) 𝜕 𝜕𝑇 𝑇𝐶3 = 0 − 𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴𝑖 𝑁 𝑖=1 𝑇2 + 1𝑇 2 ( ℎ 𝑣 𝑛𝑃 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 ) ∑ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) = 0 𝑇∗ = √ 2( 𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴𝑖 𝑁 𝑖=1 ) ℎ 𝑣 𝑛𝑃 ∑ 𝑑𝑖 2𝑁 𝑖=1 + (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 )∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 𝑇𝐶3 = √2 (𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴𝑖 𝑁 𝑖=1 ) ( ℎ 𝑣 𝑛𝑃 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 )∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) Minimizing 𝑇𝐶3 (𝑛) Means minimizing (𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴𝑖 𝑁 𝑖=1 )( ℎ 𝑣 𝑛𝑃 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + (1 − 𝐷 𝑃 + 𝐷 𝑛𝑃 )∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) Which equivalent to minimizing
- 18. 18 𝐴 𝑣𝑠 ℎ 𝑣 𝑛𝑃 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + 𝐴 𝑣𝑠 𝐷 𝑛𝑃 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 + 𝑛 ∑ 𝐴𝑖 𝑁 𝑖=1 (1 − 𝐷 𝑃 )∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 𝐴 𝑣𝑠 𝑛𝑃 (ℎ 𝑣 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + 𝐷∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) + 𝑛 (1 − 𝐷 𝑃 ) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 Applying the first difference approach 𝑇𝐶3( 𝑛) < 𝑇𝐶3( 𝑛 + 1) 𝐴 𝑣𝑠 𝑛𝑃 (ℎ 𝑣 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + 𝐷∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) + 𝑛 (1 − 𝐷 𝑃 ) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 < 𝐴 𝑣𝑠 (𝑛 + 1)𝑃 (ℎ 𝑣 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) + (𝑛 + 1)(1 − 𝐷 𝑃 ) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 𝑛( 𝑛 + 1) > 𝐴 𝑣𝑠 𝑃 (ℎ 𝑣 ∑ 𝑑𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) (1 − 𝐷 𝑃 ) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 Let 𝛽 = 𝐴 𝑣𝑠 (ℎ 𝑣 ∑ 𝑑𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) ( 𝑃−𝐷) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 𝑛2 + 𝑛 − 𝛽 > 0 𝑛 = −1 ± √1 + 4𝛽 2 Since n can’t be negative 𝑛 = −1 + √1 + 4𝛽 2 𝑇𝐶3( 𝑛) < 𝑇𝐶3( 𝑛 − 1) 𝐴 𝑣𝑠 𝑛𝑃 (ℎ 𝑣 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + 𝐷∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) + 𝑛 (1 − 𝐷 𝑃 ) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1
- 19. 19 < 𝐴 𝑣𝑠 (𝑛 − 1)𝑃 (ℎ 𝑣 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) + (𝑛 − 1)(1 − 𝐷 𝑃 ) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 𝑛( 𝑛 − 1) < 𝐴 𝑣𝑠 𝑃 (ℎ 𝑣 ∑ 𝑑𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) (1 − 𝐷 𝑃 ) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 Let 𝛽 = 𝐴 𝑣𝑠 (ℎ 𝑣 ∑ 𝑑𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ) ( 𝑃−𝐷) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 𝑛2 + 𝑛 − 𝛽 > 0 𝑛 = −1 ± √1 + 4𝛽 2 Since n can’t be negative 𝑛 = −1 + √1 + 4𝛽 2 −1 + √1 + 4𝛽 2 < 𝑛 < 1 + √1 + 4𝛽 2 𝑛2 ∗ = ⌈ −1 + √1 + 4𝛽 2 ⌉ 𝑛3 ∗ = ⌈0.5(−1 + √1 + 4𝐴 𝑣𝑠 ℎ 𝑣 ∑ 𝑑𝑖 2𝑁 𝑖=1 + 𝐷 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 ( 𝑃 − 𝐷) ∑ 𝐴𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 )⌉ Proposition 2. TC3(n3*) < TCc(nc*) Where TCc the total cost for the consecutive delivery policy proposed by Zavanella and Zanoni (2009) Consecutive delivery policy: At least one shipment will be sent to each buyer within one cycle and if there are two or more shipments they will be sent consecutively
- 20. 20 𝑇𝐶 𝑐∗ = √2 (𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴𝑖 𝑁 𝑖=1 ) ( ℎ 𝑣 ∑ 𝑑𝑖 2𝑁 𝑖=1 + ∑ ℎ 𝑏𝑖 𝑑𝑖 2𝑁 𝑖=1 𝑛𝑃 + ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 − ∑ ℎ 𝑏𝑖 𝑑𝑖 2𝑁 𝑖=1 𝑃 ) Proof. Let nc * be the optimal number of orders for the consecutive delivery policy. Then 𝑇𝐶3( 𝑛 𝑐 ∗) = √2( 𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴 𝑖 𝑁 𝑖=1 )( ℎ 𝑣 𝑛 𝑐 ∗ 𝑃 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + ∑ℎ 𝑏𝑖 𝑑 𝑖 𝑁 𝑖=1 − ∑ 𝑑𝑖 𝑁 𝑖=1 ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 𝑃 (1 − 1 𝑛 𝑐 ∗ )) 𝑇𝐶 𝑐( 𝑛 𝑐 ∗ ) = √2 (𝐴 𝑣𝑠 + 𝑛 ∑ 𝐴𝑖 𝑁 𝑖=1 ) ( ℎ 𝑣 𝑛 𝑐 ∗ 𝑃 ∑ 𝑑𝑖 2 𝑁 𝑖=1 + ∑ ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 − ∑ ℎ 𝑏𝑖 𝑑𝑖 2𝑁 𝑖=1 𝑃 (1 − 1 𝑛 𝑐 ∗ )) Given that ∑ 𝑑𝑖 𝑁 𝑖=1 ∑ℎ 𝑏𝑖 𝑑𝑖 𝑁 𝑖=1 > ∑ℎ 𝑏𝑖 𝑑𝑖 2 𝑁 𝑖=1 Then 𝑇𝐶3( 𝑛𝑐 ∗ ) < 𝑇𝐶 𝑐 ( 𝑛𝑐 ∗) and therefore 𝑇𝐶3( 𝑛3 ∗ ) < 𝑇𝐶 𝑐 ( 𝑛𝑐 ∗)