updated+hw5+sol

# Consider the case of items with family level and item

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3. Consider the case of items with family-level and item-level replenishment fixed costs. Consider the case where all items must be replenished at the time of every family replenishment, i.e., all mi’s are required to equal 1. In the case all ai= 0, we saw in a previous problem set that coordinated replenishment provides lower costs than independent replenishment of the items. But for general values of ai, it is not clear if coordinated replenishment with all mi’s equal to 1 is better than independent replenishment. a. For coordinated replenishment in this case, write the formulas for the optimal value of the family cycle length T and the associated minimum total relevant costs per unit time. ( ) ( ) + = i i i v D I a A T 2 * and + = + + = = = i i i n i i i n i i v D I a A D T Iv T a A T TRC ) ( 2 2 * * *) ( 1 1 . b. Find the best coordinated and independent replenishment strategies for each of the following two examples: Example 1 A = \$10, I = 0.2 per year i D i v i a i (Item) (units per year) (item-level replenishment fixed cost) 1 800 1 2 2 400 0.5 4 For independent control, i i i i D Iv a A EOQ TRC ) ( 2 ) ( + = , so 44 . 95 \$ 47 . 33 97 . 61 ) 400 )( 5 . 0 )( 2 . 0 )( 14 ( 2 ) 800 )( 1 )( 2 . 0 )( 12 ( 2 ) ( ) ( 2 1 = + = + = + = EOQ TRC EOQ TRC TRC

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4 For coordinated control, 00 . 80 \$ ) 200 800 )( 2 . 0 )( 4 2 10 ( 2 ) ( 2 = + + + = + = i i i v D I a A TRC so for this example coordinated control is better. Example 2 A = \$10, I = 0.2 per year i D i v i a i (Item) (units per year) (item-level replenishment fixed cost) 1 900 1 2 2 20 0.5 4 For independent control, i i i i D Iv a A EOQ TRC ) ( 2 ) ( + = , so 21 . 73 \$ 48 . 7 73 . 65 ) 20 )( 5 . 0 )( 2 . 0 )( 14 ( 2 ) 900 )( 1 )( 2 . 0 )( 12 ( 2 ) ( ) ( 2 1 = + = + = + = EOQ TRC EOQ TRC TRC For coordinated control, 32 . 76 \$ ) 10 900 )( 2 . 0 )( 4 2 10 ( 2 ) ( 2 = + + + = + = i i i v D I a A TRC so for this example independent control is better. d. Now, for general data, determine as simple a relationship as possible that must be satisfied in order for the case of coordinated control with all m i = 1 to be preferable to completely independent item control. Your relationship should be an inequality involving the a i , A , D i and v i parameters. Also, show that your relationship is satisfied if all a i = 0. For coordinated control to be superior to independent control, we must have + i i i v D I a A ) ( 2 < + i i i i D Iv a A ) ( 2 + i i i v D a A ) ( < + i i i i D v a A ) ( + i i i i i i v D A v D a A ) ( < + j j j i i i i v D A D v a A ) ( + i i A a 1 < Ο Π Ξ Μ Ν Λ + i j j j i i i v D v D A a 1
5 Let A a a i i / = ʹ′ and let = j j j i i i v D v D f . Then coordinated control is preferred when ʹ′ + i i a 1 < ( ) ʹ′ + i i i f a 1 .

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