A company produces a certain product by assembling it at an assembly plant. All the components needed to assemble the product are purchased from a single supplier. A shipment of all the components is received from the supplier each time the assembly plant needs to replenish its inventory of the components. The company incurs a shipping cost of $500 in addition to the purchase price for the components each time this is done. Each time the supplier needs to replenish its own inventory of the components, quick production runs are set up to produce the components. The total cost of setting up for these production runs is $50,000. The annual cost of holding each set of components is $50 when it is held by the supplier and $60 when it is held at the assembly plant. (It is higher in the latter case since there is more capital tied up in each set of components at this stage.) The assembly plant steadily produces 500 units of the product per month. All the assumptions of the model for a serial two-echelon system described in Sec. 18.5 apply to the joint inventory system for the supplier and the assembly plant.
(a) Suppose that the supplier and the assembly plant separately optimize their own inventory policies for the sets of components. Calculate the resulting Q*2, n*, n, and Q*1. Also calculate C*1 and C*2, the total variable cost per unit time for the supplier and the assembly plant, respectively, as well as C* = C*1 + C*2.
(b) Suppose that the supplier and the assembly plant cooperate to simultaneously optimize their joint inventory policy. Calculate the same quantities as specified in part (a) for this new inventory policy.
(c) Compare the values of C*1, C*2, and C*obtained in parts (a) and
(b). Would either organization lose money by using the joint inventory policy obtained in part (b) instead of the separate policies obtained in part (a)? If so, what financial arrangement would need to be made between these separate organizations to induce the losing organization to agree to a supply contract that follows the inventory policy obtained in part (b)? Comparing the values of C*, what would be the total net savings for the two organizations if they can agree to follow the jointly optimal policy from part (b) instead of the separate optimal policies from part (a)?