Consider a three-stage serial production system with the following data:

Stage	Holding cost rate (per dollar per unit time)	Purchase Value (dollar per unit)	setup cost
1 (first)	I	C1	S1
2 (intermediate)	I	C2	S2
3 (final)	I	C3	S3

The demand rate for the final product is D units/year and is deterministic. Note that C₁ < C₂< C_3. The production system is technologically restricted in that the lot size used for Stage 2 should be exactly the same as the lot size used in Stage 1. The problem is to determine lot sizes for each stage, assuming that the lot size in stage 2 is an integer multiple of the lot size in stage 3.

1. Define the decision variables in the problem where you want to minimize average cost while satisfying all demand without any shortage.
2. Write an expression for the average holding cost for all stages, AHC(.). You can use echelon holding costs.
3. Write an expression for the average total cost (average total cost of setup + average holding cost for all stages) ATC(.). Express lot size requirements as constraints.
4. Describe an algorithm to solve for the optimal integer multiple.