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# Inventory policy with uncertain demand and backorder

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2) A paint shop implements an inventory policy on its stock of white paint, which costs the store \$6 per can. Monthly demand for cans of white paint is normal with mean 28 and standard deviation 8. The replenishment lead time is 14 weeks. Excess demand is back-ordered, but costs \$10 per back ordered can in labor and loss of goodwill. There is a fixed cost of \$15 per order, and the holding cost is based on 30% interest rate per annum. In your computations, assume 4 weeks per month.

2.1) Write down the model name and parameters.
2.2) What are the optimal lot size and reorder points for white paint (include the formulas)?
2.3) What is the optimal safety stock (include the formula)?

3) Suppose the paint shop from the previous problem adopts a service level policy.

3.1) What are the optimal lot size and reorder points for white paint, such that 90% of the cycles are filled without backordering (include all formulas)?
3.2) What is the fill rate corresponding to the reorder policy computed in the previous part (include all formulas)?

4) Suppose the paint shop from the previous problem adopts an (s,S) policy based on monthly reviews of its inventory instead of a continuous review. Orders are made at the beginning of each month and lead times are zero. The demand in the months January to June was 37, 33, 26, 31, 14, 40, respectively, and the starting inventory in January was 26 cans.

4.1) Based on the (Q,R) solution you found in problem 2, what are the corresponding (s,S) values?
4.2) Under the (s,S) policy you found, what is the order size in each of the months (January to June)?

#### Solution Preview

2) A paint shop implements an inventory policy on its stock of white paint, which costs the store \$6 per can. Monthly demand for cans of white paint is normal with mean 28 and standard deviation 8. The replenishment lead time is 14 weeks. Excess demand is back-ordered, but costs \$10 per back ordered can in labor and loss of goodwill. There is a fixed cost of \$15 per order, and the holding cost is based on 30% interest rate per annum. In your computations, assume 4 weeks per month.
2.1) Write down the model name and parameters.
-> Let c be the cost of a paint can; c = \$6
Let µ be the mean demand per month; µ = 28
Let σ be the monthly standard deviation; σ = 8
Let L be the replenishment lead time; L = 14 weeks
Let o be the backorder cost; o = \$10 per can
Let x be the fixed ordering cost; x = \$15
Let I be the interest rate per annum; I = 30%

This is a (s, Q) model in which s is the reorder point and Q is the order quantity or the lot size. This model is also referred as (Q, R) in which Q is order lot size and R is the reorder level.

2.2) What are the optimal lot size and reorder points for white paint (include the formulas)?
-> The optimal lot size in this case can be derived using the economic order quantity concept. The total cost equation in this case consists of ordering cost, inventory holding cost and backorder cost (note that purchasing cost is not included as it is not relevant from decision making perspective).
Ordering cost = Cost per order * Number of orders
Number of orders = Annual demand/Optimal lot size
Let D be the annual demand
Ordering cost = o*D/Q (1)
Inventory holding cost = Inventory holding cost per can per annum*Average inventory
Inventory holding cost per can per annum = interest rate per annum*cost of a can
= I*c
Average inventory = ((Q-B)2/2Q) + SS, where SS is the safety stock and B is the number units backordered
Inventory holding cost = I*c*[((Q-B)2/2Q) + SS] (2)

Backorder cost = Backorder cost per unit*Number of backorders
Number of backorders = B2/2Q
Backorder cost = o* B2/2Q (3)

Combining ...

#### Solution Summary

The solution provides details help on how to derive various inventory policy parameters. The computations are shows with detailed formulas and underlying concepts. With the posting a student can address inventory policy questions.

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