# Inventory policy with uncertain demand and backorder

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.

Answer the following questions:

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.

Answer the following questions:

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.

Answer the following questions:

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)?

https://brainmass.com/business/operations-research/inventory-policy-uncertain-demand-backorder-612739

#### 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.

Answer the following questions:

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.

Supply Chain Management Exercise Question

Supply Chain Management exercise questions

1. Mike Johnson went back to work in his family business after graduating. The family business, Johnson Bicycles, is a bike shop that specializes in high quality bikes. They carried a full line of bikes, ranging from touring bikes to mountain bikes and aimed to be the premiere, full service bike shop.

Currently, Johnson Bicycles has three retail outlets that it stocks individually (and hence manages inventory separately for each location). Mike thinks that the operation is big enough to take advantage of scale economies and wants to investigate the possibility of consolidating inventory at a centralized distribution point. Mike thinks that centralizing inventory should lead to a substantial reduction in inventory costs.

The basic idea would be to stock the bikes centrally (excluding the token display model) and expedite delivery to the stores on an as needed basis. He figured if the principal of aggregation would work for the series 200 bike, then it should provide costs savings for all styles of bikes. Table 1, below, shows last year's sales history for the model 200 series at each retail location.

Series 200 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Std Dev Total

Outlet 1 9 5 15 10 12 5 3 12 23 12 8 5 5.5 119

Outlet 2 10 10 30 12 22 15 17 30 11 14 12 15 7.1 198

Outlet 3 12 22 5 15 14 23 10 4 9 15 3 5 6.7 137

Total Sales 31 37 50 37 48 43 30 46 43 41 23 25 8.9 454

As can be seen, monthly sales varied, but generally, sales remained fairly constant from year to year. Mike estimated it cost $65 every time the company placed an order for a given model at each location. The supplier's lead time is a hefty two months, but fortunately it has been constant over time. Historically, they've stocked to a 98% service level. Each bike model cost $75 per bike and Mike figures that holding cost to be 25% of the purchase price on an annual basis. With their high service level, there was naturally, a lot of extra safety stock in inventory. Mike knew every additional unit of safety stock (recall safety stock is any inventory held in excess of average demand) added directly to the total cost of inventory in the form of added holding cost. That is, total cost is equal to the cost of ordering plus the cost of holding inventory, where the cost of holding inventory is equal to the cost of holding average inventory plus safety stock.

a. Using the s, S model, what are the order quantities, reorder points, and total annual inventory costs associated with inventory management of the Series 200 line of bikes at each location?

b. What would be the order quantity, reorder point, and total annual inventory cost for a consolidated plan where bikes are stored at a central distribution center?

c. Would you recommend switching to centralized inventory management? What other factors besides costs should Mike consider?

Please see attached for full question.

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