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# Calculating optimal batch size for Channel coordination

A manufacturer of explosive gases is facing the following costs:

Set up cost: \$2500
Holding cost: \$2
Production cost: \$0.5
Whole sale price: \$0.75
Total demand: 160000 units

A retailer who buys from the manufacturer has the following costs:

Set up cost: \$100
Holding cost: \$2
Whole sale price: \$0.75
Revenue per unit \$2
Total demand: 160000 units

You may make the following assumptions:

The manufacturer's production rate is very fast and he does not incur any significant holding cost during production.

Because of safety concerns the manufacturer can't keep any inventory of gases. That is, once a production of a batch is completed it must be delivered immediately.

1.1 What is the retailer's optimal batch size?
1.2 Assuming that the manufacturer produces in batches of the size calculated in (1.1) what is the profit of the manufacturer?
1.3 What is the system optimal batch size?
1.4 In this case, what is the profit of the manufacturer?
1.5 What are the cost savings due to channel coordination? (Compare the costs when the batch size is the one you have calculated in (1.1) and (1.3)
1.6 Assuming the channel coordinated quantity. What is the profit of the manufacturer?
1.7 The manufacturer insists that he must make a profit of at least \$15000 per year. He will not sign the deal if his profits are below this level. What is the best thing the retailer can do? (No alternative supplies are available at the time).

#### Solution Preview

See the attached file. The text here may not print correctly for tables and symbols.

1.1 What is the retailer's optimal batch size?
D=Annual Demand = 160000 units
Ch=Holding Cost = \$2 per unit
Cs=Set up Cost = \$100 per setup
Now, EOQ = (2*D*Cs/Ch)^0.5 = (2*160000*100/2)^0.5=4000 units.

The optimum batch size is 4000 units

1.2 Assuming that the manufacturer produces in batches of the size calculated in (1.1) what is the profit of the manufacturer?
Number of batches produced = 160,000/4000 = 40
Set up cost = 40*2500=100000
Holding cost = 4000/2*2=4000
Production cost = 0.5*160000=80000
Selling price = 0.75*160000=120000
Profit to manufacturer = 120000-80000-100000-4000= -64000

1.3 What is the ...

#### Solution Summary

This is a very good problem to understand the basic concepts of supply chain management. It illustrates how to set up the optimal batch size for the manufacturer, for the retailer and for the overall system so that the channel coordination between the manufacturer and retailer is optimized. It also calculates the corresponding profits to the two parties and how the two can share the profits so that the decision taken by both of them is optimal. It uses mathematical equations (formulas) to arrive at the required quantities and then discusses it in text.

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