• Demand is known with uncertainty and is fixed at λ unit/unit time
• Shortages are not permitted
• Lead time for delivery is instantaneous
• No time discounting of money
• Costs include K per order (fixed cost)
• hi includes cost of capital, storage, handling and insurance

 

Time Value of Money

 

Σ  c_k α^k

K=1

 

If no discounting then α=1

 

Average cost:

 

Choose the policy that minimizes the average cost.

 

 

 

Figure 1: No backlogging allowed

 

 Making Decision: When to order and how much to order.

 

Aim: to come up with the best possible policy that minimizes average cost.

 

If backlogging then:

 

 

Figure 2: Backlogging Allowed

 

One thing is clear. We never want to order at point A (refer Figure 1)

 

So lets consider the following figure:

 

 

Figure 3: Another Modification

 

In this if the λ rate line touches the X-axis i.e. ordering when inventory becomes zero then it minimizes the cost and hence savings.

 

                               N(t) + h ∫ I(t) dt

No of times  i order

                

                   t

 

So, we narrow our search to something like this:

 

 

 

 

 

Now, intuitively we can say that the above triangles should be equal.

 

Lets take the following example:

 

 

 

After two triangles, it repeats. So for a fixed cycle time, if h1 and h2 are the heights of the triangles, then for minimum area, each triangle should be of height (h1+h2)/2

 

Alternatively, if A1 and A2 are fixed, then for the total area  be fixed, each should have an area of (A1+A2)/2 (here we have increased the cycle time)

 

So, now we have:

 

                                                                                   

 

       Q

 

 

               Q/λ

Cost/Cycle

                               K + hQ²          

               Fixed cost           2λ         

 

 

We want cost/time:

 

                                          K + hQ²/2λ

                                              Q/λ

 

 

 

 

 

Q* = √2kλ/h