Week 9 - Deterministic & stochastic inventory models

Question 1

What does Cycle length mean?

Answer 1

t=Q/d is the time it takes Q to run down at rate d

Question 2

EOQ model with no shortages allowed (deterministic)
1. Time per cycle
2. Total order cost per cycle
3. Total cost per time
4. Optimum order quantity
5. Optimum cycle length

Answer 2

1. Time per cycle
   t = Q/d
2. Total order cost per cycle
   = {immediate} order cost + HOLDING COST
   = K + cQ + hQ^2/2d
3. Total cost per time
   = dK/Q + dc + hQ/2
4. Optimum order quantity
   Q* = sqrt(2dK/h)
5. Optimum cycle length
   t* = Q*/d

Question 3

EOQ model with planned shortages (deterministic)
> S = inventory level at ORDER ARRIVAL & backlogged demands are satisfied

1. Total order cost per cycle
2. Total cost per time
3. Optimum order quantity
4. Optimum cycle length
5. S*

Answer 3

1. Total order cost per cycle
   = immediate cost + holding costs + shortfall costs
   = K + cQ + hS^2/2d + P(Q-S)^2/2d
2. Total cost per time
   = dK/Q + dc + hS^2/2Q + P(Q-S)^2/2Q
3. Optimum order quantity
   Q* = sqrt(2dK/h * (p+h)/p)
4. Optimum cycle length
   t* = Q*/d
5. S* = sqrt(2dK/h * p/(p+h))

Question 4

Stochastic inventory models

1. Optimal order quantity
2. Reorder point R - what is the service level requirement?
   ie. Reorder __ quantity when stock decreases below __ units

Answer 4

1. Optimal order quantity
   Q* from the EOQ model with planned shortages
   » assume DETERMINISTIC demand d; hence only an approximation
2. Reorder point R
   » service level requirement = the prob. of a STOCKOUT between the time an order is placed & received should not be more than a given tolerated stockout probability, q
   - P(X < R) ≤ q, so set P(X < R) = q
   - 1 - F(R) = q // apply CDF cumulative density function
   - R = F^-1(1-q)

Question 5

Perishable goods: the newsvendor problem

1. Expected total cost for order quantity S
2. Optimal service level, which gives the optimal order quantity S*
3. Unit regret (cost) of underordering
4. Unit cost of overordering

Answer 5

1. Expected total cost for order quantity S
   = cS + ___ too long! see notes!!!
2. Optimal service level
   F(S) = (p-c)/(p+h) = cu/cu+co
   F(S) = prob(demand X ≤ S)
3. Unit regret (cost) of underordering
   cu = p - c
4. Unit cost of overordering
   co = h + c

Question 6

Formula for Holding cost

Answer 6

= storage cost - salvage value

Question 7

Newsvendor problem - Let’s say the optimal solution is to order 10 or 11 bunches of orchids. How to decide whether to order 10 or 11?

Answer 7

1. Compare the EXPECTED PROFIT if order 11 bunches rather than 10
   ie. compare the incremental impact of the extra unit(s)
2. Calculate if demand > 10, we are glad of our choice by __x Cu
3. Calculate if demand ≤ 10, we regret our choice by __x Co
4. Total expected extra profit = probabilities x the values we got above
5. If expected profit is POSITIVE, we should stock 11 bunches

Question 8

Newsvendor problem - How does the answer change if there are already some inventory in stock? What 2 options to compare?

Answer 8

Compare the COSTS of:
1. Order __ units to obtain the optimal solution as calculated in an earlier question, OR
2. Don’t order any more units and ONLY SELL CURRENT STOCK, saving the fixed cost of __

Question 9

1. If shortages are not allowed, why is the Selling price irrelevant?
2. Even if shortages are allowed, why irrelevant?

Answer 9

1. The income from selling the good will always be the same
2. The loss of revenue is modelled by the Shortage cost

Question 10

Stochastic inventory model - What is Safety stock?

Answer 10

The EXPECTED inventory level just before the new stock arrives
= R − E(X)
where X is the demand during the lead time