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Universidade de Lisboa, Faculdade de Ciências
Mestrado em Matemática Aplicada à Economia e Gestão
Logística e Gestão de Operações
Módulo de Logística
Exercises – Inventory Models
1. A factory produces and sells televisions in the national and international market. In the same
factory the speakers to incorporate in the television are produced. Televisions are assembled
at a continuous rate of 8000 a month. Speakers are produced in batches. The factory wants to
know how many speakers to produce and when to produce them in order to minimize global
costs. Each television needs 2 speakers. The costs are :
Fixed producing: 5000 euros
Holding cost/speaker/month: 0. 5 euros
Unitary producing cost : 10 euros
In this inventory model shortage is not allowed.
2. A firm orders paper boxes from a factory to pack canned sardines. According to the annual
demand of canned sardines this firm needs 5000 lots of paper boxes. The fixed order cost is
125 monetary units (u.m.) and the unitary acquisition cost is 50 u.m. (cost for each lot of
paper boxes). The holding annual cost for each lot of paper boxes is 20 u.m.
a) Consider lead time equal to zero, instantaneous replenishment and no shortage
allowed. Obtain the optimal policy of replenishment.
b) Consider now lead time equal to 2 days. Obtain the order point.
3. A product is used at the rate of 20 tons per year. The fixed order cost is 250 euro . The
holding annual cost of each ton 200 euro. The acquisition cost (in euro)is a function of
the quantity ordered according to the table
c(Q)
0  Q  10
10  Q  20
20  Q  25
Q  25
12
10
7
5
Consider the hypothesis of EOQ. Obtain the optimal quantity to order.
4. The manager of a warehouse must decide whether to buy or produce an item in the factory of
the firm. The annual demand of the item is 2500 unities and no shortage is allowed. The
unitary acquisition cost is 2500 u.m. and the fixed order cost is 500 u.m.
The production rate is 10000 annual unities and the production cost is 2200 u.m. The set-up
cost is 5000 u.m. The holding cost is 10% of the acquisition cost (buy or production) by time
unity
Exercises – Inventory Models
a) Which is the optimal decision?
b) For the decision chosen in a) calculate the maximum stock level of the item and the
time it is reached.
c) Solve considering shortage.
5. A firm produces spare pieces form industrial machines. The semiannual demand 8000
unities. The unitary cost of producing a piece is 50 € and the set-up cost is 1000 €. The
machine that produces these pieces has a production rate of 30000 pieces per year. The
annual holding cost of each piece is 20 €. Obtain the optimal producing policy.
6. The firm Malver supplies bags and knapsacks for the whole country. The more popular
models are the executive for bags and the casual for knapsacks. The annual demand for the
casual knapsacks is 150 000 units, the unitary cost is 30 euros and the annual holding cost for
each unit is 20% of its cost. The executive bags have an annual demand of 100 000 units, a
unitary cost of 45 euros and an annual holding cost for each unit of 20% of its cost. In both
cases the fixed ordering cost is 250 euros. The manager decided to invest at most 75000
euros to produce both items. Use solver of Excel to obtain the optimal quantities to order.
7. Consider the storage in a factory of a chemical product to be used in the manufacture of
textiles. The annual demand of this product is estimated in 10000 liters. The factory buys the
product from a supplier being the demand during the lead a normal distribution N(300,40).
Each order has a fixed cost of 70 u.m. and each liter of product is bought at the price of 3 u.m..
The annual holding cost is estimated as 20% of acquisition the price. Demand must be
completely satisfied (even if with small delays). There is a penalty of 1500 u.m. for each liter
when there is shortage. Present an inventory for this product.
8. A firm in the building construction business has in stock a type of building material whose
daily use follows a Normal distribution with mean 1000 and standard deviation 20. The lead
time is 15 days. The fixed order cost is 1960 u.m. and the holding daily cost for each unity is 2
u.m.
c) What is the probabilistic distribution of consume during lead time?
d) Obtain the order point that gives a level of service of 95%.
e) Suppose that the shortage unitary cost is 5 u.m.. Obtain Q* and r* and the
approximated value for the average quantity in stock.
9. A tobacco shop wants to make a unique order of a special magazine number. The demand of
this special edition follows a Normal with mean 800 and standard deviation 50. Each
magazine costs 2.5 euros and is sold at a price of 5 euros. After some time the magazines not
Mestrado em Matemática Aplicada à Economia e Gestão– Logística e Gestão de Operações
Módulo de Logística 2015/2016
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Exercises – Inventory Models
sold can be returned to the editors at a salvage price of 1.5 euros each. Obtain the optimal
quantity to order and the global expected costs
10. A moll every year in the beginning of December makes a unique order of Christmas trees.
According to last year experience the manager wants to know how much to order this year so
that the expected global costs are minimized.
The manager thinks that the expected demand this year will follow an exponential
distribution with mean 4500. The estimated cost to buy a tree will be 10000 u.m. and it will
be sold by a price of 25000 u.m.. Last year there were no trees left in stock. The manager
decided to consider two cases.
a) A tree not sold is stored for next year which leads to a holding cost of 2500 u.m.
b) A tree not sold until Christmas is solved at a salvage value of 7500 u.m
Mestrado em Matemática Aplicada à Economia e Gestão– Logística e Gestão de Operações
Módulo de Logística 2015/2016
Page 3