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The longer a machine stays in service, the higher is its maintenance cost, and the lower its productivity. When a machine reaches a certain age, it may be more economical to replace it. The problem thus reduces to determining the most economical age of a machine.

**Equipment Replacement Model**

The longer a machine stays in service,
the higher is its maintenance cost, and the lower its productivity. When a
machine reaches a certain age, it may be more economical to replace it. The
problem thus reduces to determining the most economical age of a machine.

Suppose that we are studying the machine replacement problem over a span
of *n* years. At the *start* of each year, we decide whether to
keep the machine in service an extra year or to replace it with a new one. Let *r(t), c(t),* and *s(t)* represent the yearly revenue, operating cost, and salvage
value of a t-year-old machine. The cost of acquir-ing a new machine in any year
is 1.

The elements of the DP model
are

*1.
**Stage **i** *is represented by year* i, i *=* *1,2, ... ,* n.*

2. The *alternatives* at stage
(year) *i* call for either *keeping* or *replacing* the machine at the *start*
of year *i.*

3. The *state* at stage *i* is the age of the machine at the
start of year *i.*

Given that the machine is *t* years old at the *start* of year *i,* define

**Example 10.3-3**

A company needs to determine the optimal replacement policy for a
current 3-year-old machine over the next 4 years *(n* = 4). The company requires that a 6-year-old machine be replaced. The
cost of a new machine is $100,000. The following table gives the data of the
problem.

The determination of the feasible values for the age of the machine at
each stage is some-what tricky. Figure 10.6 summarizes the network representing
the problem. At the *start* of year 1, we have a 3-year-old
machine. We can either replace it *(R)* or keep it *(K)* for another year. At the start of year 2, if replacement occurs, the
new machine will be 1 year old; otherwise, the old machine will be 4 years old.
The same logic applies at the start of years 2 to 4. If a l-year-old machine is replaced at the start of year 2,3, or 4, its
replacement will be 1 year old at the start of the following year. Also, at the
start of year 4, a 6-year-old machine must be replaced, and at the end of year
4 (end of the planning horizon), we salvage (5) the machines.

The network shows that at the start of year 2, the possible ages of the
machine are 1 and 4 years. For the start of year 3, the possible ages are 1,2,
and 5 years, and for the start of year 4, the possible ages are 1,2,3, and 6
years. .

The solution of the network in Figure 10.6 is equivalent to finding the
longest route (i.e., maximum revenue) from the start of year 1 to the end of
year 4. We will use the tabular form to solve the problem. All values are in
thousands of dollars. Note that if a machine is replaced in year 4 (i.e., end
of the planning horizon), its revenue will include the salvage value, *set),* of the *replaced *machine and
the salvage value,* s*( 1), of the* replacement *machine.

Figure 10.7 summarizes the optimal solution. At the start of year 1,
given *t* = 3, the optimal decision is to
replace the machine. Thus, the new machine will be 1 year old at the start of
year 2, and *l* = 1 at the
start of year 2 calls for either keeping or replacing the machine. If it is replaced, the new machine will be 1 year old at the start of year
3; otherwise, the kept machine will be 2 years old. The process is continued in
this manner until year 4 is reached.

The alternative optimal policies starting in year 1 are *(R, K, K, R)* and *(R, R, K,
K).* The total cost is $55,300.

**PROBLEM SET **10.3C

1. In each of the following cases, develop the network, and find the
optimal solution for the model in Example 10.3-3:

a. The machine is 2 years old at the start of year l.

b. The machine is 1 year old at the start of year l.

c. The machine is bought new at the start of year l.

*2. My son, age 13, has a lawn-mowing business with 10 customers. For
each customer, he cuts the grass 3 times a year, which earns him $50 for each
mowing. He has just paid $200 for a new mower. The maintenance and operating
cost of the mower is $120 for the first year in service, and increases by 20% a
year thereafter. A l-year-old mower has a resale value of $150, which decreases
by 10% a year thereafter. My son, who plans to keep his business until he is
16, thinks that it is more economical to buy a new mower every 2 years. He
bases his decision on the fact that the price of a new mower will increase only
by 10% a year. Is his decision justified?

3. Circle Farms wants to develop a replacement policy for its 2-year-old
tractor over the next 5 years. A tractor must be kept in service for at least 3
years, but must be disposed of after 5 years. The current purchase price of a
tractor is $40,000 and increases by 10% a year. The salvage value of a
l-year-old tractor is $30,000 and decreases by 10% a year. The current annual
operating cost of the tractor is $1300 but is expected to increase by 10% a
year.

a) Formulate the problem as a shortest-route problem.

b) Develop the associated recursive equation.

c) Detennine the optimal replacement policy of the tractor over the next 5
years.

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