A company is planning its production schedule
over the next six months (it is currently the end of month 2).
The demand (in units) for its product over that timescale is
as shown below:
| Month |
3 |
4 |
5 |
6 |
7 |
8 |
| Demand |
5000 |
6000 |
6500 |
7000 |
8000 |
9500 |
The company currently has in stock: 1000
units which were produced in month 2; 2000 units which were
produced in month 1; 500 units which were produced in month
0.
The company can only produce up to 6000 units per month and
the managing director has stated that stocks must be built up
to help meet demand in months 5, 6, 7 and 8. Each unit produced
costs Mu 15 and the cost of holding stock is estimated to be
Mu 0.75 per unit per month (based upon the stock held at the
beginning of each month). The company has a major problem with
deterioration of stock in that the stock inspection, which takes
place at the end of each month regularly, identifies ruined
stock (costing the company Mu 25 per unit). It is estimated
that, on average, the stock inspection at the end of month t
will show that 11% of the units in stock, which were produced
in month t, are ruined; 47% of the units in stock, which were
produced in month t-1, are ruined; 100% of the units in stock,
which were produced in month t-2, are ruined. The stock inspection
for month 2 is just about to take place.
The company wants a production plan for the next six months
that avoids stockouts. Formulate their problem as a linear program.
Let Pt be the production
(units) in month t (t=3,...,8) ; Iit be the number of units
in stock at the end of month t which were produced in month
i (i=t,t-1,t-2) ; Sit be the number of units in stock at the
beginning of month t which were produced in month i (i=t-1,t-2);
dit be the demand in month t met from units produced in month
i (i=t,t-1,t-2)
Constraints
- production limit
Pt <= 6000
- initial stock position
I22 = 1000
I12 = 2000
I02 = 500
- relate opening stock in month t to closing stock in previous
months
St-1,t = 0.89It-1,t-1
St-2,t = 0.53It-2,t-1
- Inventory continuity equation where we assume we can meet
demand in month t from production in month t. Let Dt represent
the (known) demand for the product in month t (t=3,4,...,8)
then
Closing stock = opening stock + production - demand and
we have
It,t = 0 + Pt - dt,t
It-1,t = St-1,t + 0 - dt-1,t
It-2,t = St-2,t + 0 - dt-2,t
where
dt,t + dt-1,t + dt-2,t = Dt
- no stockouts
all inventory (I,S) and d variables >= 0
Objective
Presumably to minimize cost and this is given by
SUM{t=3 to 8}15Pt + SUM{t=3 to 9}0.75(St-1,t+St-2,t) + SUM{t=3
to 8}25(0.11It,t+0.47It-1,t+1.0It-2,t)
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