# Multiperiod Production and Aggregate Planning MSIS 22:711:580 Spring 2004

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Multiperiod Production and Aggregate Planning MSIS 22:711:580 Spring 2004
```Multiperiod Production and
Aggregate Planning
MSIS 22:711:580
Spring 2004
Last updated February 25, 2004
1
Multiperiod Production
The following example is adopted from the text
S. Albright, W. Winston, C. Zappe, “Data Analysis & Decision Making”,
Duxbury, 2004.
The Problem Statement
A company produces footballs and must decide how many footballs to produce
each month for the next 6 month. The forecasted demands for the next six
months are 10000, 15000, 30000, 35000, 25000 and 10000. The company wants to
meet these demands on time, and it has currently 5000 footballs in its inventory.
Also a month’s production can be used to meet the demands of that month. The
maximum monthly production capacity is 30000 footballs. The storage capacity
of the company is 10000. The company estimates that production costs for each
football in each of the next month is given by \$12.50, \$12.55, \$12.70, \$12.85,
\$12.85, and \$12.95. The holding cost of each football is 5% of its production cost
in each month. (This cost includes the cost of storage and the cost of money tied
up in inventory. Notice that price of footballs in each month is irrelevant since
we assume the demand in each month is met completely. So the only relevant
parameter here is the cost over 6 months which must be minimized.)
The Algebraic Formulation:
For the sake of generality, let us assume that our planning problem consists of
T periods; in this example T = 6.
1
The Decision Variables
The decision variables, those to be determined are as follows:
Production Pt : These are the amount of production of footballs in period t;
there is one for each period.
Inventory It : These are the amount of inventory in period t; there is one for
each period.
The Constraints
There are two “capacity type constraints”:
Production capacity: There is a limit of C that can be produced each month,
thus Pt ≤ C.
Inventory capacity: There is room for only S footballs to be held, thus It ≤ S
While in this example C = 30000 and S = 10000 are fixed, in general these
bounds need not be fixed over periods. You may anticipate purchase of new
machines in future, and so the capacity may increase. Similarly, you may anticipate to move into a larger facility and your storage capacity could increase.
Balance constraints
The key constraints are based on a simple observation:
In each period the inventory carried over from the previous period
plus the amount produced in the period must equal demand in that
period plus inventory carried to the next
Mathematically
It−1 + Pt = Dt + It
P1
I0
P2
1
I1
or equivalently It−1 + Pt − It = Dt
P3
2
I2
P4
3
I3
P5
4
I4
P6
5
I5
6
D
D
D
D
D
D
1
2
3
4
5
6
I6
The flow of production, demand, and inventory from period to period.
2
The costs
There are two kinds of costs we consider here. One is production cost for each
period ct , and the cost of holding inventory ht . In this example, it is stated that
cost of holding is 5% of the production cost in each month, that is ht = .05ct ,
but in general the costs can be given directly. Thus we incur two kinds of costs:
Cost of production:
T
X
ct Pt
t=1
Cost of Inventory:
T
X
ht Pt
t=1
Thus the total cost is
T
X
ct Pt +
t=1
T
X
ht Pt
t=1
The Linear Programming Model Put Together
We now have all the information to set up the linear programming. The only
thing to note is that the initial inventory, I0 is not a decision variable but is
given (in our example I0 = 5000.) The linear program, with the objective of
minimizing costs, while satisfying constraints can now be formulated as follows:
minimize
subject to
PT
PT
ct Pt + t=1 ht Pt
It−1 + Pt − It = Dt
0 ≤ Pt ≤ C
0 ≤ It ≤ S
t=1
The nonnegativity of production is obvious. In this model we also assume that
no shortages are allowed, so the inventory must be nonnegativite also.
You can now check the spreadsheet aggPlanningSimple.xls to see how to use
software to solve this problem. Also check the sensitivity sheets and make sure
you can interpret them correctly.
2
Aggregate Planning
The following example is adopted from the text
W. Winston, S. Albright, “Practical Management Science” Duxbury, 1997.
The Problems Statement
During the next four months a custom shoe store must meet the following
demands for shoes: 300, 500, 100 and 100. At the beginning of month 1 there
are 50 shoes in the inventory and the store has three workers. Each worker
3
has a salary of \$1600 a month and each worker can work up to 160 hours each
month before they get overtime payment. A worker may be required to work
up to 20 hours of overtime per month and are paid \$25 per hour for each hour
of overtime. It takes four hours of labor and \$5 of raw material to produce one
pair of shoes. At the beginning of each month, and only at the b egining of
each month, each worker can be hired or fired. Cost of hiring a worker is \$1500
and each fired worker costs \$2000. At the end of each month, a holding cost
of \$30 per pair of shoes of left in inventory is incurred. Production in a given
month can be used to meet that month’s demand. Design a production and
hiring schedule for the next four months.
This problem is more detailed than the multiperiod production problem
considered in the last section. For example, in multiperiod production problem
cost of production was given. In the aggregate planning problem, the source of
cost is also included in the model: We have, in addition to inventory costs, labor
costs and raw material cost. Furthermore, the labor costs itself is decomposed
into several categories: regular wages, overtime wages, cost of hiring and cost
of firing.
The Algebraic Formulation
Again we assume that we have T periods available; for this problem T = 4.
Decision Variables
For each period t = 1, 2, . . . , T , in addition to prediction levels Pt and inventory
levels It , we have
Workers Hired: Ht ,
Workers Fired: Ft ,
Overtime Hours: Ot ; notice this variable is measured in hours,
Workforce: Wt .
Constraints
There are several types of constraints
Overtime: Since each worker can be required to work up to 20 hours a month
we must have Ot ≤ 20Wt .
Work Balance: The workforce and the number of hired and fired must add
up, so
Wt = Wt−1 + Ht − Ft
We can use this as a constraint, but we may also use it to “eliminate”
workforce Wt as a variable in the model. In that case everywhere that Wt
occurs, we may replace it by this formula.
4
production Capacity: Since each pair of shoes requires 4 hours of labor, We
cannot possibly produce more than total number of hours available divided
by 4. Thus1
160Wt + Ot
Pt ≤
4
Production Balance: These are as in multiperiod production plan:
Pt + It−1 − It = Dt
2.1
Costs
The objective is to minimize the total cost. Thus, we have to analyze various
costs in each period and then sum over all periods.
Hiring costs: It costs \$1600 per worker hired, so hiring cost is
T
X
1600Ht
t=1
Note that we can also assume different hiring costs in different
P Hperiod. If
hiring cost in period t is CH
.
Then
hiring
cost
would
be
t
t Ct Ht .
Firing costs: For \$2000 per fired worker
T
X
2000Ft
t=1
F
If costs of firing changes in each period,
P Fthen, assuming Ct is cost of firing
in month t, total cost of firings is t Ct Ft .
Regular Salaries: Workers are paid \$1500 a month, thus regular salary costs
are
T
X
1500Wt
t=1
If salaries vary in each month (and we know what they will be at the
S
beginning of planning horizon)
P Sthen, assuming Ct is salary in month t,
total salaries are given by t Ct Wt .
Overtime Costs: Overtime work is \$25 an hour, so the overtime costs is
T
X
25Ot
t=1
If overtime wages
and CO
t is overtime rate for month t, then total
Pvary,
O
overtime cost is t Ct Ot .
1 The model presented in class had an error: It was stated that 160+20=180 hours are
available and thus, Pt ≤ 180
Wt . In fact, production is limited by the actual hours available
4
not potential hours. The spreadsheet presented in class originally was correct, however.
5
Raw Material: It costs \$5 of raw material per pair of shoes, so the cost of raw
materials is
T
X
5Pt
t=1
If raw material costs varies in each month, then if CR
is the cost of raw
tP
material in each month, the total raw material cost is t CR
t Pt .
Inventory Costs: It costs \$30 per pair of shoes to hold in storage. Thus the
cost of inventory is
T
X
30It
t=1
I
If inventory varies over months, and CP
t is the inventory cost in month t,
then total inventory costs is given by t CIt It .
In addition all decide variables are assumed to be nonnegative.
The Linear Program:
Now the linear program is given by
PT
PT
PT
PT
PT
PT
minimize
t=1 1600Ht +
t=1 2000Ft +
t=1 1500Wt +
t=1 25Ot +
t=1 5Pt +
t=1 30It
subject to Wt − Wt−1 + HT + FT = 0
OT ≤ 20WT
(W0 = 3)
Pt ≤ 160W4t +Ot
It−1 + Pt − It = Dt
(I0 = 50)
Ot ≥ 0, Pt ≥ 0, It ≥ 0, Ht ≥ 0, Ft ≥ 0, Wt ≥ 0
3
Extensions
There are unlimited opportunities to extend this model so that it includes more
morsels of reality and detail. I will list some of them below, along with their
formulation.
Integer constraints
When you run the spreadsheet model you will notice that the optimal solution
involves H1 = 6.375. It is possible to spin this result into a realistic solution.
For instance one could argue that this means that you hire 6 people full time
and then the seventh person will work part time for 37.5% of a regular full
time. Similarly it is possible that values for inventory and production turn out
to be non-integer which could be interpreted as an unfinished pair of shoes.
6
If this is satisfactory for your model, then you have nothing to do. But if
these explanations are not satisfactory (for instance your firm does not allow
part time employees, and you cannot allow unfinished products in inventory)
then you must add the integer constraints to your model. This is done in the
spreadsheet model in a different worksheet. Notice that while in the optimal
solution of linear program, no overtime hours are used, in the integer program,
some overtime has to be allocated.
Shortages are allowed
The model we have here assumes that demands are met completely in each
period and no shortages are allowed; this is a consequence of It ≥ 0 condition.
However, there are situations where shortages are desirable. For instance, in
some situations demand in an early month could be so high that no amount of
past inventory and production level can meet it2 . To allow for shortages, we can
drop the requirement that It ≥ 0 and interpret negative inventory as backlog.
This would not work, because then the cost of shortage would be negative. The
−
correct procedure is to create a new set of variables for shortages, say
and
P It I−
I−
a cost for each undelivered pair of shoes, say Ct . Then the term t Ct I−
t
should be added to cost, and the production balance constraints should be
changed to:
−
It−1 + Pt = Dt + I−
t−1 + It − It
Additional upper and lower bounds on decision variables
There are numerous scenarios where we may have additional bounds on hiring,
firing, workforce, shortage and inventory levels. For instance, we may not want
to hire more than a certain number of people per month. Or due to political
and public relations reasons, we may wish to limit firing per month to a certain
level. There may be good reason to have a minimum inventory carried every
month, for instance to guard against unexpected sudden decrease in labor force
or surge in demand.
All these constraints can be easily incorporated in the linear programming
model.
Exercise 1
Suppose there is a fixed cost of \$15 per pair of shoes for shortages. In addition we want to limit hires per month to 4, and firing to 2. And we wish to
have a minimum of 40 pair of shoes in inventory per month. Incorporate these
constraints in both linear and integer programming model in the spreadsheet
and observe the results. For the linear programming model generate sensitivity
analysis, and comment on the shadow price of each of these new constraints.
2 The way we have set up our model for the shoe problem, this can never happen, because
we have no bounds on the number of workers we can hire. But in reality, due to various
logistic reasons such as space limitations, there should be upper bound constraints on the size
of workforce in each month. In that case, it is possible to have impossible to meet demand.
7
Outsourcing and Subcontracting
Sometimes there may be economic advantage to buy part of the goods from
outside rather than producing them inhouse. Thus we need to add yet another
set of decision variables, Qt , for the amount of shoes bought from outside;
cost of each such pair of shoes is CQ
need to add the cost of these
t . Then we
P Q
outsourced shoes into the objective function:
t Ct Qt . And the production
balance constraints should now be modified to
−
It−1 + Pt + Qt = Dt + I−
t−1 + It − It
Exercise 2
Suppose up to 20% of the shoes can be purchased from a subcontractor in each
period at the price of \$42 per pair. Add these constraints and modifications
to the model and resolve the problem for both linear programming and integer programming models. For the linear program generate sensitivity analysis
report and comment on the shadow price of new constraints.
More realistic cost functions
In our model we have assumed a linear cost function throughout. This may
be appropriate in most cases, but in a couple of situations it may not be an
accurate model.
First, you may argue that for inventory you either have to lease a storage
area if you carry inventory, or you have to pay no cost if you do not have
inventory. In case you go ahead and lease the storage area then the cost stays
constant until you have filled up the storage. Anything beyond that you have
to lease another storage unit and your cost jumps. A third possibility is a fixed
lease charge, plus a linear cost proportional to the amount of inventory. These
three models are shown in the following graphs.
Inventory
Inventory
Inventory
Cost
Cost
Linear cost
fixed cost lease
8
Cost
linear plus fixed
It is possible to formulate all these kinds of cost functions using integer programming techniques. (This will discussed in future lectures.)
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