Chapter 4 Production Planning - gezabottlik.com

Chapter 4 Production Planning

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Before we can address the scheduling of detailed tasks, we need to have a long term plan

that is consistent with our resources. By resources in this context we mean those that require a

relatively long time to acquire. Examples of these are operating rooms in hospitals, automated

assembly equipment in factories, airplanes for airlines, etc. We can further segregate the time

scale for these acquisitions. For the purposes of this text we will assume that large acquisitions

such as those mentioned above are in place. Less time consuming activities, such as hiring

personnel or arranging for work to be done by a third party can be included in our approach to

planning. For the sake of convenience, we will break our planning into three phases – plans that

are made a year in advance, those that plan for about three month, and the remaining ones that

can be immediate or about up to two weeks.

This chapter concerns itself with the one year horizon. As most of the mathematics and

practices in this area developed in the context of manufacturing, much of the terminology refers

to it. However, the principles and methods are equally applicable to transportation, healthcare,

distribution, mining and any other area you can think of.

The planning process is driven by four forces:

1) Demand (orders) from customers and/or our own forecasts of requirements

2) Our available resources

3) Our ability to alter the near term resources

4) Any limitation imposed by practicality (such as work hours in a day) or management

directives (such as limits on overtime)

Each of the above may either be known or estimated to the best of our ability. Perhaps it

is best to begin our study with an example.


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Example 4-1. A manufacturing operation consists of the production of a single product. The

demand, or orders for the product are reasonably well known for the coming 12 months (our

horizon). We know how many hours of labor are required to produce each unit of product. We

also know how many workers are available at the beginning of the horizon. Further, the cost of

carrying inventory from one month to the next, the cost of laying off or hiring a worker are all

known. Our objective is to minimize the total cost of satisfying all of the demand by the end of

the horizon (a year in this case).

To put this example into further perspective, let’s examine the problem in general. For

each month we need to decide how many units of the product to produce. At its simplest this

would require producing exactly what the demand called for. There are three possibilities – the

demand is either less, the same or more than our ability to produce with our existing labor force.

We have the following choices in case it is more:

1) Forego satisfying the demand altogether

2) Hire additional workers to meet the demand

3) Convince the customer to accept a smaller quantity now, and accept the delivery of the

balance in the next month (i.e., allow shortages)

4) Have our existing staff work extra hours (overtime)

5) Sublet (farm out, offload, subcontract) the shortage to a third party

6) A combination of all of the above

If the demand and our ability to produce are the same, we do not have to take any action.

Finally, if the demand is less, the following options are open to us:

1) Let some of our staff go (lay off, fire, terminate, furlough)

2) Produce up to our ability and use the excess for future demand (Inventory)


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3) Produce more than our ability with the same actions available under the first case.

To determine which of these choices to exercise, we need to define a measurable

objective. The most straightforward is the total cost of the annual plan. Naturally, we are subject

to a number of restrictions (usually referred to as constraints), the most common of which are

listed below:

1) The total demand for the product must be satisfied by the end of the planning horizon or

the customer will not accept late shipments

2) There is a limited number of hours available in each month (usually 160 per worker)

3) Each unit of product requires a specific number of labor hours to produce it

4) The number of extra hours (overtime) that a worker can perform in a month is specified

As we shall see later, each of the ideas mentioned so far may be modified according to

the conditions of our specific problem. We are now in a position to specify our first example in

detail;

Given dt – demand for the product in month t, 1 ≤ ≥ݐ 12

Cs - Cost of subcontracting a unit of product

Wo – Initial number of workers before month 1

Io – Inventory of the product before month 1

n – Hours required to produce a unit of product

M – The number of regular hours a worker can work in a month

CI – The cost of carrying a unit of product from one month to the next

CH – Cost of hiring a worker

CF – Cost of laying off a worker

R – Cost of one regular hour of labor


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O – Cost of one hour of overtime labor

OTL – Number of overtime hours a worker may work in a month

But how are we to solve the problem? It is important to first precisely define the variables

that are under our control:

Xt – The number of units of product to produce in each month (does not include

subcontracted units)

It – The amount of inventory to carry from month t to t+1

Ht – The number of workers to hire in and for month t

Ft – The number of existing workers to terminate at the beginning of month t

Ot – The total number of overtime hours worked in month t

St – The total number of units subcontracted in month t

Figure 4-1 Data for the planning example 4-1

Once we have assigned numbers to each of the given items (Figure 4–1) we need to

decide just exactly how to minimize the total cost. Figure 4-2 shows the spreadsheet with all

variables entered, as well as all conditions satisfied and an arbitrary (guessed) solution. We also

express our previous definitions in equation form:


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Figure 4-2 Manual Spreadsheet solution for satisfying the demand exactly in each month

The size of the work force (Worker balance) W୲= W୲ି ଵ+ H୲− F୲ for each t

The inventory balance I୲= I୲ି ଵ + X୲− d୲+ S୲ for each t

The work balance, the hours required to produce the units must be less than or equal to

the available hours X୲≤ W୲(M + O୲)/n for each t

Demand satisfaction ∑ d୲= ∑ (X୲+ S୲)ଵଶ୲ୀଵ

ଵଶ୲ୀଵ

Overtime limit O୲≤ O for each t

All variables must be positive X୲, I୲, H୲, F୲, S୲≥ 0 for each t

The total cost = sum of the cost of (Regular hours, overtime hours, hiring, firing,

inventory, subcontracting)

[RW୲+ OO୲+ H୲Cୌ + F୲C + I୲C୍+ S୲Cୗ]

ଵଶ

௧ୀଵ

It is extremely unlikely that a guess will be optimal. In fact, completing the guesses is not

that simple because we have to satisfy all the conditions. At this point you should try to better


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my guess by creating the spreadsheet and producing your own estimates without recourse to any

mathematical techniques.

SOLVING FOR THE OPTIMAL COST

Fortunately there is an excellent method available, as long as all our variables and

conditions are linear, i.e., do not contain powers or cross products. The method is referred to as

Linear Programming (LP or the Simplex Method, which was developed by Dantzig in 1947). It

relies on the principle that no matter what set of values for each variable we start with, the

solution always proceeds to a better set of variables until it is unable to do so. And the stopping

point is guaranteed to be optimal. Most of our problems, however, are not linear and optimality

is not guaranteed, although usually achieved. Solutions generally depend on the starting values

of the variables, so it is a good idea to start with a more or less feasible solution. Naturally, there

is a large body of mathematics behind the method which is beyond the content of this book.

There are many references that have all these details. Suffice it to say that we have the simple

expedient to use Excel’s Solver Add-In as long as the problem is small enough. Much more

sophisticated software tools are commercially available to solve real applications. The

appropriate dialog box for our example is shown in Figure 4-3.The resulting solution is shown in

Figure 4-4. Since it is an integer problem for the work force changes and not the production

numbers (answers were rounded), there is a possibility that slightly better solutions exist.


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Figure 4-3 Excel Solver dialog box for example 4-1


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Figure 4-4 Solver solution for satisfying the demand exactly in each month for example 4-1

It is relatively straightforward to extend the basic problem to more than one product by

adding the subscript i to the variables and constraints. Many operations periodically choose a

method other than general optimization. There are two commonly used methods:

1) Chase – where each demand is exactly satisfied by varying the number of workers

2) Level – where demand is satisfied by varying the inventory and the work force and

keeping the production constant

A chase method may also modify overtime and offloading

(X୧୲+ S୧୲= d୧୲for each i and t). This has the benefit of creating zero inventory – a very

desirable outcome. Unfortunately, the method is rarely successful in practice as controlling the

variables and the available material is very difficult. The most common outcome is frequent late

deliveries (shortages) for which our model does not yet account.

The level method produces the same number of units of each product in every period

(X୧୲+ S୧୲= X୧୲+ ܵ௧ for each i and t). The obvious benefit is no disruption to the routine

production of the products and perhaps a constant work force. However, higher inventories and

shortages tend to be introduced.

Example 4-2. Level approach to example 4-1. We have to rewrite some of the equations to make

the production constant after the first period. In this instance we have also fixed the work force

after the first period. Figures 4-5 and 4-6 have the dialog box and the resulting solution. This is

an excellent example where judgement is needed in interpreting the solution. In this instance,

because of the relative value of subcontracting and terminating employees and the cost of an

hour of labor, a large quantity of product is to be subcontracted. This would result in a

substantial increase in per hour labor costs (see Appendix A) and not be in our best interest.


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Figure 4-5 Dialog box for the level solution (example 4-2)


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Figure 4-6 Solver solution for the level work force (example 4-2)

This leaves us with one important consideration – how do we treat the cost of not

satisfying a demand in the month requested. Usually, there is no immediate financial penalty –

we tell the customer that a shipment will be delayed and the customer grudgingly accepts the

condition because he has already built in a safety factor (see the chapter on inventory and

consider it from the customer’s point of view) and because more often than not it is difficult or

impossible to obtain the same product from someone else. However, frequent such occurrences

will eventually lose the customer and/or damage our reputation with other customers. Therefore

it is important to assume some penalty that will minimize such occurrences.

Example 4-3. We will redo our original scenario while allowing shortages and optimize while

not insisting or either chase or level. For this we do need one more variable for shortages, SGit,

and the monthly shortages are added as a variable and the inventory balance equation adjusted

for it. The shortage cost is assumed to be $200/unit.

The inventory balance I୲−SG୲= I୲ି ଵ + X୲− d୲+ S୲− SG୲ି ଵ for each t

The dialog box and the solution are shown in Figures 4-7 and 4-8.


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Figure 4-7 Dialog box for Example 4-3

The conditions of each operation would dictate whether one chooses level, chase, or

optimization, but if practicable, one should choose optimization.


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Figure 4-8 Solver solution for the case of shortages (example 4-3)

Chapter 4 Production Planning - gezabottlik.com

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