3. Product Mix Models

A field in which linear models find wide applicability is that of decisions related to some aspects of the production process. In particular, production mix problems correspond to the case in which it is necessary to choose, among many possible products, the quantity to produce, taking into account the availability of some scarce resources (like, e.g., available workers). An assumption frequently made in some of these models, is that of operating in a market in which the supply of finished products always finds a suitable demand in the market. This assumption might however be somewhat relaxed without necessarily having to include stochastic models of the demand.

The basic problem therefore reduces to choosing the quantities to produce to maximize the total gain, within the constraints imposed by the availability of resources. The components of these models are:

  1. a set of possible products;

  2. a set of available resources;

  3. the quantity of each resource required for the production of each product;

  4. the net gain for each product.

Also in this model we can highlight the basic elements of all optimization models and, in particular,

  • index sets: the set of possible products and the set of resources;

  • parameters: the unit gain for each product (a parameter indexed by the set of products), the availability of resources (a numeric parameter indexed by the set of resources), a table in which, for each product and for each resource, the quantity of resource needed to produce one unit of this product is given; this parameter is indeed a matrix indexed by the Cartesian product between the set of resources and the set of products; other parameters may also be present, such as lower and upper limits on the quantity to be produced;

  • decision variables: the quantities to be produced, a vector of variables indexed by the set of products. It is worth observing that there is only a difference between parameters and variables: variables are under the control of the decision maker, while parameters are considered known and constant, and cannot be modified by the solver;

  • constraints:

    • lower and upper limits on the quantities to be produced (two sets of constraints indexed by the set of products);

    • maximum resource consumption (a set of constraints corresponding to the different types of resources).

  • an objective function; in the example we assume that we wish to maximize the total gain.

It may be useful to put the elements of the model in a scheme.

model

ProductMix

Sets:

  • \(\set{Products}\) : product types

  • \(\set{Resources}\) : available resources types (work hours, machines, prime material, …)

Parameters:

  • \(\param{Gain}_p\) : unit gain for product \(p \in \set{Products}\)

  • \(\param{Avail}_r\): maximum availability of resource \(r \in \set{Resources}\)

  • \(\param{MinQty}_p, \param{MaxQty}_p\): minimum and maximum quantities to be produced, for each \(p \in \set{Products}\) ;

  • \(\param{Consumption}_{rp}\): quantity of resource \(r \in \set{Resources}\) required in order to produce one unit of \(p \in \set{ Products}\);

Variables:

  • \(\var{x}_p\): quantity to be produced, for each \(p \in \set{Products}\)

Constraints:

  • Lower and upper bounds on production:

    \[ \param{MinQty}_p \leq \var{x}_p \leq \param{MaxQty}_p \]

for every \(p \in \set{Products}\);

  • Resource consumption:

    \[ \sum_{p \in \set{Products}} \param{Consumption}_{rp} \var{x}_p \leq \param{Avail}_r \]

for each \(r \in \set{Resources}\)

Objective: Maximize the total gain:

\[ \max \sum_{p \in \set{Products}} \param{Gain}_p \var{x}_p \]

Of course this is an overly simplified model of a realistic situation; by the way:

Warning

the above statement is true for all of the models described in this volume!

Several deliberately simplified modeling assumptions are made in order to keep models easy to understand without unnecessary details. In realistic situations, complex models will be built by joining parts of different models.

In this product mix example, many additional aspects might be considered, like, e.g.:

  • fixed costs: the decision regarding the production of a particular piece must sometimes be based on the presence of a fixed cost to be paid just to start production. As an example, starting production might require “warming up” a machine or a plant until it is ready to produce, thus consuming costly energy; in other situations, hours should be initially devoted to training. This modeling aspect will be seen in the chapter on Fixed Charge models and will require the introduction of binary variables;

  • dynamic models: demand forecasts do not concern a single period, but a set of consecutive production periods, linked together through the introduction of an inventory. These models will be introduced in the section on Multi Period Models.

  • stochastic models: some parameters, like, most notably, the demand for a certain product, can be seen as realizations of random variables. Even when dealing with make-to-order systems, in which production is started based on known purchase orders, randomness is always present when dealing with future planning periods. And, of course, when the company operates in a make-to-stock way, demand needs to be somewhat anticipated and, thus, estimated.

    In many situations in which uncertainty is present decision makers simply replace random variables with an estimate, usually the expected value or the most likely one. However, optimization approaches do exist which are capable of taking into account uncertainty in an explicit way, thus accounting, e.g., not only for expected values but also for the variance of random variables. A very short and simplified introduction to these models will be presented in Chapter Stochastic linear optimization models, where it will be shown how far from optimality a policy might be when summarizing a stochastic quantity by its expected value.

    application

    Choosing among different production modes

    Sometimes a product mix model is not used to decide, among several products, which ones to produce and in which quantities. Frequently, the quantity to be produced is fixed and known; the decision problem might in this case be that of choosing among multiple production modes, instead. In other words, we are not required to decide how much to produce, but how much of the production should be allocated to different plants, with different resource consumption, different resource availability and possibly different production costs and, thus, gains. Formally the problem is identical to the standard one. As an example, consider steel rods of different diameters to be produced using different machines. The available machines can produce material of different diameters, with production rates and costs which depend on the machine.

    The data file below is an example of this situation.

    steel.dat
    param: 
    RESOURCES : Avail    Speed MinDiam MaxDiam HourlyCost:=
    mach-1       35       150     3       6            10
    mach-2       35       100     5       8            15
    mach-3       35        75     6      12            17
    ;
    param: TYPES: Qty    diameter   revenue :=
    A             218000      4      0.017
    B             114000      6      0.019
    C             111000      8      0.02
    ;
    
    let PRODUCTS := {t in TYPES, r in RESOURCES: 
                     MinDiam[r] <= diameter[t]
                     and MaxDiam[r] >= diametert]};
    
    let  {(t,r) in PRODUCTS} 
        consumption[t,r] := 1/(60 * Speed[r]);              
    # resource consumptions (in fractions of one hour)
    
    let {(t,r) in PRODUCTS} profit[t,r] := revenue[t] 
                     - HourlyCost[r] * consumption[t,r];
    # net gain per unit produced per machine
    

    Considering the data in the example, the resources in this case are machines which differ one another for their availability (hours per week) and production speed: in this example the actual availability doesn’t just depend on the available hours, but also on the speed of the machine. So, for example, if the speed of machine 1 is 150 meters per minute, in order to produce one meter \(1/(60 \cdot 150)\) hours would be required. In this case, the consumption of each resource does not depend on the product itself, but on the machine type. There is also another special type of dependence: not every machine, in fact, is capable of producing all product types. As an example, product type A can only be worked on machine 1 (its diameter is within the interval 3 to 6 which characterizes this machine), while product type B (diameter 6) can be produced both on the second or on third machine, but not on the first one. The net gain resulting from the decision to produce a type of rod on a specific machine is obtained as the difference between the revenue associated with this rod and the cost of using the machine. If we wish to formalize this model, the following scheme might be a guide.

    model

    Choice between production modes

    Sets:

    • \(\set{Resources}\): available machines

    • \(\set{Products}\): product types;

    • \(\set{Match} \subseteq \set{Resources} \times \set{Products}\): set of feasible combinations between product types and production-enabled machines.

    Parameters:

    • \(\param{MinQty}_p, \param{MaxQty}_p\): minimum and maximum quantities to be produced for each \(p \in \set{Products}\) ;

    • \(\param{Avail}_r\): resource availability \(r \in \set{Resources}\)

    • \(\param{Consumption}_{rp}\): amount of resource \(r \in \set{Resources}\) needed to produce a unit of product \(p \in \set{Products}\); this parameter can be an input data, or it can be computed as a function of other parameters, as in the case of the included file, where it is based on the production speed of each machine;

    • \(\param{Cost}_r\): hourly cost of using resource \(r \in \set{Resources}\);

    • \(\param{Revenue}_p\): unit revenue for product \(p \in \set{Products}\)

    • \(\param{Gain}_{pr}\): difference between revenues and costs

      \begin{align*} \param{Gain}_{rp} = \param{Revenue}_p - \param{Cost}_r \times \param{Consumption}_{rp} \end{align*}

    for \(p \in \set{Products}\), \(r \in \set{Resources}\), provided that \(Match(p, r)\) is true, i.e., that product \(p\) can be obtained on machine \(r\) ;

    Variables:

    • \(\var{x}_{rp}\): quantity of \(p \in \set{Products}\) to be produced on machine \(r \in \set{Resources}\);

    Constraints:

    • Minimum and maximum production:

      \begin{align*} \param{MinQty}_p \leq \sum_{r \in \set{Resources}: Match(p, r)} \var{x}_{rp} \leq \param{MaxQty}_p \end{align*}

      for each \(p \in \set{Products}\).

    • Resource availability:

      \[ \sum_{p \in \set{Products}: Match(p, r) } \param{Consumption}_{rp} \var{x}_{rp} \leq \param{Avail}_r \]

      for each \(r \in \set{Resources}\)

    Objective: Maximize the total gain:

    • Gain maximization:

      \[ \max \sum_{p, r: Match(p, r)} \param{Gain}_{rp} \var{x}_{rp} \]

    It can be seen that the structure of this model is quite close to that of the generic product mix; some variants had to be introduced just in order to take into account the characteristics of different machine and compatibility issues.

3.1. Dual of the product mix problem

Consider a simplified version of the product mix model introduced before:

\begin{align*} \max \sum_p \param{Gain}_p \, \var{x}_p & \\ \sum_p \param{Consumption}_{rp} \, \var{x}_p & \leq \param{Avail}_r & \forall \, r \\ \var{x}_p & \geq 0 & \forall \, p \end{align*}

The dual of this problem might be easily derived, either by reformulating it as a standard linear optimization model, or by considering it as the dual of a standard one; in this case the only “non standard” aspect of the problem is the non negativity of the variables, which can be traced back to a primal with inequalities in place of equations. Formally the dual of turns out to be:

\begin{align*} \min \sum_r \param{Avail}_r \mathvar{\lambda}_r & \\ \sum_r \param{Consumption}_{rp} \mathvar{\lambda}_r & \geq \param{Gain}_p & \forall \, p \\ \mathvar{\lambda}_r & \geq 0 & \forall \, r \end{align*}

This problem lends itself to an economic interpretation. A work agency would like to buy the available resources from the producer. The agency will try to purchase all the available resources of the company and will propose a unit price \(\mathvar{\lambda}_r\) for resource \(r\), trying to minimize the total cost. In order to convince the company to release manpower and other resources, the proposed prices must be competitive. The dual constraint associated with product \(p\), represents the fact that, if the manufacturer decides to sell the resources requested to produce one unit of \(p\) to the dual agency, the revenue will be at least equal to the unit gain.

Duality implies that there will be no advantage for either actor, neither the producer nor the agency: optimal values are identical and a strictly competitive price (in the dual) will be allowed only for products types which will not be produced in the optimal mix.

As a simple illustrative example consider the following data.

\[ \begin{array}{lrrrrrrrl} \text{prod:}& \text{1} & \text{2} & \text{3} & \text{4} & \text{5} & \text{6} & \text{7} & \\ \hline \text{carpentry} & 7 & 5 & 9 & 10 & 10 & 12 & 15 & \leq 2000 \\ \text{coating} & 2 & 2 & 2 & 3 & 3 & 3 & 3 & \leq 1500\\ \text{assembly} & 2 & 2 & 4 & 7 & 9 & 15 & 18 & \leq 1700\\ \text{control} & 1 & 1 & 1 & 2 & 1 & 2 & 2 & \leq 300 \\ \text{packaging} & 1 & 1 & 1 & 1 & 2 & 1 & 0 & \leq 500 \\ \hline \text{Gain} & 10 & 18 & 20 & 25 & 27 & 28 & 35 & \max \end{array} \]

It is readily seen, possibly through a suitable implementation, that an optimal solution consists of producing 200 units of product 2 and 100 of product 5, with a total gain 6300. The resources “carpentry” and “control” are used at full capacity.

Solving the dual, the variable associated to carpentry gets a price 1.8, the one associated to control 9, all the others being fixed to 0. With this price, while there is no strict convenience, for the product mixer, with respect to the production of products 2 and 5, apparently there might be a gain for what concerns other products. As an example, selling the resources needed to produce one unit of product 1 will generate a revenue of 21.6, significantly higher than 10, the unit gain on that product. However, the producer has no interest in this product, as it does not appear in the optimal solution.

We note in passing that duality helps in gaining confidence on the solution reported by a numerical solver: if we plug the above solutions in the primal and dual respectively, we can immediately check that both are feasible and satisfy complementarity, without the necessity of going through the computation of a numerical solver.

We observe that duality is an extremely powerful modeling, and not just computational, tool. Whenever a problem which models a real situation is formulated, its dual might be interpreted in the same context and sometimes gives rise to interesting models. Finding a reasonable dual interpretation might not be easy, but is surely worth the effort.

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