PRODUCTION MIX PROBLEMS: FORMULATION AND SOLUTION STRATEGIES

This paper examines the sub-optimality of generalized greedy heuristics for solut ion of a certain well-known production mix problem formulation . The drawbacks of not modelling routing in the mix problem formulation are discussed, and an alternati ve mix prob lem formulation (with assoc iated solution strategies) is proposed . OPSOMN1ING Hierdie artikel verduidelik die sub-optirnaliteit van ' n algemene gulsige heuristiek vir 'n sekere bekende produkmengsel probleemformulering. Die nadele van weglating van produk roete inligting in die formulering is bespreek, en ' n alternatiewe formulering (met geassosieerde oplossingsstrategiee) is voorgestel. I This research was supported by a visiting assistant professorship from the School of Industrial Engin eering, Purdue University, West Lafayette, Indiana , USA, and a research grant from the University of Pretoria. 107 http://sajie.journals.ac.za


INTRODUCTION
The production mix problem (PMP) that we consider is a mathematical program that aims to identify, in multi-workstation and multi-product manufacturing environments, the proportion that each product should form of the total production volume in order to maximize production revenues over a finite-length planning period .The optima) mix is constrained by market demand and workstation capacities.
Formally, the PMP is defirred as follo ws: PMP: subject to: (1) (2) (3) (4) where X ; denotes the quantity of type i product that should be produced, r, denotes the revenue accruing from production of one unit of type i product, tij denotes the capacity consumption (i.e., time) per unit of product i on workstation}, C; denotes the capacity available on workstation i. and D; denotes the maximum market demand for product i.~+ denotes the non-negative integers.
The objective function (I) requires that the total revenue be maximized.Constraint set (2) requires that the capacity constraints of each workstation be respected .Constraint set (3) requires that the market demand be treated as an upper bound on production of each product type (this will not be appropriate in some environments, notably in produce-to-stock environments; in such cases constraint set (3) is discarded).Finally, constraint set (4) requires that production volumes be integer-valued and non-negative.The assumptions are as follows: there are at least two products; there are at least two workstations; O<C j <00 'II}; O<r; <00 Vi; O~t ij < co V i ,}; 2:>ij > 0 vi .andD; >0 V i.
When constraint set (4) is relaxed to Xi E 91 + Vi we call the resultant program the relaxed PMP .The relaxed PMP is a linear program (LP).
The work of Goldratt [9,10] on the " theory of constraints" has increased general awareness of the product mix problem in the last several decades.The theory of constraints (TOC) holds that production environments can be effectivel y managed by identifying and managing constraining bottleneck workstations, and has served to popularise some well-known basic results from queueing theory among production management professionals.One of the key issues in constraint-based production management is the identification of bottleneck solutions for some instances of the PMP , and shows that the greedy heu ristic can in fact perform arbitrarily poorly (i.e., no m inimum per formance guarantee can be given for the greedy heur istic ).The obvious conclusion arising from these observations is that the PMP should be solved using an exact LP solver OIlZV.
The PMP does not model routing constraints.Ke y consequences of this ormssion are disc ussed , namel y the opportunity to optimize the product-workstation routing strategy is foregone, and the possibility of unexpected massive wo rk-in-progress accumulati on s is rais ed (drawin g on results from the theory of queueing network stability).An alternative mix problem formu lati on (the rate mi x problem (R MP), which ha s comp utationa l requirement s simi lar to that of the PMP) is described that explicitly in corporates rout in g information.Solution strateg ies for optim isation of both the routings and mi x usin g the RMP are described, and g uidelines are pro vided for th e preliminary identification of pos sible path olo gical workin-process effects in the mix and routing solutions.
In summary, the objectives of this paper are as follows: to re visit and gene ricall y and fundamentall y explain Plenert 's claim that gr eedy heuristics are a poor choice of solver for the PMP; to point to a ke y drawback in the PMP formul at ion and the con sequences thereof; to propose an alternative mix problem formulation that allows opt imi sation of both the mix and routing strategy, and to pr ovide simple pract ical guidelines for pr eliminar y identification o f pathological work-in-process accumulations.
The organization of this paper is as follows: the performance of greedy heuri stic s for mix problems is analysed in §2.The consequences of omitting routing information in mix problem formulations is presented in §3.An alternative mix problem formulation and associated solution strategies appears in §4.Guidelines to chec k system stability are provided in §5.Conclusions are rendered in §6.

GREEDY HEURISTICS
Pure greedy heuristics build a solution to a problem over sev eral iterations, assigning the numeric value of exactly one variable in each successive iteration, by optimising a surrogate objective measure (that is, a me asure other than that contained in the objective function ).The iterative process continues until all variables ' values have been fixed (possibly set equal to zero).There is no mechanism in pure greedy heuri stic s for " undo ing" the variable value assignments of preceding iterations.
Specifically, the pure greedy heuristic (minor variations are possible, but do not materially impact the structure of the solution) for an n-product and m-workstation PMP is: Pure Greedy Heuristic for the PMP Step 0: Initialize , letting C.i = q for j = 1,....m denote the remaining workstation capacities; let d, = D, for i = I, .. .,ndenote the unsatisfied market demands ; and let X i = 0 for i = 1,. ..,n denote the production levels of each product.

1:
Identify the critically-constrained (bottleneck) workstation.The bottleneck is workstationj* where : http://sajie.journals.ac.za Broadly speaking, the pure greedy heur istic (see Plcnert [23]) identifies a so lution to the PMP by iden.tifying a bottlene ck workstation, and then maximizes the throu ghput of the produ ct generatmgthebest profit per unit of bottleneck workstation capa city.In step 1 ofthe greedy heuristi c we co ns ider just one of sever al way s of identifyin g bottleneck wor kstations (there the bottleneck is the defined as the most overloaded work station ).Alt ernati ve "static" definitions are po ssible.Bottlenecks may shift (Lawrence and Bu ss [14]) dynamicall y over time if the "bottleneck" workstation at any given instant in time is defined as the most conges ted workstation.The pur e greedy heuri stic for the PMP doe s not acco unt for dynamic bottleneck s.
Luebbe and F inch [17] compare the performance of the pure greedy heuri stic to optima l solutions of an instanc e of the PMP first published by Goldratt.Luebbe and Finch' s erroneou s conclusio n, based upon the com parison of the greed y heuri stic perform an ce with the optimal solution of ju st one instance of the PMP , is essenti ally that the greed y heu risti c is not outperformed by LP solvers.Other authors have arri ved at the same incorrect con clusion (e.g., Geyser [8]).Plenert [24] co ncludes that the greedy heuristic and exact LP solvers will perform identi call y, based upon two instances of the PMP (plenert compared the performance of linear program so lvers, the greedy heuristic, and a trad itional accounting algorithm).Lee and Plenert [15] appear to have provided the first instance of the PMP where an LP solution out-performs the greedy heuri stic.Dissenting commentary on Lee and Plenert [15] was published by Mada y (18], and Posnack [25].Plenert [23] later published other instances of the PMP where the greedy heuristic is out-performed by linear (or integer) program solvers.Fredenhall and Lea [6) afterward proposed a revised greedy heuristic for the PMP bas ed upon exhaustive search in the solution neighb ourhood of the greedy heuristic solution, seemingly suggesting that heur istics are still worth a look , despite the fact that Plenert and Lee's examples provide concrete proof that the greedy heuristic returns sub -optim al solutions on some instances of the PMP (and despite the fact that an exact pol ynom ial-time solver is available).While the superiority of LP solvers over the greed y heuri stic is clear to some authors (e.g., Miltenburg [20]), it is worth revisiting the issue in more depth to understand why the greedy heuristi c may fail and to quantify its performance bounds, if any.In the next subs ection th is paper explains the sub-optimality (and potentially disastrous solution quality ) of gre edy heuristics, concluding that that the PMP should be solved using an exact (LP ) solver only .The paper later addresses modell ing issues, and suggests that even the PMP formul ation should be overlooked in favour of an alternative rate mix formulation that explicitly encodes routing information .

PERFORlVIANCE OF THE PURE GREEDY HEURISTIC
To recapitulate the importance of Lee and Plenert's findings , consider the followin g 2product , 2-workstation instance of the PMP: PMP Instan ce: http://sajie.journals.ac.za (9) Here ( 6) and ( 7) are capacity constraints corresponding to the constraint set of equation ( 2), while ( 8) and ( 9) are demand constraints corresponding to the constraint set of equation (3) .Constraint (10) corresponds to the constraints in equation ( 4) in the prototype PMP formulation .
The optimal solution for this instance is (x/, X2) = (10, 45), with a revenue total of 65.Note that the optimal solution requires 100% utilisation of both workstations.Now, consider the solutions generated by the greedy procedure: workstation I is the bottleneck (step 1), and the marginal returns per unit of bottleneck capacity (step 2a) arc 2/3 and 1 for product 1 and 2 respectively.The heuristic indicates that we should produce 50 units of product 2 (step 2b), and eventually terminates with the sub-optimal solution of (Xl, X2) = (0, 50) that generates a revenue total of only 50 (since X 2 = 50 forces x , = 0 in equation (7» .Did we identify the correct bottleneck originally?To check, suppose that workstation 2 is the bottleneck.The marginal returns per unit of bottleneck capacity are 2 and 0.5 for product I and 2 respectively, and the production of x , = 25 is ind icated.The heuristic eventually terminates with the suboptimal solution (x., X2) = (25, 0) which again produces a total revenue of only 50 (since x , = 25 forces X2 = 0 in equation (6» .Neither choice of workstation as bottleneck produces a solution that is even within 80% of the optimal revenue level for thi s instance. Artificial instances of the PMP can be constructed to make the greedy heuristic look arbitrarily bad.That is, we arc unable to find a lower bound on the performance of the greedy heuristic.Consider the following n-product and n-workstation instance of the PMP (note that m = n): let ri= 1 for i = I, .. ., n; C j = 100 for j = 1,.. .,»;D ,> 100 for i=l," .. ,/1 and let Writing this out we get:

PMP Instance:
subject to: Let E be a small constant strictly less than one , (for example, choose c = (11-1 r' and note that we assumed earlier that n 2: 2 to avoid trivialities) .All machines have the sarnci nitial load for the purposes of bottleneck identification.Arbitrarily suppose that machine k is "the" bottleneck.Any product ex cept for product k now has a return per bottleneck hour of c• l (which is greater than or equal to one , and the return per bottleneck hour for product k is exactly one).The greedy heuristic will select any product I ;c k as the most profitable product (per bottleneck hour), and ass ign x, = 100 to satisfy the demand constraint on product I and all workstation cap ac ity constraints.The heur istic eventually terminates with the solution of Xi = oVi ;c[ and X I = 100, producing a total revenue of 100 (note that the revenue is .independent of n).It is easy to confirm that the solution (X/, ... ;'xn) = (50, .. .,50) is a feasible solution for this instance, and that the total revenue from this solution is 50n (which is greater than 100 if n>2, and this is not even the optimal solution).As n ~ex) the greedy heuristic produces arbitrarily poor solutions for thi s instance relati ve to the optimal solution (since the heuristic produces a fixed revenue, while the optimal revenue grows as a function of 11) .This example pro ves that no minimum performance guarantees can be found for the greedy heuristic .
Greedy heuristics are principal-direction (axis) edge-following algorithms, and the greedy heuristic for the PMP is no exception.The variable assignment trajectory of the PMP greedy heuristic (which commences at the origin, with all variables initiall y unassigned) is graphically depicted in Figure 1 using thi ck black lines for two samp le relax ed PMP pol ytopes.The relaxed PM? pol ytope is located in the positive orthant, and is generally comprised of a hypercube induced by constraint set (3) cut by facets indu ced by con stra int set (2).That is, when multiple workstation bottlenecks exist, the greedy heur istic may return sub-' optimal solutions.
The behaviour of the gree?y heuristic can be parti all y ch aracterized: let •. , 11 .
(F D denotes the hypercube induced by constraint set (3» , and let: (F?,\./p denotes the relaxed PMP polytope).Let clix) denote the clo sure of set x, and A 1B={x EA: x f!B) where A and B are sets .
Claim 1: if cl(F D \ Fp M P) is convex then the greedy heuristic w ill return the optimal PMP solution.
The proof of the claim appears in the appendix.The claim is highly instructive since it implies that at least two workstations must be utilised in the optimal PMP solution (to ensure that cl(FD \ F PMP) is non-convex) before the greedy heuristic may return a sub-optimal solution.Moreover, these workstations must be "connected" by product routings in the sense that at least two different products are produced on each of these workstations in the optimal mix , i.e., the optimal mix must involve at least two bottleneck workstation s before the greed y heuristic may fail.Hopefully, this situation has not been frequentl y encountered in practice!Unfortunately, there is at least one industrial plant with multiple bott lene cks (where the greedy heuristic under-performed), and it seems plausible that there may be man y more (Bischoff [2]) .
Although the greedy heuristic may perform well on many PMP insta nces, there can be no justification for its use in view of the proliferation of software in office suite software for optimally solving LPs, and the potentially disastrous financial con sequences of not find ing the optimal production mix.
3. The PMP does not incorporate product routing constraints or strategies.This omission has at least two potentially cr ucial consequences:

MIX PROBLEM FORMULATIONS: ISSUES
1. the PMP cannot be used to optimise with respect to routing strategy when routing alternatives exist, and 2. the impl ied routing must be inferred from the PMP solution and technological considerations.It is not hard to see that multiple routing configurations can potentially be inferred from a single PMP instance.For example, the two networks in Figure 2 (which differ only in the order of workstations visited by product 2) have the same PMP formulation.Unfortunately, this is not a tri vial observation because an unstable (this is explained below) routing strategy may be extrapolated from the PMP solution.
A queueing network is said to be unstable if the network does not possess a stationary queue length distribution.In practice an unstable queueing network will experience highly undesirable and wildly oscillating queue length sizes that eventually drift off to infinity.Informally, thi s phenomenon would be characterized in a production setting by pathological and increasingly massi ve fluctuations (oscillations) in work-in-process inventory at two or more workstations, and by a steady overall increase in the work in process (toward infinity) as time increases if the facility were left unmanaged.In practice there is considerable capacity management intervention in an y production facilit y that averts su ch disasters.Nevertheless, it is natural to prefer a stable mix and routing strategy design over an unstable design as a stable design will reduce the need for, and frequency of management interventions , and ma y potentially result in a higher maximal achievable facility throughput.
Until recently it was thought that a queueing network would be stable for any workconserving (workstations may not be idle when work is awaiting service there) service discipline if the utilisation of every station is less than 100% (if the required utilisation exceeds 100% the ensuing inventory explosion is obvious).Unfortunately, research on the stability of queueing networks has shown that networks exist that are unstable even though the utilisation of every station is strictly less than 100%, for certain scheduling policies.
Suppose that til = t i : = 0 and t u = t2l = 0.5 in both networks in Figure 2. It turn s out (see Dai [5]) that Network A in Figure 2 is stable under any work-conserving scheduling policy, whereas Network B is unstable for several scheduling policies.For example, the priority policy (with product type 2 having higher priori ty than product type ] at workstation I, and product 1 having higher priority at workstation 2 in Network B, as indicated in the figure) results in an unstable network..ln fact, Network A has a stable achievable capacity of] 00% at both workstations, while no more than 50% utilisation can be achieved at either station in Network B (beca use high priority service effectively blocks low priority work from entering the system)!Service polic ies pla ya huge role in determining whether a queueing network will be stable or otherwise, and it is interesting to note that Bramson [3] and others have shown that even the extremely common first-in-first out (FIFO) scheduling policy when implemented at all machines, does not guarantee network stability in all networks!Unfortunately, no general technique is known at the present time that will identify unstable mix problem solutions.For the practitioner the easiest way to eliminate the p-osSlbility of an unstable system is to generate a product mix and routing strategy solution, and then check if it is stable by: a) checking to see if (or forcing ) the routing structure meets certain structural requirements that guarantee stability.Guidelines for performing this check are set out in §5; or b) employing carefully screened (using simulation) work sequencing and allocation policies at every workstation that guarantee stability irrespective of the routing structure.This possibility is delicate and many companies prefer to rather simplify the routing structure to guarantee stability (Louis [16]) .
Fortunately, the deficiencies of the PMP formulation in respect of routing modelling can be substantially remedied.The next section describes an alternative mix problem formulation based upon an infinite-length planning period and production rates (rather than volume s) that incorporates explicit probabilistic routing constraints.Solution strategies for obtaining both the optimal mix and routing strategy solutions using this formulation are discussed in §4.

FORMULATION
A finished product in general requires a series of processing operations (and possibly assembly or disassembly operations also); assign each such operation in the routing of a product a unique class identifier so that each class is served at exactly one workstation.We now view the environment as a network of m workstations that together process M classes of "customers" (intermediate or end products).Clearly, there are in general more classes in the network than products in the corresponding PMP formulat ion (i.e, M 2n) .Write OJ for the set of classes served at workstation}.
Assume for now that routings are known and probabilistic: every time a class-i customer completes service it is routed for service as a c1ass-j customer with probability pij.This is a simplified routing model that incorporates the widely-used special , case of deterministic routing (w here Pij E{O.J) .'vi,) , but that cannot approximate state-dependent (e.g., inventorydependent) routing strategies.Let P = (Pij ) denote the M xM routing matrix.The probability thatacustomer leavesthe network after class-i service is 1-" M p .. .Assume P to be L..J ,=I 1) transient.This means that I -P (I is the identity matrix) is invertible, or in other word s, all products entering the product ion facil ity require processing on at most a fin ite number of workstations before lea ving the netw ork with probability one .This assumption implies that the network is open.Let b = (b;) denote the M-dimensional vector of unit class pro cessin g times with hi denoting the unit processing time of class i (to relate thi s to our previous notation, note that hi = Ii) where i ECJ J .
The PMP is a finite-horizon problem (that is, the Ds and Cs in the PMP formulation are specified for a fixed-len gth planning period).A rate-based approach facilitates the formulation of an associated infinite-length per iod problem that we call the rat e mi x problem (RMP ), which is defined as follows : RMP: L i.: -», Vi w here 'i' is the marginal un it revenue for class-i production (usuall y, r/ = 0 for an y class i that is not an end product), a, (a variable in the problem ) is the exogenous arrival rate of clas si customers to the net work (i.c., customers that are completely new to the system), and Ai(a variable) is the effective arrival rate of class-.customers at the workstation that serves class-z customers.The effective arr ival stream of an y class is the superposition of exogenous and endogenous (routed) streams of customers from other work stations.The load P~' is the maximal permissible utilisation (expressed as a fraction) of stat ion i, and /1.7 is the maximal market demand rate for class-i production.Let The first constraint set (12) specifies the dependence of the effective arrival rate of class-.customers on the rate of exogenous and probabilistically-routed endogenous customer st ream s.The second constraint set ( 13) requires that the load on each workstation be strictly less than the maximal workstation load capacity (w hic h must be less than or equal to unity by assum ption).The third constraint set (14 ) requires that all exogenous arrival rates are non-negative.Constraint set (15) requires that the departure rates of a class be less than the rate that the market (demand) can absorb.The objecti ve function (11) calls for the maximization of the total marginal rate of return by optimising with respect to the variables, a, Vi and ~Vi.

RMP SOLUTION STRATEGIES
The RMP formulation (with pre-specified probabalistic routings) is an LP and the optimal mix can be obtained in polynomial time (in practice, even extremely large RMPs will be optimally solved in a negligible amount of time on a computer using an LP solver, ne ver a heuristic) .
One of the most attractive features of the RMP formulation is that it may be adapted to explore and optimise the maximal achievable throughput in 'a production facility, and the sensitivity thereof to routings.While routing options may be limited in many production settings, they playa key role in determining the achievable throughput capacity in a variety of industries with multiple workstations of the same type.An example of such an environment is a brewery, where there are typically a number of fermenting vessels (of possibly dissimilar size) that can produce a variety of beer products, and a number of storage tanks (of possibly dissimilar size) in which beer is stored after fermentation, and prior to bottling.The routing (and the timing thereof) of beer from a fermentation vessel to (possibly multiple) storage tanks profoundly impacts the achievable throughput at some breweries (Meyer [19]).
Consider expanding the RMP which is obtained by adding the constraint set 0 ~Pi} ~1,V i, j to the RMP and treating the routing probabilities (pijs) and mix variables (ajs and "tiS) as optimisation variables.The expanded RMP is a non-linear program (since constraint set ( 12) is now non-linear) that apparently cannot be efficiently solved in large dimensions.One workaround for this problem is to enumerate some, or possibly all feasible routings, and treat each product routing alternative as a unique phantom product in the RMP .Minor modifications to the RMP formulation are required to ensure that the phantom products associated with a real physical product do not exceed the demand rate constraints (constraint set (15»).This approach requires that we again need only solve an LP.Although the workaround is exponentially explosive in the number of workstations and products, industrial settings usually offer only limited scope for flexible routings (due to plant layout or technological considerations of the equipment).Alternatively, it is easy to devise an iterating scheme that alternately solves for the mix (wh ile fixing the routings) and for properties of the routings (e.g., identification of highway routings, or dispersion routings) 'while fixing the mix .
Finite planning periods are largely a result of the convenience thereof for accounting practices and computerized planning software systems that have been used over the last several decades.In general there is usually no compelling manufacturing reason to.insist on using finite production planning periods.If an RMP solution must be converted to a finite planning period volume, then the period volume can normally be found by simply multiplying the optimal product rates by the length of the planning period.Unfortunately, this approach is confounded by requirements for specified large batch sizes (such as those found in breweries, where whole tanks/vessels of beer product must be routed).Meyer [19] has addressed this problem in a case study, scheduling the finite-period production volumes (subject to a variety of additional practical constraints such as due dates, sequence-dependent .setupsand batching considerations) using a shifting bottleneck (SB) procedure (see, for example, Adams, Balas and Zawack (1], Pinedo and Singer [22], and Uszoy and Wang (26]) to minimize the fullschedule makespan.Meyer found, in limited experimentation, that the solutions obtained in this ITlannerproquce required makespans that exceeded the actual planning 'period by only 7% on average.Given that the finite-period schedules are not optimal (SB solution techniques heuristically decompose the production environment into a collection of one-machine scheduling sub-problems, and the order in which the sub-problems are solved impacts the final schedule), Meyer's techniques seem promising. .

QUEUEING NETWORK STABILITY GUIDELINES
Several quick checks can be effected to evaluate whether a PMP or RMP solution is potentially unstable : if the routing is of the feed-forward type (sec Dai [5]) (that is, it' the routing graph, when sketched in similar fashion to Figure 2, with workstations represented by vertices and routing requirements by directed edges in the graph, contains no dicycles), then the solution of the PMP is attainable in the sense that the network will be stable.This result was proven by Dai [5] among others .A feed-forward routing structure permits no rework loops , and work may never reenter a production line from a downstream workstation.
Unfortunately, non-feed-forward routings are unavoidable in some situations due to extremely high machine costs and processing requirements that require that some products revisit a particular machine as many as 30 times (this is typical in semi-conductor wafer fabrication) .
In such cases service policies can be specified that guarantee stability: if the unit class processing times are identical at a station, and if this is true at all stations (the processing times may differ between, but not within a workstation) then the network will be stable if the FIFO service discipline is employed at all workstations (Bramson [3]) .
The application of universally stable (Warren and O 'Cinneide [27]) scheduling and sequencing policies (such as the head-of-line-processor-sharing (HOL-PS) policy (Bramson [4]» at every station will ensure that the optimal solution of the PMP is attainable and that a stable network will result, irrespective of the routing topology. Unfortunately, priority policies , which are commonly used in practice, are particularly frag ile from a stability perspective, and extreme care must be exercised when priority, polling or FIFO policies are to be used in networks with non-feed-forward routings.While it is possible to stably implement any feasible PMP or R!v1P mix and routing solution through the careful choice of scheduling and sequencing policies, the matter is delicate, and simulations are recommended in practical settings.The blunt approach of using a feed-forward routing strategy (circumventing the need for analysis) is preferred and most widely used in practice, particularly in assembly plants, even though it comes with a potential associated reduction in maximal achievable throughput capacity.

CONCLUSIONS
The use of greedy heuristics for solving the PMP is fundamentally flawed .Given that significant revenue may be foregone by using the greedy heuristic, the use of an exact solver is unavoidable.
The PMP formulation fails to account for routing.The routing strategy has to be inferr ed from PMP solutions, and the PMP formulation does not permit the optimisation of routing strategies.
The RMP is an alternative .to the PMI' formulation that explicitly models assembly, disassembly, and probabilistic (including deterministic) routing in a rate-based formulation that avoids possible bin-packing (Nemhauser and Wol se y 1998) effects that must be dealt with at planning boundaries in revolving implementations of PMP solutions.As in the case of the PMP, though, the RMP ignores a variety of practical aspects such as sequence-dependent setups, due dates, and batching.The incorporation of many such practical issues may force a formulation that would be hard or impossible to solve to optimality in large dimensions.
When optimising for the mix only, the RMP is an LP with computational complexity similar to that of the PMP , and solution using any (exact) LP solver is straightforward and can be expected to consume a negligible amount of computer (PC) time, even for the largest practical production mix problems.As in the case of the PMP, RMP solutions must also be checked to ensure that they do not produce an unstable production environment.
Optimisation strategies are discussed that permit optimisation of both the mix and the routing strategy.Deeper investigation of this possibility is merited .
Fi gure 1: Sample PM? pol ytop es(b)    Th e limitations of the greedy heuri stic arc im mediately evident in FigureI: the greedy heuristic can never reach a solution via an extremal edge that is not a principal direct ion .In http://sajie.journals.ac.za polytope (b) the opt ima l solution (marked x *) is thus unreachable by a greed y heuristic.It is precisely the failure to accommodate the possibility of optimisation of several variables simultaneously that accounts for the sub-optimality of the greedy heuri stic on polytope(b).