Understanding Some Unexpected Failures of the Theory of Constraints Product Mixes Autoria: Alexandre Linhares Abstract: The theory of constraints proposes that, when production is bounded by a single bottleneck, the best product mix heuristic is to select products based on their ratio of throughput per constraint use. This is not true for cases when production is limited to integer quantities of final products. We demonstrate four facts, all of which go directly against current thought in the TOC literature. For example, there are cases on which the optimum product mix includes products with lowest product margin and lowest ratio of throughput per constraint time, simultaneously violating the margin heuristic and the TOC-derived heuristic. We start with a simple product mix example previously analyzed in the literature and introduce a new product which dramatically alters the results obtainable from a theory of constraints perspective. It is thus crucial to understand the nature of these unexpected failures to obtain optimum product mixes. Such failures are due to the combinatorial complexity (NP- Hardness) of the product mix decision problem, a fact that is demonstrated here. 1. Introduction The theory of constraints is a remarkably successful operations management philosophy, centered on the idea of focusing managerial attention to the local constraints which inhibit the global performance of an entire system (Goldratt and Cox 1984; Goldratt and Fox 1986; Goldratt 1990a; Goldratt 1990b). Over the last two decades it has gathered much momentum, with the creation of organizations such as the Goldratt Institute and the TOC Center in Dayton, Ohio. It has also spawned a host of products, such as the OPT software and an innovative bestselling management novel. Blackstone (2001) reviews some of its core ideas and fields of application. The focus of our study will be on the problem of selecting the optimum product mix under the theory of constraints, which is deemed as an improvement over traditional practices (Gupta et al 2002; Kee and Schmidt 2000; Wahlers and Cox 1994). Consider a facility with a set of products to build, but without the capacity (i.e., a fixed time horizon) required to meet the demand for all of them. Let us suppose that this facility must deal with integer quantities of final products. In this case a product mix decision must be made, with the obvious tradeoff of prioritizing some product lines at the expense of others. A traditional method for selection of the product mix is given by selecting the products with highest individual product margin with higher priority, regardless of the time spent on the bottleneck(s). Let us name this method as the margin heuristic. The theory of constraints, however, proposes that product lines be selected according to their ratio of throughput per time spent on the system constraint(s). Let us refer to this approach as the TOC-derived heuristic. The heuristic has been formally stated in numerous TOC publications (e.g., Goldratt 1990a); for the convenience of the reader let us follow from Fredendall and Lea (1997): TOC-derived Product Mix Heuristic “Step 1: Identify the system’s constraint(s): (a) Calculate the required load on each resource to produce all the products. The constraint or bottleneck (BN) is the resource whose market demand exceeds its capacity. Step 2: Decide how to exploit the system’s constraint(s): (a) Calculate the contribution margin (CM) of each product as the sales price minus the raw material (RM) costs; (b) Calculate the ratio of the CM to the products processing time on the bottleneck resource (CM/BN); (c) In descending order of the products’ CM/BN, reserve the BN capacity to build the product until the BN resource’s capacity is exhausted; (d) Plan to produce all the products that do not require processing time on the bottleneck (i.e., the ‘free’ product) in descending order of their CM.” (Fredendall and Lea 1997, p. 1535-1536) It was believed, when originally suggested, that this TOC-derived heuristic would obtain the best combination of products for all cases. In fact, it does obtain optimum solutions when production is allowed to be fragmented. However, studies by Plenert (1993) and by Lee and Plenert (1993) clearly demonstrated – by comparing the product mixes obtained with the TOC-derived heuristic with those obtained with integer linear programming – that when production must be done over integer quantities the TOC-derived heuristic could fail to find the optimum product mix in the case of multiple constraining resources. These rather interesting results launched an increasing series of studies for improved policies (Fredendall and Lea 1997; Hsu and Chung 1998) and for advanced heuristics such as genetic algorithms (Onwubolu and Mutingi 2001a; Onwubolu and Mutingi 2001b) and tabu search (Onwubolu 2001), providing new product mix methods for the case of multiple constrained resources. After all, this multiple constraining resources scenario framed the original context on which Plenert (1993) demonstrated the shortcomings of the TOC-derived heuristic. The situation, however, is in fact more complex than it seems. In this paper, we demonstrate the following facts, all of which go directly against current thought in the theory of constraints literature: FACT 1. There are cases on which the TOC-derived heuristic fails even with a single bottleneck. FACT 2. There are cases on which the TOC-derived heuristic fails to obtain a higher profit than the traditional product margin heuristic. FACT 3. There are cases on which the optimum product mix includes products with lowest product margin and lowest ratio of throughput per constraint time, violating both the traditional heuristic and the TOC-derived heuristic. FACT 4. There are strong reasons to believe that an efficient and optimum heuristic is simply impossible. 2. Review of the Blackstone (2001) example Let us start the discussion with an example from Blackstone (2001). Figure 1 presents an hypothetical facility capable of producing three products: product X, product Y, and product Z. Product X sells for $90 and has a weekly demand of 50 units; product Y sells for $100 and has a weekly demand of 75 units, and product Z sells for $70 and has a weekly demand of 100. The facility has five work stations (A, B, C, D, E) and the products are devised from four types of raw materials (RM1, RM2, RM3, and RM4). 2 Production of X is started with two units of RM2 processed at station A for 10 minutes each. One of them is taken to station C for 15 minutes of processing while the other is taken to station D also for 15 minutes. These materials are then joined, along with a new unit of RM1, at station E, in a 5 minute process. Hence X has $40 of material cost. X Y Z $90 $100 $70 50/Week 75/Week 100/Week E E E 5 min 10 min 5 min RM1 D D $10 15 min 10 min C C 15 min 5 min A A B A 10 min 10 min 10 min 5 min RM2 RM2 RM3 RM4 $15 $15 $15 $10 Figure 1. After Blackstone (2001): a hypothetical facility. Production of Y is started with a unit of RM2 processed at station A for 10 minutes, and a unit of RM3 processed at station B also for 10 minutes. After station A is finished, the resulting material flows to station D for 15 minutes and then to station E for an additional 10 minutes. After station B processes RM3, the resulting material is taken to stations C (5 minutes), D (10 minutes), and finally it is joined with the material resulting originally from RM2, in station E, in a 10 minute process. Material costs for Y equal $30. Capacity requirements per unit Station Load (minutes) Product X Product Y Product Z Product X Product Y Product Z Total requirement Station A 20 10 5 1000 750 500 2250 Station B 0 10 10 0 750 1000 1750 Station C 15 5 5 750 375 500 1625 Station D 15 25 10 750 1875 1000 3625 Station E 5 10 5 250 750 500 1500 Demand 50 75 100 Selling price 90 100 70 Raw materials 40 30 25 Labor 10 10 5 Overhead 30 30 15 Product cost 80 70 45 Product margin 10 30 25 Table 1. Summary data of the Blackstone (2001) example 3 Product Z is made out of one unit of RM3 and one unit of RM4. RM3 is initially processed at B for 10 minutes, then followed by 5 minutes at C, 10 minutes at D, and 5 minutes at E, where it will be joined with the unit of RM4 processed for 5 minutes at A. The material costs for Z are $25. A word of caution: the reader might have noticed that there are some incorrect data in Blackstone’s paper. For example, in that paper, figure 6 reveals that product Z requires 15 minutes of processing at station C, while the next figure will place that number at only 5 minutes. It is impossible to tell from that text which numbers were the correct ones, but this does not affect the arguments in either way. In here we are thus referring to the data from the matrix in Blackstone (2001)’s figure 7, and not from the facility scheme of Blackstone (2001)’s figure 6. We thus have on table 1 a summary of the load on the stations where it may be concluded that there is not enough capacity in station D, which clearly makes it the only system constraint. Now a decision must be made: which products should be produced? This is the product mix problem under a bottleneck.