Constraints - These represent how each decision variable would use limited amounts of resources. Data - These quantify the relationships between the objective function and the constraints.
Synonyms Linear Optimization. Share this Term. Tech moves fast! Stay ahead of the curve with Techopedia! Join nearly , subscribers who receive actionable tech insights from Techopedia. Thank you for subscribing to our newsletter! Based on this information obtained about the customer, the car dealer offers a loan with certain characteristics, such as interest rate, loan amount, and length of loan repayment period. Linear programming can be used as part of the process to determine the characteristics of the loan offer.
The linear program seeks to maximize the profitability of its portfolio of loans. The constraints limit the risk that the customer will default and will not repay the loan. The constraints also seek to minimize the risk of losing the loan customer if the conditions of the loan are not favorable enough; otherwise the customer may find another lender, such as a bank, which can offer a more favorable loan. Consider the example of a company that produces yogurt. There are different varieties of yogurt products in a variety of flavors.
Yogurt products have a short shelf life; it must be produced on a timely basis to meet demand, rather than drawing upon a stockpile of inventory as can be done with a product that is not perishable. Most ingredients in yogurt also have a short shelf life, so can not be ordered and stored for long periods of time before use; ingredients must be obtained in a timely manner to be available when needed but still be fresh. Linear programming can be used in both production planning and scheduling.
To start the process, sales forecasts are developed to determine demand to know how much of each type of product to make. There are often various manufacturing plants at which the products may be produced. The appropriate ingredients need to be at the production facility to produce the products assigned to that facility.
Transportation costs must be considered, both for obtaining and delivering ingredients to the correct facilities, and for transport of finished product to the sellers. The linear program that monitors production planning and scheduling must be updated frequently - daily or even twice each day - to take into account variations from a master plan.
Over cities worldwide have bikeshare programs. Although bikeshare programs have been around for a long time, they have proliferated in the past decade as technology has developed new methods for tracking the bicycles. Bikeshare programs vary in the details of how they work, but most typically people pay a fee to join and then can borrow a bicycle from a bike share station and return the bike to the same or a different bike share station. Over time the bikes tend to migrate; there may be more people who want to pick up a bike at station A and return it at station B than there are people who want to do the opposite.
Once other methods are used to predict the actual and desired distributions of bikes among the stations, bikes may need to be transported between stations to even out the distribution. Bikeshare programs in large cities have used methods related to linear programming to help determine the best routes and methods for redistributing bicycles to the desired stations once the desire distributions have been determined.
Learning Objectives In this section, you will learn about real world applications of linear programming and related methods. Shape optimization seeks to make a shock-free airfoil with a feasible shape.
Linear programming therefore provides engineers with an essential tool in shape optimization. Transportation systems rely upon linear programming for cost and time efficiency. Bus and train routes must factor in scheduling, travel time and passengers. Airlines use linear programming to optimize their profits according to different seat prices and customer demand. Airlines also use linear programming for pilot scheduling and routes.
Optimization via linear programming increases airlines' efficiency and decreases expenses. Manufacturing requires transforming raw materials into products that maximize company revenue. Each step of the manufacturing process must work efficiently to reach that goal. For example, raw materials must past through various machines for set amounts of time in an assembly line.
To maximize profit, a company can use a linear expression of how much raw material to use. Constraints include the time spent on each machine. Any machines creating bottlenecks must be addressed. The amount of products made may be affected, in order to maximize profit based on the raw materials and the time needed. Modern energy grid systems incorporate not only traditional electrical systems, but also renewables such as wind and solar photovoltaics. They used QP to calculate stepwise the changes in diet gradually limit the percentage of animal protein to 50, 25, Although QP is an optimization method 14 , the goal was to find a diet that encounter the dietary guidelines per scenario with the lowest number of changes in the menu retaining the typical diet for each country.
The four applied scenarios resulted in reductions for the blue water footprint of 4, 6, 9, and The original diet was assigned as the optimization objective. QP resulted in estimated cost for any scenario.
Therefore, the result was close to the traditional, culturally acceptable diet and fulfilled the nutritional constraints QP has advantages over LP when the goal is to find small changes on population level. The number of nutritional constraints vary from 5 to 37, which could have a major impact on the results of the studies: the lesser the number of constraints, the higher the risk of inadequacy of nutrient intake of the nutrients not considered.
Even with a high number of nutritional constraints, bioavailability of nutrients e. This could partly be solved by adding constraints on certain food groups rich in phytochemicals, e. Evaluating the limited number of studies using LP on diets, we conclude that the studies of Wilson et al.
Future LP diet studies should combine all three of these constraints. The most important challenge to improve future LP diet studies with ecological constraints, is to build bigger databases with more foods and more environmental data, with improved quality and consistency of the data. Although the papers cited above 33 , 66 observed substantial reductions in GHGEs, it is striking that they found much higher emission levels—in absolute terms—than the Dutch study The found emission of 1.
On the other hand, one of Wilson et al. These differences may be explained by different methods used to calculate GHGEs per product or variances in food cultures and preferences. Table 2 makes it clear that the studies differ in number of food items 13—; an indication of the completeness of diets , the size of the population, the number of nutritional constraints 5—33; an indication of the nutritional quality of the diets , the selected economic and ecological constraints, and the solutions to make the outcomes culturally acceptable.
One of the attempts to make outcomes culturally acceptable, is the introduction of acceptability constraints. Six studies demonstrated good examples of those constraints. From the first studies of Dantzig to date, researchers have struggled with the unrealistic outcomes of LP solutions.
It was expected that adding acceptability constraints could help to prevent this. A good example is Maillot et al. For any given food, upper limit on the quantity was defined by the 95th percentile of consumer intake. The vitamin D constraint was the most difficult to fulfill, followed by sodium, magnesium, and saturated fatty acids.
However, the use of cultural acceptability constraints limits finding solutions. In Parlesak et al. They calculated five different cost-minimized food baskets for a family of four. The food baskets that met food based dietary guidelines was twice the price.
Introducing cultural acceptability constraints increased the cost three times. So, variety in the diet and cultural acceptability has a price Thompson et al. They also put an upper bound on most foods and removed foods with smaller amounts in the diet, as well as less healthy options such as full-fat milk.
They applied lower bounds of consumption, particularly on popular foods. For example, bread, potatoes and pasta have comparable GHGEs and prices, but the model will try to optimize one of the products for cultural reasons: for instance, consumption of potatoes was limited in Spain and pasta in Sweden Other examples of the improvement of LP methodology were demonstrated in literature by using more nutritional constraints 47 and selecting most frequently consumed foods 4 , For example, in the men's diet 50 of the 83 products were kept unchanged in number of calculated portions, and in the women's 55 of the 73 products.
The Optimeal tool calculated a change in portions for 8 foods for men and 7 for women. Finally, 9 new food items were added to the men's diet and 8 to the women's unsalted peanuts, pear, kale, sauerkraut, lentils, marrowfats, soy drink, mackerel, and mussels. Nevertheless, the diet was almost vegetarian, with less portions of meat and dairy. Likewise, new products such as soy drink, marrowfats and lentils were added to the diet, which are not consumed by the majority of the Dutch population A reality check is needed to determine if this would be acceptable for consumers.
Tyszler et al. The metric for changes was measured by a penalty score based on the popularity of foods. The reasoning behind this modeling is that diets which are like the current one is more likely to be accepted by most of the population than more extreme diets. The figure indicates that, if the goal of the optimization is a diet with lower environmental load Vegetarian or Vegan are not the only options.
There are many other solutions to this diet problem with a smaller number of adaptations in the diet Figure 3. Example of the application of acceptability constraints and the effects on the environmental impact of different diet scenarios M, males; F, females. The lower the penalty score is, the closer the diet is to the current diet and the more acceptable Although the Diet Problem has a long history, most diet solutions are from or later, as computers with larger calculation capacity became widely available and LP tools were developed.
The literature shows that LP can be applied to a variety of diet problems: from food aid, national food programs, dietary guidelines, to individual solutions. In supporting dietary guidelines, LP has proven its value in many ways. Most studies have used nutritional constraints combined with cost constraints.
However, even when the number of constraints is increased, LP is not always able to find solutions. Nutritional constraints should reflect at least the national dietary guidelines. In defining affordable diets and investigating the relationship between cost and health, LP studies provided insightful contradictions. LP shows that cheaper and healthier foods can be found easily, but when price becomes a constraint, often a shift occurs to unusual food unless the right constraints are chosen.
LP can produce solutions that are not realistic for the population, especially when cultural acceptability is not considered. Introducing acceptability constraints is recommended, but none of the studies provide the ultimate solution for calculating acceptability. LP can play a role in the future developments on acceptance of changes and personalized food. Table 2 demonstrated that the analyzed studies are not always clear about the choice of their programming tool and objective function.
Arnould et al. It should be expected that the methods are clearly described. The older software tools Rglpk package, R stat software and Solver in Excel are still in use and seem to function well, but because of the complexity of the diet problem, more sophisticated and tailor-made tools are built for specific application Optimeal and Cost of the Diet-tool.
Further development is needed to implement acceptability constraints. Quadratic Programming has many advantages over LP when you want small changes on population level. QP differs from LP in that the functions are not linear but quadratic. An inherent limitation of LP is that it limits the amount of changes, while sometimes a wider range of small changes in products can give more useful solutions, e.
QP have this advantage above LP. LP also demonstrated to be an applicable tool to conscientiously convert predefined nutrient constraints into diets with unpredictable food combinations.
Only 12 studies applied and introduced ecological constraints and of these, only two also included cost constraints. These studies showed that the environmental impacts of diets can be halved, staying within the existing nutritional constraints. LP makes it possible to propose diets with lower impacts than diet scenario studies.
In other words, LP is an important tool for environmental optimization and has a lot of potential. Important is consistency in methodology to derive environmental figures full scope and completeness of constraints.
Future possibilities lie in finding LP solutions for diets by combining nutritional, cost, ecological, and acceptability constraints.
LP is clearly a very helpful instrument for finding solutions to a variety of very complex diet problems. The author confirms being the sole contributor of this work and approved it for publication.
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Thanks to Harry Aiking and Hans Blonk for their critical comments on the concept of this paper. How to Feed the World in Rome: FAO Toward an integrated approach to nutritional quality, environmental sustainability, and economic viability: research and measurement gaps. Ann N Y Acad Sci. Linear programming: a mathematical tool for analyzing and optimizing children's diets during the complementary feeding period.
J Pediatr Gastroenterol Nutr. Individual diet modeling translates nutrient recommendations into realistic and individual-specific food choices. Am J Clin Nutr. Macdiarmid JI. Is a healthy diet an environmentally sustainable diet? Proc Nutr Soc. Linear Programming 1: Introduction. Google Scholar. Smith VE. Linear programming models for the determination of palatable human diets.
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