Linear Programming: A Supportive Decision Making Tool in the Food ...

Global Journal of Logistics and Business Management 1(2013), 40-44 ISSN: 23050098

Linear Programming: A Supportive Decision Making Tool in the Food Industry Juma Makweba Ruteri Global Outsourcing and Management Consult Email: [email protected]

Abstract A linear programming technique (LP) is used in this article for aiding decisions in the food industry that has limited resources. The limited resources make processors to have few options in business operation activities. Because of capital investment, the processor has three options to buy and install only one production line. Through LP analysis subjected to constraints that were given, the processor has to procure and install production line that showed to generate high profit. LP technique is illustrated by comparing profits of tomato juice, tomato ketchup and tomato sauce. Tomato ketchup seems to generate more profit than other products. However, decisions that only based on LP results in food product perspective might not be enough and therefore other factors that influence consumers’ purchase have to be considered. Key Words: Linear programming, decision making, food industry 1.

Introduction

As it is in any other business, food industry sector also needs innovation to stay ahead of the competition. No business can continue to offer the same unchanged product forever; otherwise sales and profits would decrease. There are a number of reasons which have been pointed out on why a new product has to be developed. Some of the reasons include:  Consumer needs and wants  The product has reached the end or it is at the maturity stage of its life cycle  Environmental changes which the company wants to capitalize on  Forces from competitors  Last not least if all of the products are experience poor sales or products are suffering from a negative reputation. If any business experience one of the reasons mentioned above it is a high time to improve or change the product. However, product development comes under cost and no guarantees can ever be offered for a new food product’s success. The implementation of carefully orchestrated plan and the right decision made significantly increase the probability of success. 1.1

Linear programming

Linear programming has provided a novel view of operations it induced research in the mathematical analysis of the structure of individual systems and also it has become an important tool for business and industrial management for improving the efficiency of their operations. According to Dantzig (1991) the food processing industry is perhaps the second most active user of LP after the petroleum refinery industries. Additionally he reported that in 1953 a major producer of ketchup used LP to determine shipping from six plants to 70 warehouses. Furthermore, he reported a major meat packer who determined the most economical mixer of animal feeds by means of LP. Williams and Redwood (1974) presented a LP model for aiding buying and operating decisions in the food industry. The major focus in their study was on the company which was buying raw oil, refining and then blending them together in two factories to produce a number of brand foods. Each factory had a limited capacity available for refining processes, thus forcing the management to work on algorithms that might optimize available resources. Briend et al (2001) used LP to analyze the price of fortified food supplements. By recognizing the importance of LP Mitsos

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Global Journal of Logistics and Business Management 1(2013), 40-44 ISSN: 23050098 and Barton (2009) presented a study that focused on mixed-integer linear programs with a single unknown parameter simultaneously affect the right-hand side vector, the cost vector and the design matrix. Flisberg et al (2009) used LP technique to solve an operational routing problem to decide the daily routes of logging trucks in the factory. LP technique was employed because it determines the way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model and also given some list of requirements or constraints represented as linear equations. It has been mentioned that there are five essential conditions in a problem situation for linear programming to pertain. First, there must be limited resources (such as a limited number of workers, equipments, finances, and raw materials); otherwise there would be no problem. Second, there must be an explicit objective (such as maximize profit or minimize cost). Third, there must be linearity (two is twice as much as one; if it takes three hours to make a part, then two parts would take six hours, and three parts would take nine hours). Fourth, there must be homogeneity (products produced on a machine are identical, or all the hours available from a worker are equally productive). Fifth is divisibility: Normal linear programming assumes that products and resources can be subdivided into fractions. If this subdivision is not possible (such as flying half an airplane or hiring one-fourth of a person), a modification of LP, called integer programming has to be employed. This paper illustrates how LP technique can be used to analyze optimal profits of each product expected to be produced on the proposed new production line. The company is in a dilemma on which product to produce. The new line can be used for production of tomato juice, tomato ketchup or tomato sauce. The company formulates, develops and produces products for the market, but also the company is aware of their limited resources. In addition to other factors that influence the performance of the product in the market, the company has to make the right decision on which product to produce under the help of analytical tools like LP. In the case of the dilemma the company has, LP might be useful as it will indicate an optimal profit that will be generated from each proposed product. It has to be reminded that the purpose of this paper is not to derive or to prove LP equations but rather to show its importance in final decision making. 2.

Linear Programming in food product development perspective

In other words a linear programming problem may be defined as the problem of maximizing or minimizing a linear function subject to linear constraints or entails an optimizing process in which nonnegative values for a set of decision variables 1 , 2 .., n are selected so as to maximize or minimize

z z z

and an objective function is in the form: Maximize or minimize

z c x  c x ... c 1 1

2

2

n

xn

[1]

The constraints may be equalities or inequalities. In addition to food processing machineries and human resources, the processors need to have enough primary and secondary raw materials. Primary raw materials we refer to are materials that constitute a large portion (body) of the product while secondary raw materials we refer to are the food ingredients or additives. Food ingredients or additives are substances added to preserve food from spoilage, preserve flavor or enhance its taste and appearance. Some additives have been used for centuries and some have been banned. As previously mentioned above, this paper is intended to use LP as a tool to support the processor to make final decision when meeting dilemma on which product to produce. The decision in this case will be made based on the optimal profit that can be generated from each proposed product. The resources available on each product (assumed formulation) for one batch of each product are as shown in Table 1 and Table 2 bellow:

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Global Journal of Logistics and Business Management 1(2013), 40-44 ISSN: 23050098 Table 1: Types of products and the amount of ingredients needed Tomato Juice Tomato Ketchup Tomato Sauce Ingredients Amount Ingredients Amount Ingredients Amount (kg) (kg) (kg) Sugar 20 Sugar 50 Sugar 25 Preservative 2 Preservative 2 Preservative 1 Food color 2 Thickening agent 1 Food color 1 Emulsion 3 Flavor 1 Thickening agent 1 Flavor 2 Flavor 1 Table 2: Available raw materials Materials Sugar Preservative Food color Thickening agent Emulsion Flavor

Amount (Kg) 1500 100 1500 250 200 50

It is assumed that each unit of tomato juice, tomato ketchup and tomato sauce sold yield a profit of $10, $8 and $5 respectively 2.1 Solution to the Problem In this project, the processor would like to produce products that yield high profit and therefore this is the problem that needs to be optimized. The linear objective function to be optimized in this case is the profit. Mathematically it can be expressed as follows:

Pt

j

x10  t k x8  t s x5

where P = an expected profit,

t

[2]

j

= Tomato juice,

t

k

= Tomato ketchup and

t

s

= Tomato sauce, which we

refer to as the decision variables and these are the number of each product to be produced from the limited resources (constraints) mentioned in Table 2 above. These constraints are expressed as linear constraints to facilitate the analysis, and are mathematically described below. It has to be remembered that tomato juice, tomato ketchup and tomato sauce are made up of a combination of different amount of primary and secondary raw materials while meeting local or international standards. In a mathematical term this is expressed by the set of limitations:

t

j

x20  t k x50  t s x25  1500Sugar

[3]

t

j

x2  t k x2  t s x1  100Pr eservative

[4]

t

j

x2  t k 0  t s x1  150Foodcolour

[5]

t

j

x0  t k x1  t s x1  250Thickeningagent 

[6]

t

j

x3  t k x0  t s x0  200Emulsion

[7]

t

j

x2  t k x1  t s x1  50Flavour

[8]

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Global Journal of Logistics and Business Management 1(2013), 40-44 ISSN: 23050098 All these constraints are linear constraints, because they consist of a simple sum of products. Linear programming is based on a mathematical iterative approach involving multiple calculations of products and sums, which can be quickly performed by a personal computer. Calculations presented below were done with the Excel 2003 (Microsoft) spreadsheet, which has a linear programming function called a “solver function” in all its recent versions. 2.3. Interpretation and Implication of the Result in Decision Making To illustrate the use of LP in decision making perspective we consider the number of units and profit that will be obtained by optimizing resources that the processor had. Table 3 indicates that the resources that the processor have will enable him to produce 13 units of tomato juice and 24 units of tomato ketchup, these units all together will generate a total profit of $322 and no tomato sauce product will be produced. Since the result has already eliminated tomato sauce production option, then the processor will remain with two products to concentrate on i.e. to produce tomato juice or tomato ketchup. Table 3: Excel solver results Resource available Sugar 1500kg Preservative 100kg Food Color 150kg Thickening agent 250kg Emulsion 200kg Flavor 50kg

T. Juice 20 2 2

T. ketchup 50 2 0

0 3 2

1 0 1

20 2 2

T. ketchup 50 2 0

0 3 2

1 0 1

1 0 1

13

24

0

10

8

5

T. Juice

Units Produced Price per unit Objective function

T. sauce 25 1 1 1 0 1 T. sauce 25 1 1

Resource used 1460 ≤ 74 ≤ 26 ≤ 24 39 50

≤ ≤ ≤

Resource Available 1500 Sugar 100 Preservative 150 Food Color Thickening 250 agent 200 Emulsion 50 Flavor

322

Further analysis reveals that, 97.3% and 74% of Sugar as the main ingredient and preservative respectively will be used in production of the two types of products and no flavor will remain. About 90.4% of thickening agent, 82.7% of flavor and 80.5% of emulsion will remain. The remained amount of thickening agent, flavor and emulsion can be used in production process of 9, 5 and 4 batches respectively if new order of sugar and preservative are procured. Capital investment is the main problem that limits the processor to procure and install two production lines. The only option he has is to select one of the two products. The profit generated being an indicative factor for establishment of a new production line then further profit analysis for the two products has to be conducted. Subjected to further LP analysis, tomato ketchup will generate higher profit than tomato juice i.e. $192.00 and $130 respectively. Based on the profit generated, tomato ketchup production line has to be procured and installed for the purpose. Ingredients used in each product are as shown in Table 4. 80% and 17.3% of the total available sugar will be used in the production of one batch of tomato ketchup and tomato juice respectively. In the case of food preservative and food flavor, 48% of the total available preservative

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Global Journal of Logistics and Business Management 1(2013), 40-44 ISSN: 23050098 will be used in tomato ketchup production and only 26% will be used in tomato juice, while for the flavor, 48% and 52% of the total available will be used in tomato ketchup and tomato juice production respectively. Table 4: Distributions of used ingredients in the two products Tomato Juice Tomato Ketchup (kg) Total used raw materials (kg) Ingredients Amount (kg) Ingredients Amount (kg) Sugar 260 Sugar 1200 1460 Preservative 26 Preservative 48 74 Food color 26 Food color 0 26 Thickening agent 0 Thickening agent 24 24 Emulsion 39 Emulsion 0 39 Flavor 26 Flavor 24 50 By producing only tomato ketchup product, the flavor available will be enough for two batches production. Other food ingredients such as food color and emulsion will be eliminated from the company’s purchase list because these items are not among the ingredients needed for tomato ketchup production. Therefore the fund allocated for these items should be allocated to increase the quantity of other needed ingredients. 3. Conclusion and Limitations Mathematically, LP is a powerful tool that can help the processor to reach into a solid decision. Only general principles and selected applications of the approach pertaining decision making on the dilemma the processor had have been presented. From LP result, tomato ketchup was said to generate more profit than other products and the food processor has to procure and install this line. However, as customers’ consumption behavior is concerned, other factors that influence the purchase of food products have to be incorporated in LP or other supporting tools in product development have to be conducted. Sensory evaluation and product sensitivity have to be conducted as they may help to avoid unnecessary losses that can happen because of consumer rejection or product expiration. References Briend, A., Ferguson, E., and Darmon N. (2001). Local food price analysis by linear programming: A new approach to assess the economic value of fortified food supplements. Food and Nutrition Bulletin, Vol. 22(2), 184-189. Dantzig, G.B. (1991). Industrial application of Linear programming in: Linear programming and extensions. Princeton University Press. New Jersey. 28-29 Mitsos, A., and Barton, P. I. (2009). Parametric mixed-integer 0-1 linear programming: The general case for a single parameter. European Journal of Operational Research. 194, 663–686. Flisberg, P., Lidena, B., and Ronngvist, M. (2009). A hybrid method based on linear programming and tabu search for routing of logging trucks. Computers & Operations Research. 36, 1122 – 1144. Williams, H.P., and Redwood, A.C. (1974). A structured Linear programming model in the food industry. Operational Research quarterly, Vol. 25(4), 517-527.

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