Keywords

1 Introduction and Review

Worldwide, the food industry has recently drawn much attention due to issues related to human health and safety. The performance of such a food supply system is heavily based upon the interactive activities between other business entities. It follows the similar chain structure as the manufacturing industries but the structure is more complex and the variety of the food products is much more diverse. The value of food products includes two parts, the nutritional value, and the physical senses value. Previous models include zero-order reaction kinetics, first-order reaction kinetics, fractional conversion kinetics, the Bigelow model, and non-linear microbiological death model [1,2]. Zero-order reaction kinetics is the traditional model, with simple calculation but with larger errors in estimation. First-order reaction kinetics and fractional conversion kinetics are models based on the experiments by changing the content during certain stages for food storage. The Bigelow model and non-linear microbiological death model are models that have been used to illustrate the changes of nutrients within the food product during a more complex situation, including the effects of changes in temperature during food cooking and after [3]. In most literatures, the costs due to the loss of product value is considered as result of food deterioration, and is normally modeled with linear or exponential deterioration rate to illustrate the cost of such loss. Their assumption is that, the reduction of inventory level is a result of joint operation of demand and deterioration. Two models were developed by Fujiwara [4] with linear deterioration rate, and Chung and Huang [5] with an exponential deterioration rate. Quantification of risk is part of the risk assessment process. Huss et al. [6] developed a semi-quantitative assessment system to evaluate the risk of seafood products. Van Gerwen et al. [7] developed a SIEFE system, and by setting different scale of risk factors on each risk level, the overall quantitative risk could be obtained. Ross and Summers [8] developed a model with nine risk input values. The most important part of their model is the concept of comparative risk. This risk contains the evaluation of probability of illness over all servings, annual exposures per person in a daily basis, and the hazard severity factor.

2 Design and Methodology

Notation and Assumptions:

D, d :

Annual (D) and daily demand (d) for a food product at the retailer, d = D/365.

Q :

Order quantity, Q = d*n, where n is days for each ordering period.

C o :

Ordering cost, all cost associated with the placement of an order.

C s :

Setup cost, all cost associated with the setup of product for each batch

Q p :

Size for each production batch.

S r :

Retail price for food product at the retailer.

S p :

Price for food product at the producer.

y :

Producer’s quality level (in Taguchi Quality Loss function concept).

h :

Holding cost rate at the retailer, during the storage and on shelf period before being purchased by consumers.

Q 10 :

Food deterioration parameter, used in the model for loss of nutrition.

F 1 :

Food product life labelled, which is based on the storage temperature of T 1 .

F 2 :

Food product real shelf life, which is based on the storage temperature of T 2 .

k :

Food product deterioration rate or quality loss rate.

k 1 :

Deterioration rate from micro-organisms’ activities.

k 2 :

Deterioration rate from enzymes’ activities.

k NL :

Food product nutrition quality loss.

k PS :

Food product physical senses quality loss.

L NL :

Nutrition loss.

L PS :

Physical senses loss.

P p :

Production cost per unit of food product.

P q :

Cost of quality per unit, a sum of internal failure cost, prevention cost and appraisal cost.

P t :

Cost of transportation per unit.

Q i :

Probability of occurrence for a certain risk related activity i.

To facilitate the modeling of the food chains, we made the following assumptions:

  1. 1.

    Replenishment rate is infinite and lead time is zero. And shortage is not allowed.

  2. 2.

    The food products follow the general form of deterioration. Deterioration process starts when retailers receive product, and no deterioration during transportation.

  3. 3.

    All cost parameters are known in advance.

  4. 4.

    Demand for a food item is assumed to be deterministic. No seasonal effect.

  5. 5.

    During storage, transportation, and on shelf period, the environment, such as temperature, lighting, packaging quality, are assumed to be steady and unchanged.

2.1 Cost of Quality, and Food Quality Loss

According to research literature in food science as discussed in the review section, quality factors for food could be divided into two major categories: (1) nutrition and energy supply, and (2) quality related to physical senses. Nutrition and energy supply serves the basic function of food: to supply energy and bio-chemical needs to maintain the survival and functions of human body. Quality senses includes the physical senses customer received from the food product. Such senses include sight, touch, smell, taste, and even hearing. In general, these senses are grouped into appearance factors, textural factors, and flavor.

2.1.1 Nutrition and Energy Supply Loss

To quantify the food nutrition loss over the time factor, a widely used method of Q 10 in food science is adopted in our model. The equation of Q 10 is used to describe the duration of storage to reach the same nutrition level under different temperature.

$$ {f_2}={f_1}*{Q_{10}}^{{\frac{\Delta}{10}}} $$
(1)

In Eq. 1, f 1 is the reference duration at reference temperature T 1 , f 2 is the duration of the targeting temperature T 2 . \( \Delta \) is the difference between targeting temperature and sample temperature of. Notice that this ‘potential’ nutrition loss for the customers is in proportion to the food retail price and the ratio of actual shelf period over the labeled shelf life. For example, a food item has been labeled for a shelf life of F 2 days, when this item has been purchased before on F 1 (F 1  < F 2 ), a certain portion (F 1 /F 2 ) of nutrition loss can be expressed as NL F1 .

$$ N{L_{{{F_1}}}}=a\cdot \frac{{{S_r}}}{{{F_2}}}\cdot {F_1} $$
(2)

2.1.2 Physical Senses Loss

According to food science, there are (1) appearance factors, (2) textural factors, and (3) the flavor factors, that are considered physical senses by consumers on food products. Appearance factors usually include size, shape, wholeness, color, consistency for liquid, and so on. Textural factors include hand feel and mouth feel of firmness, softness, juiciness, chewiness, grittiness. Flavor factors include both taste and odor. The loss of physical senses for food products is normally the result of food deterioration. Major causes for food deterioration include (1) Growth and activities of microorganisms, such as bacteria, yeast and so on; (2) Activities of natural food enzymes; (3) Insects, parasites, and rodents; (4) Temperature; moisture and dryness; air (particularly oxygen); and light; and (5) Time duration.

2.1.2.1 Growth and Activities of Microorganisms

According to food science, if the original number of microorganisms in food product is A, then the number of microorganisms in food after time t will be:

$$ N=A\cdot {t^2} $$
(3)

And the quality loss can be formulated as in proportional to the number of microorganisms in the food. If the loss rate is k, the loss from microorganisms is:

$$ {L_{micro }}=k\cdot N=k\cdot A\cdot {t^2}={k_1}\cdot {t^2} $$
(4)
2.1.2.2 Activities of Natural Food Enzymes

According to Potter [3], bacteria or microorganisms are the greatest factors in food deterioration, and the activity of enzyme is the second greatest. We can define the food quality loss due to enzymes as the following:

$$ {L_{enzyme }}={k_2}\cdot {t^2} $$
(5)
2.1.2.3 Insects, Parasites, Rodents, Temperature, Moisture, Dryness, Oxygen, and Light

With proper packing and storage of food items in modern food retailer facilities, these factors are not considered in our models.

2.1.2.4 Time

It is clear that one of the most important factors for food quality loss calculation due to food storage and food shelf life is dominated by the time factor. The quality loss due to physical senses for one product unit can be expressed as:

$$ {L_{PS }}={L_{micro }}+{L_{enzyme }}={k_1}\cdot {t^2}+{k_2}\cdot {t^2}={k_{PS }}\cdot {t^2} $$
(6)

2.2 Quality Loss Functions and Food Quality Loss Due to Physical Senses

Taguchi’s quality loss function is originally designed for the manufacturing industry. In our case, the longer a product has been stored, the greater the loss customer will be suffering. Equation 6 can be modified according to the Taguchi format as:

$$ {L_{PS }}={k_{PS }}\cdot {t^2}=\frac{{{S_r}}}{{F_1^2}}\cdot {t^2} $$
(7)

Within each ordering period, food items may be stored on shelf from 0 to n days, waiting to be purchased. The Total Quality Loss due to physical senses as:

$$ {L_{PS }}=\int\limits_0^n {(d)\cdot } ({k_{PS }}\cdot {t^2})dt=\frac{{d\cdot {k_{PS }}\cdot {n^3}}}{3}=\frac{{d\cdot {S_r}\cdot {Q^3}}}{{3F_1^2{d^3}}}=\frac{{{S_r}\cdot {Q^3}}}{{3F_1^2{d^2}}} $$
(8)

2.3 Food Risks Associated with Time

One of the main drawback in Ross and Summers model [8] is the assumption of the constant risk levels for food items, regardless of the time spent on the shelf. In our model, we combine all the risk factors in Ross and Summers model and eliminate the frequency of consumption, market share, as well as population and exposure distribution. The overall risk for a certain period of t is:

$$ \bar{R}=\int {{k_{fr }}} {t^2}dt=\frac{1}{3}{k_{fr }}{n^3}=\frac{1}{3}{k_{fr }}\frac{{{Q^3}}}{{{d^3}}} $$
(9)

2.4 Objective Function for the Food Supply Chain Model

The Total Cost including all direct costs (production, transportation, shortage, inventory holding, etc.), quality loss cost(nutrition loss and physical sense loss), as well as cost associated with food risk factors is summarized as:

Total Cost = Production cost + Setup cost + Yield and Defective cost + Transportation cost + Ordering cost + Inventory holding cost + Nutrition loss cost + Physical senses loss cost + Food risk loss (converted cost)

$$ \begin{array}{llll} \mathrm{ Total}\;\mathrm{Cost}= \; {P_p}\cdot D+{C_s}\cdot \frac{Q}{{y\cdot{Q_F}}}+{P_Q}\cdot D\cdot \frac{(1-y) }{y}+{P_t}\cdot D+{C_o}\frac{D}{Q}+h\cdot {S_p}\cdot \frac{Q}{2} \\ +\frac{{Q\cdot{S_r}\cdot D\cdot Q}}{{2{F_2}\cdot d}}+\frac{{b\cdot {S_r}\cdot D}}{{3F_1^2\cdot {d^2}}}\cdot {Q^2}+\frac{{{k_{fr}}{Q^3}}}{3{d^3}}\end{array} $$
(10)

Similar to the traditional EOQ model, optimization technique is then applied to find the optimal ordering quantity that minimizes the Total Cost. Due to the limit of paper length. We will not present the solution procedure here.

3 Conclusion

In this paper, we present a model for food supply chain procurement decision making by considering the traditional EOQ factors as well as additional factors such as food nutrition loss, physical senses quality loss, and food risk financial loss. Preliminary results on four different food items, not presented in the paper, demonstrated that the feasibility of adding the additional factors in a food supply chain decision making process. Our future work will be to refine the food risk financial loss factor and to expand the model for different scenarios such as including distributors, and producers.