food production chains

formulating a costs/profits model

explicit models- general models that contain all the numbers that you need to take into account, ex. the prices of your potatoes, tons of availability of potatoes, far easier than implicit models but take more time

implicit models- summaries of explicit models, only contain data in the form of letters which stand for the different numbers in your model, far smaller but more complicated

creating an explicit model

1. decision variables- these variables are calculated by the program to give maximized profit or minimized costs, want to name them as you need to know what you want to change- Decision variables: XA1, XA2, …, XC4: Amount of potato from supplier A-C purchased in season 1-4.

2. objective function- decides what you want to minimize or maximize and how much something will cost in the end, in a case where you buy potatoes from a supplier…in season… you want to buy the amount you need for the least amount of money —→ Min{w = 80*XA,1 + 72*XA,2 + 90*XA,3…+140*XC,4}

3. constraints- set of rules which the program needs to take into account; Availability XA,1 <= 40 XA,2 <= 60 … XC,4 <= 10

creating an implicit model

all the explicit rules count, but not working with numbers

1. decision variables- For an implicit model, decision variables can look like this: Xi,j refers to tons sourced from supplier i in season j. In our example we got 3 suppliers and 4 seasons. So i = 3 and j = 4.

2. objective function- Implicit model: -> Min{ w = ∑(i) ∑(j) CI,j * XI,j }, shows the sum of the potatoes bought from a supplier… in season…. times the cost of these potatoes, represents total costs

3. constrains- Availability XI,j <= AvI,j for all I,j The Avi,j stands for the availability of potatoes a supplier has in a season. This constraint applies to all suppliers and all seasons, so that’s why ‘for all i,j’ is added to the constraint

non negativity constraints

saying your variables can’t be less than 0

Explicit model: XA,1,XA,2,...,XC,4 >= 0 Implicit model: Xi,j >= 0 for all i,j

hard and soft constraints

hard constraints- always active, always apply and turned on Hard constraints examples: ∑(i) Ai*Xi <= b ∑(i) Ai*Xi >= b ∑(i) Ai*Xi = b

soft constraints- sometimes can be worthwhile to violate the available inventory level- Where ∑(i) Ai*Xi <= b regulates your normal inventory level (where b is your maximum storage) you can basically turn this off by saying:∑(i) Ai*Xi - s(+) <= b, s(+) stands for a number that is not negative, this number Is large and will exceed the ∑(i) Ai*Xi, so the left side is very small and will always be lower than b, so the inventory is not longer a problem in the model to be taken into account

can also be viewed like this- ∑(i) Ai*Xi <= b + s(+) Your s(+) is large, your right side gets very large, ∑(i) Ai*Xi cannot exceed the right side. Inventory level isn’t a problem anymore

example hard and soft constraint

1. Hard: ∑(i) Ai*Xi >= b

Soft: ∑(i) Ai*Xi + s(-) >= b

Because of the large s(-) the left side will always be larger than b. The constraint is very easily turned off in principle. s(-) is called a SLACK variable, its still non negative

2. Hard: ∑(i) Ai*Xi = b Soft: ∑(i) Ai*Xi -s(+) + s(-) = b Normally when the constraint is hard, you want it to be exactly number b. When it’s soft, s(+) can be subtracted or s(-) can be added up to make it a different number from b: 50 = 50, 50 + 1 = 50, 50 - 1 = 50

If- then constraints/selection problems (BIG M constraints)

closely linked to semi continuous variables

regulating the upper bound of a process, M is just a big number

If yj = 1, activity j is performed

If yj = 0, activity j is NOT performed and something else happens

xj <= M * yj for all j, if yj is 0 it will not be performed, if it is 1, the big number M will be bigger than whatever xj is and it will be performed

semi continuous variables

You want your variable to be above a certain number, below a certain number or be equal to 0. This constraint requires making binary variables. Binary variables are variables that can ONLY be 0 or 1. That means:

Y(i) ⋲ {0,1} Y = 0 if x = 0

Y = 1 if x >= L or x<= M, L and M stand for any given number, but where M is a big number. L is any given number, it’s only the lower bound of the production line f.e.

You need to produce at least 20 potatoes. So L = 20 This can give you these constraints: x <= M y(i) x >= L y(i)

minimax

Minimizing the maximum

Min z s.t.

∑(j) Ckj*xj<= z for all k

Because you want to minimize the maximization, ∑(j) Ckj*xj should be lower than value z. You minimize z basically, and your sum(j)C(kj)*x(j) (Your costs for example) should be lower than z, even though this would normally be maximized

Your costs could be maximized for example if you want to see how much a whole process costs, but in the end you still want to minimize this, so you won’t spend extra money when not needed

maximin

The way to write it down is this:

Max z s.t.

∑(j) Ckj*xj>= z for all k

costs are minimized normally but they want to maximize quality, but here the ∑(j) Ckj*xj should be higher than Z, because z (Quality) is maximized, and the costs should be atleast higher than that value, whilst still being minimized

discrete valued variables

It sometimes occurs that variables are restricted to take values in a specified discrete set {β1, ... , βk) in which βj ⋲ N+ for example j:={0, 10, 20}. you give your variable x a certain number you want it to be

x-(β1*y1+...+βk*yk) = 0

y1+...+yk = 1

yk⋲{0,1} for all k

Imagine your β1 for example is 20, and your β2 is 40. Your x needs to be 20 or 40 to be exactly 0 as the right side. Because y1 is binary, it will be 1 or 0. If y1 is 0, 20*0 = 0. β1 is not activated. β2 on the other hand will be activated, because y1 + y2 needs to be equal to 1 (Second line). So your x will be 40, because: 40 - (20*0+40*1) = 0

either or constraints

For example: You want, if you have 3 total constraints, only 2 constraints to work. At least k (2) of the constraints (1) to (3) must hold (k < r). sum(i) yi = 3 - 2 = 1

The one y that is 1, will activate M (The big number). The b in the example is your upper bound. If M (A big number) is added, the upper bound doesn’t matter anymore as it will be large enough anyways, and the constraint is turned off. The other 2 constraints will still hold, because the other y’s are 0. It’s basically turning constraints on and off, if it’s needed in the program to maximize income or minimize costs.

non negativity constraint example

if then and semicontinuous variable constraints

minimax example

maximin

discrete valued variables

either or constraints

so in 4. if Z constraint is relaxed, no products are selected and when the constraint is active, 2 products are selected, coefficient 2 because RHS should be zero

in 3. you have to give two relaxation options which you cannot get if only one Z, so introduce Z1 to subtract 1 and Z2 to add 1, which allows RHS to move from 1 to 0 to 2

the production chain shows

how fruit juice moves from raw agricultural production to the consumer ;

farmers, distribution centres, factory, supermarket, consumer

ex. locations, brazil is production/concentration, netherlands is processing and packaging

processing steps in the fruit juice chain

juice production, concentration, pasteurisation, packaging, mixing

intermediate and final products; juice, pulp, juice concentrate, packaged fruit juice

classification of the food industry

divided into 3 categories

1- fresh food production

2- ingredient production

3- product assembly

the general production chain - primary production, harvested product, transport to wholesaler, transport to retailer, product at consumer

raw materials include- plant crops, animals (meat), animal products (dairy and eggs)

fresh food production

goal- convert freshly harvested products directly into consumer products

ex. 1 cleaning and packaging fresh fruits- 1) cleaning, 2) sorting, 3) coating/slicing/shaping 4) packaging, output; consumer product

ex. 2 canning fruits 1) cutting 2) packaging 3) heating, output; canned product

ingredient production

goal- convert harvested products into ingredients for later use in food manufacturing

ex. 1 wheat to flour; 1) dry cleaning 2) successive milling 3) sieving, outputs; flour, bran, removed impurities

ex. 2 flour to starch; 1) dough kneading 2) filtering 3) washing 4) rinsing via hydrocyclones 5) drying, outputs; powder starch, wet starch, gluten, side streams

ex. 3 tomato paste production; 1) sorting 2) cleaning 3) cooking 4) evaporation 5) packaging

outputs; tomato paste

product assembly

purchase ingredients and assemble then into final consumer products

ex. tomato sauce production

inputs- tomato paste, other ingredients

operations- formulation, structuring, stabilisation, packaging, output; tomato sauce

types of operations in food processing

4 major operation categories- seperation, conversion, structuring, stabilisation

seperation operations

purpose- seperate material into 2 or more streams

levels of seperation-

A. cleaning- remove contamination such as stones, dirt, soil

methods- dissolving, sieving, rinsing, sedimentation

B. fractionation- obtain different fractions from one stream, increase concentration

methods- filtration, centrifying, extraction, evaporation, sedimentation

C. purification- purpose- increase purity/concentration of components, remove molecular components

methods- dilution, precipitation, ultrafiltration, nanofiltration, reverse osmosis

conversion operations

purpose- change molecular composition

ex. fermentation, acidification/neutralization (pH changes), enzymatic reactions, heating reactions, gelatinization, inactivation of microbes or enzymes

applications- enzymatic synthesis of oligosaccharides from lactose for infant formula, addition of rennet in cheese production, alcoholic beverage production

structuring operations

change the positioning/arrangement of ingredients

like mixing, emulsification, crystallisation, freezing

involve phase changes often

food stabilisation and preservation

non thermal preservation- methods- high hydrostatic pressure, pulsed electric fields, magnetic fields, UV light pulses

thermal preservation- heating pasteurisation, sterilisation, UHT cooling-chilling, chilled drain freezing, lower water activity

reduction of water activity/antimicrobial methods; salting, sugar addition, drying, fermentation, acidulation, causticity, hurdle technology

combined nature of food operations

operations can occur at different scale; removing stones vs seperating molecules

one operation can have different aims- heating for conversion, heating for preservation

one operation may combine multiple purposes; conversion and structuring, seperation and stabilisation

process flow diagrams

rectangle represents a process, arrow represents material flow between processes, text shows input and output materials

mass flow analysis

conservation of mass- mass can neither be created nor destroyed

accumulation= in - out + production

accumulation represents change over time, if nothing changes with time; accumulation = 0

types of processes

batch- no input or output during operation

semi batch- inputs or outputs added periodically

continuous- continuous input and output

steady state- variables such as temperature and flow rate do not change with time, typical for continuous processes

transient- process variables change with time, typical for semi batch processes

special cases of mass balances

batch processes- since input and output are zero; accumulation=production

no conversion occurring, then; production=0

continuous steady state process without production, then- in=out

overall mass balance

for a steady state process with no accumulation-

m1+m2=m3+m4, this is the total mass balance

component balances

a seperate balance can be written for each component

for component A-

mass fraction, molar fractions and ratios

mass fraction- mass of a compounds divided by total mass

simple mass balance example

fat seperator example

a seperator produces

recycle and bypass streams

reasons for recycle streams- increase yield from chemical reactions, conserve energy, reduce inlet concentration of ingredients, ex. recycling air in a dryer

reasons for bypass streams- reduce size of unit operations, improve product quality, safety measures such as coding

large recycle streams are often the largest streams in a process

recycling example

continous processes

for continous systems- mass product flow is represented by

component balances are written using flow rates and compositions

mass flow analysis-

heating

direct- steam injection, batch process

indirect- heat exchangers, plate, tubes and pipes, scraped surfaces, continous processes

evaporation

removing water from a liquid feed to produce a concentrated liquid product

ex. fruit juices (orange juice concentrate), vegetable juices (tomato paste), falling film

evaporation, mass balances

$X_{w,F}$: The mass fraction of water ($w$) in the Feed ($F$) stream.

sankey diagram

visualising energy/mass flows

conservation of mass/energy

representation of flow by arrows

arrow width= scale factor * M

focus on streams, width of the arrow is related to size of stream

for mass, energy, all conserved quantities

all flows in go into the same side, all flows out go into opposite side

for conserved variables, cumulative in should be equal to cumulative out

use one color for each conserved variable

rules sankey diagram

use one color for each conserved variable

main input (raw material) usually in at left side

main output (product) usually out at right side

waste (unwanted output) goes down, emissions go up

clear/short name to each process step

short labels to all arrows

recycles can be made explicit, large recycles usually mean inefficient processes

only simple rectangles

do not split arrows when they represent one stream

do not use different scales in one graph

do not use colors for streams (ex. milk, concentrated milk) use them for components for which a mass balance can be made (solid, water, fat)

food supply chain analysis

to compare production chains quantitatively→ analyse material flows, measure water and raw material efficiency, create flow diagrams sankey diagrams and mass balance calculations

mass balance

for any process, mass entering = mass leaving;

mass flow analysis

evaporation calculation

removes water to concentrate food products

• The feed starts with $12\%$ solids ($x_{d,sF} = 0.12$).

• The final product is concentrated to $42\%$ solids ($x_{d,sP} = 0.42$).

drying of food products

removes water from solid/liquid feeds using hot air

types of dryers- cabinet/tray/belt dryers; applications- vegetables, fruits, breakfast cereals

bin dryers; grain

spray drying; converts liquids into powder, milk powder, instant coffee

fluidized bed dryers; peas, vegetables, grain, powders

freeze drying; chicken meat, mushrooms for soup mixes

spray drying products

products made using spray drying; milk powders, instant coffee, tea drinks, powdered egg products

important quality parameter; resolubilisation of powder

dryer mass balances

drying systems involve- feed stream, air inlet stream, product stream, air outlet stream

total mass balance, air stream mass, dry matter balance, water balance

seperation processes

centrifugation- industrial centrifugation is usually continous