Un breve esbozo de los modelos teóricos sobre el duelo

To augment its competitiveness a small business wishes to improve the production of its best selling products. One of its main activities is the production of cards with microchips and electronic badges. The company also produces the components for these cards and badges. Good planning of the production of these components therefore constitutes a decisive success factor for the company. The demand for components is internal in this case and hence easy to anticipate.

For the next six months the production of four products with references X43-M1, X43-M2, Y54-N1, Y54-N2 is to be planned. The production of these components is sensitive to variations of the level of produc- tion, and every change leads to a non-negligible cost through controls and readjustments. The company therefore wishes to minimize the cost associated with these changes whilst also taking into account the production and storage costs.

The demand data per time period, the production and storage costs, and the initial and desired final stock levels for every product are listed in the following table. When the production level changes, readjustments of the machines and controls have to be carried out for the current month. The cost incurred is proportional to the increase or reduction of the production compared to the preceding month. The cost for an increase of the production isBC1 per unit but only BC0.50 for a decrease of the production level.

Table 8.10:Data for the four products

Demands Cost Stock levels

Month 1 2 3 4 5 6 Production Storage Initial Final X43-M1 1500 3000 2000 4000 2000 2500 20 0.4 10 50

X43-M2 1300 800 800 1000 1100 900 25 0.5 0 10

Y54-N1 2200 1500 2900 1800 1200 2100 10 0.3 0 10 Y54-N2 1400 1600 1500 1000 1100 1200 15 0.3 0 10

What is the production plan that minimizes the sum of costs incurred through changes of the production level, production and storage costs?

8.4.1

Model formulation

The mathematical model developed for the problem in Section8.2has many similarities to this model. We denote the variables and constants in the same way. Let PRODS be the set of components, MONTHS = {1, . . . , NT} the set of planning periods, DEMptthe demand for product p in time period t, CPRODp and

CSTOCKprespectively the cost of producing and storing one unit of product p, and ISTOCKpand FSTOCKp the initial and final stock levels of product p.

The variables producept and storept represent respectively the quantity produced and the amount of product p in stock at the end of period t. With the convention storep0 = ISTOCKp the stock balance constraints are given by (8.4.1) and the final stock levels are guaranteed by constraints (8.4.2).

∀p ∈ PRODS, t ∈ MONTHS : storept= storep,t−1+ producept− DEMpt (8.4.1)

∀p ∈ PRODS : storep,NT ≥ FSTOCKp (8.4.2)

To measure the changes to the level of production, we need to additional sets of variables, reducet and

addtthat represent the reduction or increase of the production in time period t. A change the the level of production is simply the difference between the total quantity produced in period t and the total quantity produced in period t − 1. If this difference is positive, we have an increase in the production level, and a reduction otherwise. The change is expressed through the value of addt− reducet.

Since the level of production cannot increase and reduce at the same time, one of the two variables is automatically at 0; whilst the other represents either the increase or the reduction of the production. This leads to the equation (8.4.3).

∀t ∈ {2, . . . , NT} : X p∈PRODS

producept−

X

p∈PRODS

producep,t−1= addt− reducet (8.4.3)

The objective function is like the one of problem8.2with, in addition, the cost of changes to the produc- tion level. We write CADD and CRED for the cost of increasing and reducing the production respectively. These costs are proportional to the changes. They form the second part of the objective function (8.4.4).

minimize X

p∈PRODS

X

t∈MONTHS

CPRODp· producept+ CSTOREp· storept
+

NT

X

t=2

CRED · reducet+ CADD · addt
(8.4.4)

Note that there are no variables reduce1 or add1 because there is no change at the beginning of the planning period.

The complete linear program contains the lines (8.4.4) to (8.4.3), to which we need to add the non- negativity constraints (8.4.9) and (8.4.10) for all variables:

minimize X p∈PRODS

X

t∈MONTHS

CPRODp· producept+ CSTOREp· storept
+

NT

X

t=2

CRED · reducet+ CADD · addt
(8.4.5)

∀p ∈ PRODS, t ∈ MONTHS : storept= storep,t−1+ producept− DEMpt (8.4.6)

∀p ∈ PRODS : storep,NT ≥ FSTOCKp (8.4.7)

∀t ∈ {2, . . . , NT} : X p∈PRODS

producept−

X

p∈PRODS

producep,t−1= addt− reducet (8.4.8)

∀t ∈ {2, . . . , NT} : reducet≥0, addt ≥0 (8.4.9)

∀p ∈ PRODS, t ∈ MONTHS : producep,t≥0, storep,t≥0 (8.4.10)

8.4.2

Implementation

As we have noted, the model of this problem is close to the one of the problem in Section8.2. A similar likeness can be observed between the two implementations with Mosel.

model "C-4 Electronic components" uses "mmxprs"

declarations

NT = 6 ! Number of time periods (months) MONTHS = 1..NT

PRODS = 1..4 ! Set of components

DEM: array(PRODS,MONTHS) of integer ! Demand of components per month CPROD: array(PRODS) of integer ! Production costs

CSTOCK: array(PRODS) of real ! Storage costs

CADD,CRED: real ! Cost of additional/reduced prod. ISTOCK,FSTOCK: array(PRODS) of integer ! Initial and final stock levels produce: array(PRODS,MONTHS) of mpvar ! Quantities produced per month store: array(PRODS,MONTHS) of mpvar ! Stock levels at end of every month reduce: array(MONTHS) of mpvar ! Reduction of production per month add: array(MONTHS) of mpvar ! Increase of production per month end-declarations

initializations from ’c4compo.dat’ DEM CPROD CSTOCK CADD CRED ISTOCK FSTOCK end-initializations

! Objective: total cost of production, storage, and change of prod. level Cost:= sum(p in PRODS, t in MONTHS) (CPROD(p)*produce(p,t) +

CSTOCK(p)*store(p,t)) + sum(t in 2..NT) (CRED*reduce(t) + CADD*add(t))

! Stock balance constraints (satisfy demands) forall(p in PRODS, t in MONTHS)

store(p,t) = if(t>1, store(p,t-1), ISTOCK(p)) + produce(p,t) - DEM(p,t) ! Changes to the level of production

forall(t in 2..NT)

sum(p in PRODS) (produce(p,t) - produce(p,t-1)) = add(t) - reduce(t) ! Guarantee final stock levels

forall(p in PRODS) store(p,NT) >= FSTOCK(p) ! Solve the problem

minimize(Cost) end-model

8.4.3

Results

When solving the problem we obtain a total cost ofBC683,929. The production level for the first four periods is 7060 units in total, in period 5 it decreases to 6100 units and remains at this level for the last period. The following table lists the corresponding production plan (there are several possible plans).

Table 8.11:Optimal production plan

1 2 3 4 5 6 X43-M1 1490 3000 2000 4000 2000 2550 X43-M2 1300 800 800 1000 1100 910 Y54-N1 2150 1500 3640 1060 1200 2130 Y54-N2 2120 1760 620 1000 1800 510 Total 7060 7060 7060 7060 6100 6100

The stock levels are relatively low. With the exception of the last period where a final stock is required, products X43-M1 and X43-M2 are never held in stock. The product Y54-N1 is held in stock in periods 3 and 6 (740 and 30 units), and the product Y54-N2 is held in stock in the periods 1, 2, 5, and 6 with 720, 880, 700, and 10 units respectively.