chapter 2 pmor

SS 2015 Production Management & Operations Research 2 - 1

Lehrstuhl fr Management Science

http://www.mansci.ovgu.de

2: Determination of Output (Product) Quantities

2.1. Outline of the Problem

2.2. Goals and Restrictions

2.2.1. Goals

2.2.2. Restrictions

2.3. Classification of Planning Situations

2.4. Output Quantity Planning for a Single Product

2.4.1. External Bottleneck

2.4.2. Internal Bottleneck

2.5. Output Quantity Planning for Multiple Products I:

External and Single Internal Bottlenecks

2.5.1. External Bottlenecks

2.5.2. Single Internal Bottleneck

2.6. Output Quantity Planning for Multiple Products II:

The Classic Product-Mix Problem

2.6.1. The Bicycle Manufacturers Problem

2.6.2. A Model for the Bicycle Manufacturers Problem

2.6.3. Graphical Solution

2.6.4. Shadow Prices and Opportunity Costs

2.6.5. Sensitivity Analysis

2.6.6. A General Model for the Classic Product-Mix Problem

2.7. Output Quantity Planning for Multiple Products III:

Set-Up Processes

2.8. Multiple-Period Output Planning




2: Determination of Output (Product) Quantities

Literature

Nahmias, S. (2005), 154-182

Krajewski, L. J.; Ritzman, L. P. (2002), 693-729

Heizer, J.; Render, B. (2006), 691-721




Classification of planning situations

- number of sub-periods

- number of product types

- degree of (expected) capacity utilization

single period multiple periods

single

product

type

multiple

product

types

neglectable

set-ups

non-neglectable

set-ups

external bottleneck

=

under-utilization of

production capacity

internal bottleneck(s)

=

full utilization of

production capacity

single (internal)

bottleneck

multiple (internal)

bottlenecks




Classification of planning situations, single-period case

expected

capacity

utilization

number of

product types

under-

utilizationfull utilization

(external

bottlenecks)

single internal

bottleneck

multiple internal

bottlenecks

single product

multiple products,

neglectable

set-up processes

multiple products,

non-neglectable

set-up processes




Period profit in relation to different product quantities




Exercise 2.1: A bicycle manufacturers product-mix problem

A bicycle manufacturer produces four types of bicycles, mens bicycles (A),

women's bicycles (B), childrens bicycles (C), and mountain bikes (D). Due to the

fact that there is a lack of qualified mechanics, the bicycle assembly stage has

proven to be a long-term bottleneck of production. The company is far from being

able to provide all the products which are requested by the customers. Table 2.1.

shows the relevant production, sales and accounting data.

Table 2.1: Data for the production program planning problem of the bicycle

manufacturer

For the forthcoming week the production manager would like to know how many

units of each product type should be produced in order to maximize the profit. The

capacity available at the bottleneck stage during the planning period will be 2,000

minutes.

Determine the optimal production program for the planning period! What will the

gross profit of that particular week be like?

product type

mens

bicycle

(A)

womens

bicycle

(B)

childrens

bicycle

(C)

mountain

bike

(D)

contribution margin

[/unit]75 60 -10 70

processing time per unit

required

at the bottleneck stage

[mins/unit]

1.5 1.0 0.8 2.0

minimum sales quantity

[units/period]200 150 25

maximum sales quantity

[units/period]1,000 750 200 375




A bicycle manufacturers product-mix problem: Non-optimal solution

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A bicycle manufacturers product-mix problem: Optimal solution

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Exercise 2.2: Another bicycle manufacturers product-mix problem

A bicycle manufacturer produces two types of bicycles, men's bicycles (M) and

women's bicycles (W). The production process is a rather simple one: Most of the

bicycle parts (frames, gear-shifting system, lamps, dynamo, brakes, etc.) are

bought on the procurement markets and assembled in a two stage process. On

the first stage (pre-assembly, carried out in department D1) the parts are prepared

for the assembly process. Here, the frames are fixed on roll-pallets, holes are

drilled into the frame which are later needed for the installation of the lighting

system, etc. On the second stage (final assembly, carried out in department D2)

the actual assembly process (installation of light and gear-shifting system, brakes,

handle-bar, mud-guards, etc.) is carried out. Each unit of product type M occupies

D1 for 10 minutes and D2 for 30 minutes. For product type W these operation

times amount to 15 minutes in D1 and 30 minutes in D2. The available (time)

capacities per month are 320 hours for D1 and 800 hours for D2, respectively.

Due to rather limited space at the down-town location of the production facilities it

is neither possible to store material, work-in-progress, nor the final products. Only

what can be sold will be produced and immediately delivered to the customers.

Likewise, the parts necessary for production are received from the suppliers on a

daily (just-in-time) basis. For most of these articles there is an oversupply

available in the markets, thus no shortages have to be expected and the prices

are likely to remain constant for the next few months.





Each unit of product type M provides a profit contribution (gross profit) of 200

Euros. The corresponding profit contribution of W is 250 Euros.

The production manager wants to determine the number of bicycles of type M and

W which should be produced during the forthcoming month (planning period). The

marketing manager expects that 1,200 units of M and 1,500 units of W (but not

more!) can be sold at the current prices (which have to be considered as fixed

because they have just been advertised in the newspapers).

1. Formulate a model which can be used by the production manager to

determine the optimal production quantities of each product type! Do not

forget to define the symbols used in the model formulation!

2. Determine an optimal production program graphically! What will the gross

profit of the planning period be like?

3. What are the shadow prices of an additional capacity unit for department D1

and D2, respectively? What are the corresponding opportunity costs?

4. One worker from department D2 is willing to work five hours overtime during

that particular month. He gets payed a regular salary of 25 Euros per hour,

the overtime bonus is 60 percent. Additional labour costs (fringe benefits etc.)

are calculated at 120 percent. Does it pay of for the company to have him

work the suggested five hours overtime? What other costs may be relevant

for this decision?

5. Carry out a sensitivity analysis with respect to the contribution margin of

product type M and product type W!





WM

pre

-asse

mb

ly

(D1)

asse

mb

ly

(D2)

10 [m

ins/u

nit]

30 [

min

s/u

nit]

15 [m

ins/u

nit]

30 [

min

s/u

nit]

32

0 [h

/pe

r]8

00

[h

/pe

r]

1,2

00

[u

nits/p

er]

1,5

00

[u

nits/p

er]

20

0 [

/unit]

25

0 [/u

nit]




Another bicycle manufacturers product-mix problem: Graphical solution

1,500

1,000

500

500 1,000 1,500

x W

x M




x0= 200x

M+ 250x

W

10xM+ 15x

W+ s

1= 19,200

30xM+ 30x

W+ s

2= 48,000

xM

+ s3

= 1,200

xW

+ s4= 1,500

xM

, xW

, s1, s

2, s

3, s

4 0

x0

Max!

Exercise 2.2 Model

Model:




A linear programming system (product-mix problem)

x0 xM xW s1 s2 s3 s4 RHS




Exercise 2.2 Optimal simplex tableau


1 0 0 10 31

30 0 352,000

0 0 11

5

1

150 0 640

0 1 0 1

5

1

100 0 960

0 0 01

5

1

101 0 240

0 0 0 1

5

1

150 1 860




Exercise 2.2 Sensitivity analysis w.r.t. uM


0 0 11

5

1

150 0 640

0 1 0 1

5

1

100 0 960

0 0 01

5

1

101 0 240

0 0 0 1

5

1

150 1 860




Exercise 2.2 Sensitivity analysis w.r.t. uW


0 0 11

5

1

150 0 640

0 1 0 1

5

1

100 0 960

0 0 01

5

1

101 0 240

0 0 0 1

5

1

150 1 860




variables

(a) goal variable

x0 : period profit contribution (in monetary units such as , $, , etc.);

(b) decision variables

xj : number of units which have to be produced of product type j (j= 1, ,n).

constants

Bk : maximal number of units of part or material type k (k = 1, ,h) which can be obtained from the procurement market(s);

Sj : minimal sales volume of product type j; minimal number of units which have to be produced of product type j (j = 1, ,n);

Sj : maximal sales volume of product type j; maximal number of units which have to be produced of product type j (j = 1, ,n);

Ti : (time) capacity of production stage (production sub-department) i

(i = 1, ,m) in time-units (secs, mins, hours, etc.);

bkj : number of units of part or material type k (k = 1, ,h) which are necessary for the production of one unit of product type j (j = 1, ,n);

tij : number of time-units which are consumed on stage i (i = 1, ,m) to process one unit of product type j (j = 1, ,n);

uj : unit contribution of margin of product type j (j = 1, ,n).

A general model for the classic product-mix problem (1)




A general model for the classic product-mix problem (2)

objective function

(2.1) x0 = j=1

n

uj xj Max!

production (capacity) constraints

(2.2) j=1

n

tij xj Ti, i = 1, ,m;

procurement constraints

(2.3) j=1

n

bkj xj Bk, k = 1, ,h;

sales constraints

(2.4) Sj xj Sj, j = 1, ,n;

non-negativity constraints

(2.5) xj 0, j = 1, ,n.




Exercise 2.3: A bicycle manufacturers product-mix problem with setups

For the bicycle manufacturer of Exercise 2.1 we now consider the problem of

determining a profit-optimal production plan for a different planning period (week

#2). The relevant data is depicted in Table 2.2. The available time capacity will be

2,400 minutes.

Formulate a model from which the optimal production program for that week and

the corresponding profit contribution can be determined.

Table 2.2: Data for the production planning problem of the bicycle manufacturer

in week #2

product type

mensbicycle

(A)

womensbicycle

(B)

childrensbicycle

(C)

mountain

bike

(D)

contribution margin

[/unit]75 30 20 70

processing time per unit

required at the bottleneck stage

[mins/unit]

1.5 1.0 0.8 2.0

minimum sales quantity

[units/period]0 100 0 50

maximum sales quantity

[units/period]400 800 200 400

time per setup

[mins]120 80 160 20

(variable) costs per setup

[]1,800 1,600 2,800 600




Exercise 2.3 Model

Ma

x!

2,4

00 0 0 0 0

40

0

80

0

20

0

40

0 0

{0,1

}

60

0

D

20

D

M

D D

- + -

2,8

00

C

16

0

C

M

C C,

- + -

1,6

00

B

80

B

MB

- + -

1,8

00

12

0 M

,

- + -

70

2.0

xD

+ +

20

0.8

+ +

30

xB

1.0

xB

xB

xB

xB,

+ +

75

xA

1.5

xA

xA

xA

xA,

=

x0 0

10

0 0

50




Exercise 2.3 LINGO solution

Global optimal solution found.

Objective value: 77400.00

Extended solver steps: 0

Total solver iterations: 3

Model title: PMOR

Variable Value Reduced Cost

XA 400.0000 0.000000

XB 780.0000 0.000000

XC 0.000000 4.000000

XD 400.0000 0.000000

YA 1.000000 5400.000

YB 1.000000 4000.000

YC 0.000000 7600.000

YD 1.000000 1200.000

M 10000.00 0.000000

Row Slack or Surplus Dual Price

1 77400.00 1.000000

2 0.000000 30.00000

3 9600.000 0.000000

4 9220.000 0.000000

5 0.000000 0.000000

6 9600.000 0.000000

7 0.000000 30.00000

8 20.00000 0.000000

9 200.0000 0.000000

10 0.000000 10.00000

11 400.0000 0.000000

12 680.0000 0.000000

13 0.000000 0.000000

14 350.0000 0.000000

15 0.000000 0.000000

ABCD




A general model for the product-mix problem with setups (1)

variables

(a) goal variable



xj : number of units which have to be produced of product type j (j = 1, ,n);

j : binary variable, j = 1, if xj > 0,0, else,

j = 1, ,n.

constants

Sj : minimal sales quantity of product type j; minimal number of units which have to be produced of product type j (j = 1, ,n);

Sj : maximal sales quantity of product type j; maximal number of units which have to be produced of product type j (j = 1, ,n);

uj : unit contribution of margin of product type j (j = 1, ,n);

M : sufficiently large number;

T : (time) capacity available at the internal bottleneck (in time-units);

cjsetup : set-up costs for product type j (j = 1, ,n) (in monetary units);

tjsetup : number of time-units necessary for setting up the facilities for the

production of product type j (j = 1, ,n) at the internal bottleneck-stage;

tj : number of time-units which are necessary for processing one unit

of product type j (j = 1, ,n).




A general model for the product-mix problem with setups (2)

objective function

(2.6) x0 = j=1

n

uj xj j=1

n

cjsetup j Max!


(2.7) j=1

n

tj xj + j=1

n

tjsetup j T;

auxiliary constraints

(2.8) xj M j 0, j = 1, ,n;

sales constraints

(2.9) Sj xj Sj, j = 1, ,n;

binary constraints

(2.10) j {0,1}, j = 1, ,n;


(2.11) xj 0, j = 1, ,n.




A general model for the multi-period product-mix problem (1)

variables

(a) goal variable



xjp : number of units to be produced of product type j (j = 1, ,n) in period p (p = 1, ,q);

yjp

: number of units available as inventory of product type j (j = 1, ,n) at thebeginning of period p (p = 1, ,q);

sjp : number of units of product type j (j = 1, ,n) to be sold during period p (p = 1, ,q).





constants

Bkp : maximal number of units of part or material type k (k = 1, ,h) in period p (p = 1, ,q);

Sjp : minimal sales quantity of product type j (j = 1, ,n) in period p (p = 1, ,q);

Sjp : maximal sales quantity of product type j (j = 1, ,n) in period p (p = 1, ,q);

Tip : (time) capacity of production stage i (i = 1, ,m) in period p (p = 1, ,q) (in time-units);

bkj : number of units of part or material type k (k = 1, ,h) which are necessary for the production of one unit of product type j (j = 1, ,n);

tij : number of time-units which are necessary for processing one unit of

product type j (j = 1, ,n) at stage i (i = 1, ,m);

rj : sales price (of one unit) of product type j (j = 1, ,n);

cjprod : variable cost of production of one unit of product type j (j = 1, ,n);

cjinv : variable inventory cost per unit of product type j (j = 1, ,n);

costs of storing one unit of product type j over one time (sub-)period;

IjB : number of units of product type j (j = 1, ,n) which are available

as inventory at the beginning of the planning period;

IjE : number of units which have to be available as inventory of product type j

(j = 1, ,n) at the end of the planning period.





objective function

x0 = p=1

q

j=1

n

rj sjp j=1

n

cjprod xjp

j=1

n

0.5 cjinv (yjp + yjp+1) Max!


j=1

n

tij xjp Tip, i = 1, ,m; p = 1, ,q;

inventory (balance) constraints

xjp + yjp yjp+1 sjp = 0, j = 1, ,n; p = 1, ,q;

yj1 = IjB, j = 1, ,n;

yjq+1 = IjE, j = 1, ,n;

procurement constraints

j=1

n

bkj xjp Bkp, k = 1, ,h; p = 1, ,q;

sales constraints

Sjp sjp Sjp, j = 1, ,n; p = 1, ,q;


xjp 0, j = 1, ,n; p = 1, ,q;

yjp 0, j = 1, ,n; p = 1, ,q;

sjp 0, j = 1, ,n; p = 1, ,q.




Rolling planning horizon

1 2 53 4 6 7

weeks months




bottleneck Engpass

break-even point Gewinnschwelle

constraint Restriktion (in einem Modell)

contribution per unit engpassspezifischer

of the limiting factor Deckungsbeitrag (eines Produktes)

depreciation Abnutzung

direct costs Einzelkosten

equality Gleichung

graphic solution graphische Lsung

inventory Lager (-bestand)

iso-profit line Iso-Gewinnlinie

linear programming Lineare Optimierung, Lineare

Programmierung

non-negativity constraints Nichtnegativittsbedingungen

objective function Zielfunktion

opportunity costs Opportunittskosten

period (profit) contribution Periodendeckungsbeitrag

planning horizon Planungshorizont

process industry Prozess-Industrie

product-mix problem Produktprogramm-Planungsproblem

product type Produktart

production stage Produktionsstufe

profit Gewinn

Glossary




random variable Zufallsvariable

restriction Beschrnkung, Restriktion

revenue Ertrag

sensitivity analysis Sensitivittsanalyse

set-up Einrichtung, (Maschinen-) Rstung

set-up costs Rstkosten

shadow price Schattenpreis

solution Lsung

unit contribution margin Stckdeckungsbeitrag

variable gross profit variabler Brutto-Periodengewinn,

per period Periodendeckungsbeitrag

Glossary

chapter 2 pmor

Documents

output quantity planning

bicycle manufacturers

production program

production manager

relevant production

bicycle manufacturers

classic productmix problem2

planning situations