Outline Resource Constrained Multi-Product Inventory Systems Fund constraint Space constraint

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Outline

• Resource Constrained Multi-Product Inventory Systems– Fund constraint– Space constraint

LESSON 14INVENTORY MODELS (DETERMINISTIC) RESOURCE CONSTRAINED SYSTEMS

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Resource Constrained Multiple Product Inventory Systems

• So far, we have discussed inventory control models assuming that there is only one product of interest.

• Often, there can be multiple products that may compete for the same resource such as fund, space, etc.

• In such a case, the EOQ solutions may be satisfactory if the fund/space required by the EOQ solutions is less than that available.

• However, The EOQ solutions cannot be implemented if the fund/space required by the EOQ solutions is more than that available.

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• An important observation on the resource constrained models is the following:

If the following ratio

is the same for all products, an optimal order quantity of each product can be obtained by reducing its EOQ value by a constant multiplication factor.

• We shall discuss two cases:– Fund constraint (satisfies the above condition)– Space constraint (may not satisfy the condition)

yearper unit per cost Holding

unit per required Resource

Resource Constrained Multiple Product Inventory Systems

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Fund ConstraintSame Interest Rate For All Products

• If the same interest rate is applied on all products, the ratio

is the same for all products. • Consequently, an optimal order quantity of each

product can be obtained by reducing its EOQ value by a constant multiplication factor.

Rate Interest Rate) restunit)(Inte per (Cost

unit per Cost


unit per Cost

1

5

Fund ConstraintSame Interest Rate For All Products

Steps

1. Compute the EOQ values and the total investment required by the EOQ lot sizes. If the investment required does not exceed the budget constraint, stop.

2. Reduce the lot sizes proportionately. To do this, multiply each EOQ value by the constant multiplier,

Sizes Lot EOQby Required Investment Total

Budgetm

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Example: Fund ConstraintSame Interest Rate For All Products

Example 4: A vegetable stand wants to limit the investment in inventory to a maximum of $300. The appropriate data are as follows:

Tomatoes Lettuce Zucchini

Annual demand 1000 1500 750

(in pounds)

Cost/pound $0.29 $0.45 $0.25

The ordering cost is $5 in each case and the annual interest rate is 25%. What are the optimal quantities that should be purchased?

7

22

2

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22

11

1

1111

11

2

2

2

1

Qc

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KEOQQ

Ich

i

Qc

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KEOQQ

Ich

i

required Fund

Lettuce, for nComputatio

required Fund

Tomatoes, for nComputatio

8

required fund Total

available Fund

factor, tionmultiplica constant the

by value EOQ each reduce

required, than less is available fund Since

available Fund

required fund Total

required Fund

Zucchini, for nComputatio

m

Qc

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KEOQQ

Ich

i

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mQQ

mQQ

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*

*

*

:Zucchini forquantity order Optimal

.

:Lettuce forquantity order Optimal

:Tomatoes forquantity order Optimal

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Example: Fund ConstraintSame Interest Rate For All Products

Tomatoes Lettuce ZucchiniIndex, i 1 2 3Annual Demand, i 1000 1500 750Ordering/Set-up Cost, Ki 5 5 5Holding cost/unit/year, hi 0.0725 0.1125 0.0625Cost/unit, wi 0.29 0.45 0.25Lot sizes, QiFund requiredTotal fund requiredFund available 300Constant multiplier, mQi reduced proportionately, Q'i

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Space Constraint

• For each product, compute the following ratio

• If the above ratio is the same for all products, a procedure similar to the one for the budget constraint may be applied.

• Assume that the above ratios are different for different products (a reason may be that space requirement is not necessarily proportional to costs).


unit per required Space

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Steps

1. Compute the EOQ values and the total space required by the EOQ lot sizes. If the space required does not exceed the space constraint, stop.

2. By trial and error, find a value of such that the space required by the following lot sizes equals the space available:

Space Constraint

ii

iii wh

KQ

2

2*

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Space Constraint

iw

ih

i

iK

iQ

i

i

i

i

i

product for unit per required Space

product for yearper unit per cost Holding

product for demand Annual

product for cost up-Set

product forquantity Order

Where,

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Example: Space Constraint

Example 5: A vegetable stand has exactly 500 square feet of space. The appropriate data are as follows:

Tomatoes LettuceZucchini

Annual demand 1000 1500 750

(in pounds)

Space required 0.5 0.4 1

(square feet/pound)

Cost/pound $0.29 $0.45 $0.25

The ordering cost is $5 in each case and the annual interest rate is 25%. What are the optimal quantities that should be purchased?

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2 Step to go

available, than more is required space Since

required space Total

Zucchini,

Lettuce,

Tomatoes,

space more require values EOQ if Check 1: Step

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2

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,

,

16

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41.3467373.0

75052

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150052

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100052

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500

sizes lot EOQ theby reuired Space

available Space

values possible of range the Find

:2 and 1 Steps between step optional An

17

0.1181 and 0.0262 between

search to sufficient is it Hence,

on bound upper An

on bound lowerA

values possible of range the Find

:)(continued 2 and 1 Steps between step optional An

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,,max

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0262.0,1181.0,0609.0min

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18value? trial the decrease or increase you Will:Question


required Space

required Space

required Space

of value Trial :2 Step

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19value? trial the decrease or increase you Will:Question


required Space

required Space

required Space


33

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22

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11

11

111

22

22

22

Qw

wh

KQ

Qw

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20process? error and trial the stop youdo When:Question


required Space

required Space

required Space


33

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11

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22

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Qw

wh

KQ

Qw

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Trial value of Tomatoes Lettuce Zucchini

Index, i 1 2 3Annual Demand, i 1000 1500 750Ordering/Set-up Cost, Ki 5 5 5Holding cost/unit/year, hi 0.0725 0.1125 0.0625Space requirement/unit, wi 0.5 0.4 1Lot sizes, QiSpace requiredTotal space requiredSpace available 500Conclusion

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Trial value of 0.1Tomatoes Lettuce Zucchini

Index, i 1 2 3Annual Demand, i 1000 1500 750Ordering/Set-up Cost, Ki 5 5 5Holding cost/unit/year, hi 0.0725 0.1125 0.0625Space requirement/unit, wi 0.5 0.4 1Lot sizes, Qi 240.77 279.15 169.03Space required 120.39 111.66 169.03Total space required 401.0748Space available 500Conclusion Decrease trial value (why?)

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Index, i 1 2 3Annual Demand, i 1000 1500 750Ordering/Set-up Cost, Ki 5 5 5Holding cost/unit/year, hi 0.0725 0.1125 0.0625Space requirement/unit, wi 0.5 0.4 1Lot sizes, Qi 328.80 341.66 270.50Space required 164.40 136.66 270.50Total space required 571.5639Space available 500Conclusion Increase trial value (why?)

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Index, i 1 2 3Annual Demand, i 1000 1500 750Ordering/Set-up Cost, Ki 5 5 5Holding cost/unit/year, hi 0.0725 0.1125 0.0625Space requirement/unit, wi 0.5 0.4 1Lot sizes, Qi 294.41 319.66 224.93Space required 147.21 127.86 224.93Total space required 499.9997Space available 500Conclusion ?

25

33

22

11

QQ

QQ

QQ

*

*

*

:Zucchini forquantity order Optimal

:Lettuce forquantity order Optimal

:Tomatoes forquantity order Optimal


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READING AND EXERCISES

Lesson 14

Reading:

Section 4.8 , pp. 221-225 (4th Ed.), pp. 212-215

Exercise:

26, 28, p. 225 (4th Ed.), p. 215

Outline Resource Constrained Multi-Product Inventory Systems Fund constraint Space constraint

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