; CHAPTER 4 INVENTORY MODELS _DETERMINISTIC_20114724841
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# CHAPTER 4 INVENTORY MODELS _DETERMINISTIC_20114724841

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```									             LESSON 14
INVENTORY MODELS (DETERMINISTIC)
RESOURCE CONSTRAINED SYSTEMS

Outline

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

1
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.
2
Resource Constrained
Multiple Product Inventory Systems

• An important observation on the resource
constrained models is the following:
If the following ratio
Resource required per unit
Holding cost per unit per year
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) 3
Fund Constraint

• If the same interest rate is applied on all products, the
ratio
Cost per unit
Holding cost per unit per year
Cost per unit                  1
                                
(Cost per unit)(Interest Rate) Interest Rate
• Consequently, an optimal order quantity of each
product can be obtained by reducing its EOQ value
by a constant multiplication factor.
4
Fund Constraint

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,

Budget
m
Total Investment Required by EOQ Lot Sizes

5
Example: Fund Constraint

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?
6
Computation for Tomatoes,i  1
h1  Ic1 
2 K11
Q1  EOQ1         
h1
Fund required c1Q1 

Computation for Lettuce,i  2
h2  Ic2 
2 K 2 2
Q2  EOQ2             
h2
Fund required c2Q2 

7
Computation for Zucchini, i  3
h3  Ic3 
2 K 3 3
Q3  EOQ3           
h3
Fund required c3Q3 

Total fund required
Fund available 
Since fund available is less than required,
reduce each EOQ value by
the constant multiplication factor,
Fund available
m                       
Total fund required                       8
Optimal order quantity for Tomatoes:
Q1*  mQ1 

Optimal order quantity for Lettuce:
Q2  mQ2  .
*

Optimal order quantity for Zucchini :
Q3  mQ3 
*

9
Example: Fund Constraint
Tomatoes    Lettuce   Zucchini
Index, i                                  1         2          3
Annual Demand, i                     1000      1500        750
Ordering/Set-up Cost, Ki                  5         5          5
Holding cost/unit/year, hi          0.0725    0.1125     0.0625
Cost/unit, wi                          0.29      0.45       0.25
Lot sizes, Qi
Fund required
Total fund required
Fund available                         300
Constant multiplier, m
Qi reduced proportionately, Q'i
10
Space Constraint

• For each product, compute the following ratio

Space required per unit
Holding cost per unit per year

• 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).
11
Space Constraint

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:
2 K i i
Q 
*

hi  2  wi
i

12
Space Constraint

Where,
Qi  Order quantity for product i
K i  Set - up cost for product i
 i  Annual demand for product i
hi  Holding cost per unit per year for product i
wi  Space requiredper unit for product i

13
Example: Space Constraint

Example 5: A vegetable stand has exactly 500 square
feet of space. The appropriate data are as follows:
Tomatoes     Lettuce       Zucchini
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?
14
Step 1: Check if EOQ values requiremore space
Tomatoes,i  1, h1  Ic1 
2 K11
Q1  EOQ1         
h1
Lettuce,i  2, h2  Ic2 
2 K 2 2
Q2  EOQ2           
h2
Zucchini, i  3, h3  Ic3 
2 K 3 3
Q3  EOQ3           
h3
Total space required w1Q1  w2Q2  w3Q3 

Since space requiredis more than available,
15
go to Step 2
An optional step between Steps 1 and 2 :
Find the range of possible  values
Space available            500
m                                            0.7373
Space reuiredby the EOQ lot sizes 678.16
1  2Kii               
i                      hi 
2 wi  m  EOQi 2

1         2  5 1000           
1                               0.0725  0.0609
2  0.50  0.7373 371.392

1         2  5 1500           
2                               0.1125  0.1181
2  0.40  0.7373 365.152

1         2  5  750           
3                              0.0625  0.0262
2 1.00  0.7373 346.412

16
An optional step between Steps 1 and 2 (continued) :
Find the range of possible  values
A lower bound on 
 min 1 ,  2 , 3 
 min 0.0609,0.1181,0.0262
 0.0262
An upper bound on 
 max1 ,  2 , 3 
 max0.0609,0.1181,0.0262
 0.1181
Hence, it is sufficient to search 
between 0.0262 and 0.1181

17
Step 2 : Trial value of  
2 K11
Q1            
h1  2w1
Space required w1Q1 
2 K 2 2
Q2              
h2  2w2
Space required w2Q2 
2 K 3 3
Q3            
h3  2w3
Space required w3Q3 
Total space required

Question: Will you increase or decreasethe trial value?18
Step 2 : Trial value of  
2 K11
Q1            
h1  2w1
Space required w1Q1 
2 K 2 2
Q2              
h2  2w2
Space required w2Q2 
2 K 3 3
Q3            
h3  2w3
Space required w3Q3 
Total space required

Question: Will you increase or decreasethe trial value?19
Step 2 : Trial value of  
2 K11
Q1            
h1  2w1
Space required w1Q1 
2 K 2 2
Q2              
h2  2w2
Space required w2Q2 
2 K 3 3
Q3            
h3  2w3
Space required w3Q3 
Total space required

Question: When do you stop the trial and error process?
20
Example: Space Constraint
Trial value of 
Tomatoes   Lettuce Zucchini
Index, i                          1         2       3
Annual Demand,  i             1000     1500      750
Ordering/Set-up Cost, Ki          5         5       5
Holding cost/unit/year, hi   0.0725   0.1125   0.0625
Space requirement/unit, wi      0.5       0.4       1
Lot sizes, Qi
Space required
Total space required
Space available                 500
Conclusion                                       21
Example: Space Constraint
Trial value of                  0.1
Tomatoes Lettuce Zucchini
Index, i                           1           2          3
Annual Demand,  i              1000       1500         750
Ordering/Set-up Cost, Ki           5           5          5
Holding cost/unit/year, hi    0.0725 0.1125        0.0625
Space requirement/unit, wi       0.5         0.4          1
Lot sizes, Qi                 240.77 279.15 169.03
Space required                120.39 111.66        169.03
Total space required        401.0748
Space available                  500
Conclusion                 Decrease trial value (why?)
22
Example: Space Constraint
Trial value of                  0.02
Tomatoes Lettuce Zucchini
Index, i                            1           2         3
Annual Demand,  i               1000       1500        750
Ordering/Set-up Cost, Ki            5           5         5
Holding cost/unit/year, hi     0.0725 0.1125        0.0625
Space requirement/unit, wi        0.5         0.4         1
Lot sizes, Qi                  328.80 341.66 270.50
Space required                 164.40 136.66 270.50
Total space required         571.5639
Space available                   500
Conclusion                 Increase trial value (why?)23
Example: Space Constraint
Trial value of              0.04287
Tomatoes    Lettuce Zucchini
Index, i                           1         2       3
Annual Demand,  i              1000     1500      750
Ordering/Set-up Cost, Ki           5         5       5
Holding cost/unit/year, hi    0.0725   0.1125   0.0625
Space requirement/unit, wi       0.5       0.4       1
Lot sizes, Qi                 294.41   319.66 224.93
Space required                147.21   127.86 224.93
Total space required        499.9997
Space available                  500
Conclusion                 ?                      24
Example: Space Constraint

Optimal order quantity for Tomatoes:
Q1*  Q1 

Optimal order quantity for Lettuce:
Q2  Q2 
*

Optimal order quantity for Zucchini:
Q3  Q3 
*

25
Lesson 14