A Company Is Trying To Decide Between Using TL Truck Load Ve

A Company Is Trying To Decide Between Using Tl Truck Load Versus

A company is trying to decide between using TL (Truck Load) versus LTL (Less than truck load) for ordering a certain item. The item cost is $50 per unit, the ordering cost is $100 per order, and the holding cost is 20% per year. The trucking cost is $400 for TL with a capacity of 750 units, and $1 per unit for LTL. The annual demand for this item is given as 2000 units, 4000 units, and various levels; the company seeks to determine the most economical ordering quantity (Q) for both options, the total costs associated with these options, and the demand level at which both options will cost the same.

Paper For Above instruction

In inventory management, companies aim to optimize their ordering and holding costs to minimize total expenses related to stock procurement. The decision between utilizing Truck Load (TL) and Less Than Truck Load (LTL) options depends on various factors, including demand levels, costs, and capacity constraints. This analysis evaluates the most economical ordering quantities (Q) and associated total costs for both TL and LTL options across different demand scenarios, specifically for annual demands of 2000 and 4000 units, as well as the breakeven demand point where both options incur the same cost.

Economic Order Quantity for TL and LTL

The classical inventory model for calculating the optimal order quantity (Q) is given by the Economic Order Quantity (EOQ) formula:

\[

Q^* = \sqrt{\frac{2DS}{H}}

\]

Where:

• D = annual demand
• S = ordering cost per order
• H = annual holding cost per unit

However, the presence of different transportation costs and capacities requires adjustments to this model for TL and LTL options.

Analysis of TL Option

For TL, the fixed transportation cost per truck load is \$400, with a maximum capacity of 750 units. The ordering cost per order is \$100. The optimal order quantity, Q_TL, is constrained by the capacity of the truck. Since the EOQ may exceed capacity, the company must consider the minimum of EOQ and truck capacity, i.e.,

\[

Q_{TL}^* = \min \left( \sqrt{\frac{2DS}{H \times 0.2}} , 750 \right)

\]

where H is derived from the purchase price and holding rate:

\[

H = \text{Unit cost} \times \text{holding rate} = 50 \times 0.2 = \$10

\]

Calculating EOQ for demand D:

• At D=2000 units:

\[

Q_{EOQ} = \sqrt{\frac{2 \times 2000 \times 100}{10}} = \sqrt{\frac{400,000}{10}} = \sqrt{40,000} \approx 200

\]

Since 200

• At D=4000 units:

\[

Q_{EOQ} = \sqrt{\frac{2 \times 4000 \times 100}{10}} = \sqrt{80,000} \approx 283

\]

Again, 283

Cost Calculation for TL

The total annual cost (TAC_TL) includes purchase cost, ordering cost, holding cost, and transportation cost:

• Purchase Cost: \( D \times 50 \)
• Ordering Cost: \( \frac{D}{Q} \times 100 \)
• Holding Cost: \( \frac{Q}{2} \times 10 \) (assuming average inventory = Q/2)
• Transportation Cost: \( \text{Number of trucks} \times 400 \), where the number of trucks needed is \( \frac{D}{Q} \):

\[

TAC_{TL} = D \times 50 + \left( \frac{D}{Q} \times 100 \right) + \left( \frac{Q}{2} \times 10 \right) + \left( \frac{D}{Q} \times 400 \right)

\]

Calculations:

• At D=2000, Q=200:

\[

TAC_{TL} = 2000 \times 50 + \left( \frac{2000}{200} \times 100 \right) + \left( \frac{200}{2} \times 10 \right) + \left( \frac{2000}{200} \times 400 \right)

\]

\[

= 100,000 + 10 \times 100 + 100 \times 10 + 10 \times 400

\]

\[

= 100,000 + 1,000 + 1,000 + 4,000 = 106,000

\]

• At D=4000, Q=283:

\[

TAC_{TL} = 4000 \times 50 + \left( \frac{4000}{283} \times 100 \right) + \left( \frac{283}{2} \times 10 \right) + \left( \frac{4000}{283} \times 400 \right)

\]

Calculating:

\[

200,000 + (14.11 \times 100) + (141.5 \times 10) + (14.11 \times 400)

\]

\[

= 200,000 + 1,411 + 1,415 + 5,644 = 208,470

\]

Analysis of LTL Option

For LTL, fixed costs are minimal, but the per unit transportation cost is \$1, with no strict capacity limit discussed directly (assuming capacity is sufficient or second to cost considerations). The EOQ for LTL is similarly computed:

\[

Q_{LTL}^* = \sqrt{\frac{2DS}{H_{LTL}}}

\]

Where \( H_{LTL} \) is as above, \$10.

Calculations for D=2000:

\[

Q_{LTL} = \sqrt{\frac{2 \times 2000 \times 100}{10}} \approx 200

\]

Similarly, for D=4000:

\[

Q_{LTL} \approx 283

\]

Cost Calculation for LTL

The total annual cost for LTL involves purchase, ordering, holding, and transportation costs:

• Purchase Cost: \( D \times 50 \)
• Ordering Cost: \( \frac{D}{Q} \times 100 \)
• Holding Cost: \( \frac{Q}{2} \times 10 \)
• Transportation Cost: \( D \times 1 \), as each unit costs \$1 for LTL

Hence,

\[

TAC_{LTL} = D \times 50 + \left( \frac{D}{Q} \times 100 \right) + \left( \frac{Q}{2} \times 10 \right) + D

\]

At D=2000, Q=200:

\[

TAC_{LTL} = 100,000 + (10 \times 100) + (100 \times 10) + 2000

\]

\[

= 100,000 + 1,000 + 1,000 + 2,000 = 104,000

\]

At D=4000, Q=283:

\[

TAC_{LTL} = 200,000 + (14.11 \times 100) + (141.5 \times 10) + 4000

\]

\[

= 200,000 + 1,411 + 1,415 + 4,000 = 206,826

\]

Comparison and Breakeven Analysis

Comparing the total costs for different demand levels shows that at D=2000, LTL has a slightly lower total cost (\$104,000) than TL (\$106,000). For D=4000, LTL again has a lower total cost (\$206,826) compared to TL (\$208,470). To find the demand level where both options cost the same, set the total costs equal and solve for D, considering the dependency of Q on D in EOQ calculations.

Equating total costs:

• \[

  D \times 50 + \frac{D}{Q} \times 100 + \frac{Q}{2} \times 10 + \frac{D}{Q} \times 400 = D \times 50 + \frac{D}{Q} \times 100 + \frac{Q}{2} \times 10 + D

  \]

Simplify by subtracting common terms:

• \[

  \frac{D}{Q} \times 400 = D

  \]

Divide both sides by D (assuming D > 0):

• \[

  \frac{1}{Q} \times 400 = 1

  \]

Therefore,

• \[

  Q = 400

  \]

Using EOQ formula to find the D where Q is approximately 400 units, with the earlier EOQ calculation giving values around 200-283, suggests that the breakeven demand occurs when the optimal Q approaches 400 units. From the EOQ formula, this would correspond to a demand of approximately:

\[

D = \frac{Q^2 \times H}{2S} = \frac{400^2 \times 10}{2 \times 100} = \frac{160,000 \times 10}{200} = 8,000

\]

Thus, at roughly 8,000 units of annual demand, both options would incur similar total costs, indicating a crossover point where the company could switch between TL and LTL depending on other operational considerations.

Conclusion

In conclusion, for demands of 2000 units and 4000 units, LTL consistently presents a slightly lower total cost profile than TL due to lower transportation costs per unit despite higher variable shipping expenses. The optimal order quantities for both options are approximately 200 and 283 units, respectively, given the EOQ constraints and capacity considerations. The breakeven demand level where both options incur equal costs is approximately 8,000 units annually, primarily driven by the relationship between transportation costs and demand volume. These insights assist businesses in making strategic decisions regarding freight options—favoring LTL at lower to moderate demand levels and considering TL when demand surpasses certain thresholds, particularly when capacity or other logistical factors come into play.

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