# Inventory Model with Shortages and Quantity Discount ## **Given Data** - **Annual demand (D):** 30,000 units - **Ordering cost (S):** \$2,500 - **Unit price (p):** - \$60 if \( Q < 500 \) - \$54 if \( Q \geq 500 \) - **Holding cost (h):** .2p - **Shortage cost (p):** \$45 - **Total cost function:** \[ TC = Dp + \frac{DS}{Q} + h \frac{(Q-B)^2}{2Q} + p \frac{B^2}{2Q} \] - **Constraint:** \( Q \leq 800 \) --- ## **Tasks** ### **1. Derive Expressions for \( Q^* \) and \( B^* \)** The EOQ model with allowed shortages gives the following optimal order quantity (\( Q^* \)) and backorder level (\( B^* \)): \[ Q^* = \sqrt{ \frac{2DS}{h} \cdot \frac{h+p}{p} } \] \[ B^* = Q^* \cdot \frac{p}{h+p} \] Where: - \( D \) = annual demand - \( S \) = ordering cost - \( h \) = holding cost per unit per year - \( p \) = shortage cost per unit per year --- ### **2. Determine Optimal Order Quantity Considering Discount** #### **Case 1: \( p = \$60, \ h = .2 \times 60 = \$12 \) (No discount, \( Q < 500 \))** \[ Q^* = \sqrt{ \frac{2 \cdot 30000 \cdot 250}{12} \cdot \frac{12 + 45}{45} } \] \[ = \sqrt{ \frac{150,000,000}{12} \cdot \frac{57}{45} } \] \[ = \sqrt{12,500,000 \cdot 1.267} \] \[ = \sqrt{15,837,500} \] \[ \approx 398 \] \[ B^* = 398 \cdot \frac{45}{57} \approx 398 \cdot .789 \approx 314 \] - Since \( Q^* = 398 < 500 \), this is feasible for the no-discount case. --- #### **Case 2: \( p = \$54, \ h = .2 \times 54 = \$10.8 \) (Discount, \( Q \geq 500 \))** \[ Q^* = \sqrt{ \frac{2 \cdot 30000 \cdot 250}{10.8} \cdot \frac{10.8 + 45}{45} } \] \[ = \sqrt{13,888,889 \cdot 1.24} \] \[ = \sqrt{17,222,222} \] \[ \approx 415 \] \[ B^* = 415 \cdot \frac{45}{55.8} \approx 415 \cdot .807 \approx 335 \] - But \( Q^* = 415 < 500 \), **not eligible** for discount. So, try at \( Q = 500 \) (lowest value to get discount). --- ### **3. Compute Total Annual Cost for Each Case** #### **Case 1: No Discount (\( Q^* = 398, B^* = 314, p = 60, h = 12 \))** \[ TC_1 = Dp + \frac{DS}{Q} + h \frac{(Q-B)^2}{2Q} + p \frac{B^2}{2Q} \] \[ TC_1 = 30000 \times 60 + \frac{30000 \times 250}{398} + 12 \times \frac{(398-314)^2}{2 \times 398} + 45 \times \frac{314^2}{2 \times 398} \] - \( Dp = 1,800,000 \) - \( \frac{DS}{Q} = \frac{75,000,000}{398} \approx 18,844 \) - \( Q - B = 840 \) - \( \frac{(Q-B)^2}{2Q} = \frac{840^2}{796} = \frac{705,600}{796} \approx 88.7 \) - \( h \times 88.7 = 12 \times 88.7 \approx 1,064 \) - \( \frac{B^2}{2Q} = \frac{314^2}{796} = \frac{9,859,600}{796} \approx 1,239 \) - \( p \times 1239 = 45 \times 1239 \approx 55,755 \) \[ TC_1 \approx 1,800,000 + 18,844 + 1,064 + 55,755 = \boxed{1,875,663} \] --- #### **Case 2: Discount (\( Q = 500, B = 500 \times \frac{45}{55.8} \approx 4031, p = 54, h = 10.8 \))** - \( Q = 500, B = 4031 \) \[ TC_2 = 30000 \times 54 + \frac{30000 \times 250}{500} + 10.8 \times \frac{(500-4031)^2}{2 \times 500} + 45 \times \frac{4031^2}{2 \times 500} \] - \( Dp = 1,620,000 \) - \( \frac{DS}{Q} = \frac{75,000,000}{500} = 15,000 \) - \( Q - B = 969 \) - \( \frac{(Q-B)^2}{2Q} = \frac{969^2}{10000} = \frac{939,961}{10000} = 93.9961 \) - \( h \times 93.9961 = 10.8 \times 93.9961 \approx 1,016 \) - \( \frac{B^2}{2Q} = \frac{4031^2}{10000} = \frac{16,248,961}{10000} = 1,624.90 \) - \( p \times 1,624.90 = 45 \times 1,624.90 \approx 73,120 \) \[ TC_2 \approx 1,620,000 + 15,000 + 1,016 + 73,120 = \boxed{1,709,136} \] --- ### **4. Should the Discount be Accepted?** Compare total costs: - **No Discount:** \$1,875,663 - **With Discount:** \$1,709,136 **Discount yields a lower total annual cost.** --- ## **Final Summary** - **Optimal order quantity without discount:** \( Q^* = 398 \), **Total cost:** \$1,875,663 - **Optimal order quantity with discount:** \( Q = 500 \), **Total cost:** \$1,709,136 - **The discount should be accepted** because it results in a lower total annual cost. --- ### **Key Formulas** \[ Q^* = \sqrt{ \frac{2DS}{h} \cdot \frac{h+p}{p} } \] \[ B^* = Q^* \cdot \frac{p}{h+p} \] \[ TC = Dp + \frac{DS}{Q} + h \frac{(Q-B)^2}{2Q} + p \frac{B^2}{2Q} \] --- **If you need a step-by-step derivation for \( Q^* \) and \( B^* \), let me know!**

# Summary of Inventory Model Analysis ## Step 1: Derive Expressions for \( Q^* \) and \( B^* \) The optimal order quantity (\( Q^* \)) and backorder level (\( B^* \)) are derived using the EOQ model for shortages. The formulas incorporate demand, ordering costs, holding costs, and shortage costs. ## Step 2: Determine Optimal Order Quantity Considering Discount Two scenarios are evaluated: one without a discount (unit price \$60, \( Q < 500 \)) and one with a discount (unit price \$54, \( Q \geq 500 \)). In the first case, \( Q^* \) is feasible, while in the second case, \( Q^* \) must be tested at the minimum discount threshold of 500. ## Step 3: Compute Total Annual Cost for Each Case Total costs for both cases are calculated, revealing that the no-discount scenario results in a total cost of \$1,875,663. The discounted scenario yields a lower total cost of \$1,709,136 when evaluated at \( Q = 500 \). ## Step 4: Should the Discount be Accepted? Comparison of total costs indicates that accepting the discount leads to significant savings. Hence, the discount option is recommended due to its lower overall annual cost. --- **Overall, the analysis concludes that the discount should be accepted for cost efficiency.**

# Inventory Model with Shortages and Quantity Discount ## **Given Data** - **Annual demand (D):** 30,000 units - **Ordering cost (S):** \$2,500 - **Unit price (p):** - \$60 if \( Q < 5,000 \) - \$54 if \( Q \geq 5,000 \) - **Holding cost (h):** \( 0.2p \) - **Shortage cost (p):** \$45 - **Total cost function:** \[ TC = Dp + \frac{DS}{Q} + h \frac{(Q-B)^2}{2Q} + p \frac{B^2}{2Q} \] - **Constraint:** \( Q \leq 8,000 \) --- ## **Tasks** ### **1. Derive Expressions for \( Q^* \) and \( B^* \)** Using the EOQ model with shortages, the optimal order quantity (\( Q^* \)) and backorder level (\( B^* \)) can be expressed as: \[ Q^* = \sqrt{ \frac{2DS}{h} \cdot \frac{h+p}{p} } \] \[ B^* = Q^* \cdot \frac{p}{h+p} \] Where: - \( D \) = annual demand - \( S \) = ordering cost - \( h \) = holding cost per unit per year - \( p \) = shortage cost per unit per year --- ### **2. Determine Optimal Order Quantity Considering Discount** #### **Case 1: \( p = \$60, \ h = 0.2 \times 60 = \$12 \) (No discount, \( Q < 5,000 \))** \[ Q^* = \sqrt{ \frac{2 \cdot 30000 \cdot 250}{12} \cdot \frac{12 + 45}{45} } \] Calculating results in \( Q^* \approx 398 \), which is feasible since \( 398 < 5000 \). \[ B^* = 398 \cdot \frac{45}{57} \approx 314 \] --- #### **Case 2: \( p = \$54, \ h = 0.2 \times 54 = \$10.8 \) (Discount, \( Q \geq 5,000 \))** Assuming \( Q \) at the lowest discount threshold of 5,000: \[ Q^* = \sqrt{ \frac{2 \cdot 30000 \cdot 250}{10.8} \cdot \frac{10.8 + 45}{45} } \] Calculating results in \( Q^* \approx 415 \), which is still less than 5,000 and indicates it is not eligible for the discount. --- ### **3. Compute Total Annual Cost for Each Case** #### **Case 1: No Discount (\( Q^* = 398, B^* = 314, p = 60, h = 12 \))** Calculating total cost: \[ TC_1 = 30000 \times 60 + \frac{30000 \times 250}{398} + 12 \times \frac{(398-314)^2}{2 \times 398} + 45 \times \frac{314^2}{2 \times 398} \] Results in \( TC_1 \approx 1,875,663 \). --- #### **Case 2: Discount (\( Q = 5,000, B = 5,000 \times \frac{45}{55.8} \approx 4,031, p = 54, h = 10.8 \))** Calculating total cost: \[ TC_2 = 30000 \times 54 + \frac{30000 \times 250}{5000} + 10.8 \times \frac{(5000-4031)^2}{2 \times 5000} + 45 \times \frac{4031^2}{2 \times 5000} \] Results in \( TC_2 \approx 1,709,136 \). --- ### **4. Should the Discount be Accepted?** Comparison of total costs shows: - **No Discount:** \$1,875,663 - **With Discount:** \$1,709,136 The discount option results in lower total annual cost, hence it should be accepted. --- ## **Final Summary** - **Optimal order quantity without discount:** \( Q^* = 398 \), **Total cost:** \$1,875,663 - **Optimal order quantity with discount:** \( Q = 5,000 \), **Total cost:** \$1,709,136 - **Conclusion:** The discount should be accepted for cost efficiency due to lower overall annual cost. --- ### **Key Formulas** \[ Q^* = \sqrt{ \frac{2DS}{h} \cdot \frac{h+p}{p} } \] \[ B^* = Q^* \cdot \frac{p}{h+p} \] \[ TC = Dp + \frac{DS}{Q} + h \frac{(Q-B)^2}{2Q} + p \frac{B^2}{2Q} \]

# Summary of Inventory Model Analysis with Quantity Discounts ## Step 1: Derive Expressions for \( Q^* \) and \( B^* \) The EOQ model for shortages provides expressions for the optimal order quantity (\( Q^* \)) and backorder level (\( B^* \)). These expressions incorporate key parameters such as demand, ordering costs, holding costs, and shortage costs. ## Step 2: Determine Optimal Order Quantity Considering Discount Two scenarios are evaluated based on the unit price: one for orders less than 5,000 units at \$60, and another for orders of 5,000 units or more at \$54. Calculations show that \( Q^* \) must be tested at the minimum discount threshold of 5,000. ## Step 3: Compute Total Annual Cost for Each Case Total costs are calculated for both scenarios, revealing that the no-discount scenario results in a total cost of \$1,875,663. The discounted scenario at \( Q = 5,000 \) yields a lower total cost of \$1,709,136. ## Step 4: Should the Discount be Accepted? A comparison of total costs indicates that accepting the discount leads to significant savings. Therefore, the discount option is recommended due to its lower overall annual cost. --- **Overall Conclusion:** The analysis concludes that the discount should be accepted for enhanced cost efficiency.