Lesson Notes By Weeks and Term v3 - Senior Secondary 2
Inventory model
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Inventory model
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Subject: Further Mathematics
Class: Senior Secondary 2
Term: 1st Term
Week: 9
Theme: Introduction To Operating Research
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Explain the concept of in ventory Define important terms in in ventory Compute the optional quantity in in ventory model
Introduction To Operating Research rent, insurance premiums, cost of capital tied up in inventory (opportunity cost), spoilage (e.g., perishable goods like tomatoes), obsolescence (e.g., outdated phone models), pilferage/shrinkage, cost of electricity for cold storage, security costs (e.g., night guards).
Lead Time: The duration between placing an order and receiving the ordered goods.
Shortage Cost (Stock-out Cost): The penalty incurred when there is insufficient inventory to meet demand. This can include lost sales, loss of customer goodwill, emergency ordering costs, and production delays. Optimal Quantity (Q) or Economic Order Quantity (EOQ): This is the ideal order quantity that minimizes the total annual inventory costs (i.e., the sum of annual ordering costs and annual holding costs). It represents the most cost-effective amount to order each time. 2.3 The Economic Order Quantity (EOQ) Model The EOQ model is a classic inventory management technique used to determine the optimal order quantity that minimizes the total cost of inventory. It aims to strike a balance between ordering costs (which decrease as order quantity increases) and holding costs (which increase as order quantity increases).
Assumptions of the Basic EOQ Model:
1. Demand is constant and known.
2. Ordering cost per order is constant.
3. Holding cost per unit per year is constant.
4. Lead time is zero or known and constant.
5. No stock-outs are allowed.
6. The entire order quantity is received at once.
Formula Derivation/Explanation: Total Annual Cost (TC) = Annual Ordering Cost + Annual Holding Cost
1. Annual Ordering Cost: If `D` is the annual demand and `Q` is the order quantity, then the number of orders per year is `D/Q`. If `Co` is the ordering cost per order, then: Annual Ordering Cost = (Number of Orders per Year) × (Ordering Cost per Order) Annual Ordering Cost = (D/Q) Co
2. Annual Holding Cost: Assuming inventory is consumed at a constant rate, the average inventory level is `Q/2` (when an order arrives, inventory is Q, it goes down to 0, average is Q/2). If `Ch` is the holding cost per unit per year, then: Annual Holding Cost = (Average Inventory Level) × (Holding Cost per Unit per Year) Annual Holding Cost = (Q/2) Ch Therefore, the Total Annual Cost (TC) = (D/Q) Co + (Q/2) Ch To find the optimal quantity (EOQ, denoted as Q), one differentiates TC with respect to Q and sets the derivative to zero.
This yields the EOQ formula: Q = √(2 D Co / Ch)
Where: `Q` = Economic Order Quantity (Optimal Order Quantity) `D` = Annual Demand (units per year) `Co` = Ordering Cost per order `Ch` = Holding Cost per unit per year Worked
Example: A supermarket in Lagos sells an average of 1200 cartons of a popular brand of soft drink per year. The cost of placing an order with the distributor is N2,
5
0
0. The cost of holding one carton in inventory for a year (including storage, electricity for chiller, and spoilage allowance) is N
5
0
0. Calculate the optimal quantity of soft drinks the supermarket should order each time to minimize total inventory costs.
Solution: Given: Annual Demand (D) = 1200 cartons/year Ordering Cost (Co) = N2,500 per order Holding Cost (Ch) = N500 per carton per year Using the EOQ formula: Q = √(2 D Co / Ch) Q = √(2 1200 2500 / 500) Q = √(2400 5) Q = √12000 Q ≈ 109.54 Since the order quantity must be a whole number of cartons, the optimal quantity is approximately 110 cartons.
3. Teaching and Learning Activities 3.1 Teacher Activities Introduction (10 minutes): Begin by reviewing the concept of businesses and their need to stock goods (inventory). Ask students to list types of goods various businesses (e.g., provision stores, tailors, farmers, pharmacists) would keep. Introduce the idea of "inventory" as these stocks of goods.
Pose a problem: "What happens if a shop orders too much stock? What if it orders too little?" This leads into the concept of balancing costs.
Concept Explanation (20 minutes): * Explain the concept of inventory in detail, using examples relevant to Nigerian Introduction (10 minutes): Begin by reviewing the concept of businesses and their need to stock goods (inventory). Ask students to list types of goods various businesses (e.g., provision stores, tailors, farmers, pharmacists) would keep. Introduce the idea of "inventory" as these stocks of goods.
Pose a problem: "What happens if a shop orders too much stock? What if it orders too little?" This leads into the concept of balancing costs.
Concept Explanation (20 minutes): Explain the concept of inventory in detail, using examples relevant to Nigerian businesses and daily life.
Introduce and define the key terms: Demand, Ordering Cost, Holding Cost, Lead Time, Optimal Quantity, providing clear Nigerian examples for each. Use visual aids (e.g., simple diagrams showing a warehouse, a truck for transport costs, etc.). Facilitate a short brainstorming session for students to identify examples of holding and ordering costs in typical Nigerian small businesses. EOQ Model Explanation and Derivation (20 minutes): Introduce the EOQ model as a tool to find the "best" order quantity. Clearly present the components of total inventory cost (annual ordering cost and annual holding cost). Guide students through the logic of how each cost is calculated. Present the final EOQ formula Q = √(2 D Co / Ch), explaining each variable. While a full calculus derivation might be beyond SS2 for every student, explaining the goal of minimizing costs by balancing the two cost types is crucial.
Worked Example Demonstration (15 minutes): Present the worked example from Section 2.3 (supermarket soft drinks). Solve the problem step-by-step on the board, explaining each stage clearly. Emphasize units and proper interpretation of the result (e.g., rounding up to a whole unit).
Facilitate Discussion and Q&A (5 minutes): Address any misconceptions or questions from students. Encourage students to think about how these costs might vary in different Nigerian business contexts. 3.2 Student Activities Brainstorming: Students participate in brainstorming examples of inventory and associated costs in different Nigerian business scenarios.
Group Definition: In small groups, students define the key terms in their own words and provide their own Nigerian examples.
Note-Taking: Students take detailed notes on definitions, formulas, and worked examples.
Problem-Solving Participation: Students actively follow the teacher's worked examples, asking questions for clarification.
Paired/Group Discussion: Students discuss potential implications of poor inventory management (e.g., expired goods, lost customers).
4. Guided Practice (With Solutions)
Question 1: A tailor in Aba uses approximately 800 yards of a specific type of fabric annually. The cost to order the fabric from a supplier in Kano is N3,000 per order (including transport and bank charges). The cost of holding one yard of fabric in storage for a year is N
1
5
0. Find the optimal quantity of fabric the tailor should order each time.
Solution 1: Given: Annual Demand (D) = 800 yards/year Ordering Cost (Co) = N3,000 per order Holding Cost (Ch) = N150 per yard per year Using the EOQ formula: Q = √(2 D Co / Ch) Q = √(2 800 3000 / 150) Q = √(1600 20) Q = √32000 Q ≈ 178.89 Therefore, the optimal quantity of fabric the tailor should order each time is approximately 179 yards.
Commentary: This calculation helps the tailor minimize the total cost associated with ordering and storing fabric throughout the year.
Question 2: A small pharmaceutical store in Port Harcourt sells 3,600 packets of pain relievers per year. The cost to place an order with the drug manufacturer is N4,
5
0
0. The holding cost for one packet of pain reliever for a year is N100 (considering storage, security, and potential expiry). a) Calculate the Economic Order Quantity (EOQ). b) What is the total annual inventory cost at this optimal quantity?
Solution 2: Given: Annual Demand (D) = 3600 packets/year Ordering Cost (Co) = N4,500 per order Holding Cost (Ch) = N100 per packet per year a)
Calculate the EOQ: Q = √(2 D Co / Ch) Q = √(2 3600 4500 / 100) Q = √(7200 45) Q = √324000 cost for one packet of pain reliever for a year is N100 (considering storage, security, and potential expiry). a) Calculate the Economic Order Quantity (EOQ). b) What is the total annual inventory cost at this optimal quantity?
Solution 2: Given: Annual Demand (D) = 3600 packets/year Ordering Cost (Co) = N4,500 per order Holding Cost (Ch) = N100 per packet per year a)
Calculate the EOQ: Q = √(2 D Co / Ch) Q = √(2 3600 4500 / 100) Q = √(7200 45) Q = √324000 Q = 1800 The Economic Order Quantity (EOQ) is 180 packets. b) Calculate the total annual inventory cost at this optimal quantity: Annual Ordering Cost = (D/Q) Co = (3600 / 180) 4500 = 20 4500 = N90,000 Annual Holding Cost = (Q/2) Ch = (180 / 2) 100 = 90 100 = N9,000 Total Annual Cost (TC) = Annual Ordering Cost + Annual Holding Cost = 90,000 + 9,000 = N18,000 The total annual inventory cost at the optimal quantity is N18,
0
0
0. Commentary: Note that at the EOQ, the annual ordering cost and annual holding cost are approximately equal. This is a characteristic of the basic EOQ model.
Question 3: A wholesaler in Onitsha deals in bags of cement. Their annual demand for a particular brand is 4,000 bags. The cost of processing an order from the factory, including transport, is N6,
0
0
0. The cost of holding one bag of cement in the warehouse for a year (including rent, security, and slight spoilage) is N
1
2
0. Calculate: a) The optimal number of bags to order per batch. b) The number of orders that will be placed annually.
Solution 3: Given: Annual Demand (D) = 4000 bags/year Ordering Cost (Co) = N6,000 per order Holding Cost (Ch) = N120 per bag per year a) Optimal number of bags to order per batch (EOQ): Q = √(2 D Co / Ch) Q = √(2 4000 6000 / 120) Q = √(8000 50) Q = √400000 Q = 632.45 The optimal number of bags to order per batch is approximately 632 bags. b) Number of orders that will be placed annually: Number of orders = Annual Demand (D) / Optimal Order Quantity (Q) Number of orders = 4000 / 632.45 Number of orders ≈ 6.32 orders The wholesaler will place approximately 6 or 7 orders annually. (It's practical to consider 6 orders with a larger final order or 7 smaller orders, but the calculated value is ~6.32).
Commentary: Rounding is important in practical application. While 6.32 orders isn't possible, it suggests roughly one order every two months.*
5. Independent Practice (Questions Only)
1. Explain, in your own words, what inventory means to a small-scale farmer in Benue State who sells yams.
2. List three types of inventory that a furniture manufacturer in Lagos might keep.
3. A bakery in Kano uses 7,200kg of flour annually. The cost to order flour from the mill is N8,000 per order. The cost of holding 1kg of flour for a year is N
5
0. Calculate the optimal order quantity for flour.
4. Define 'holding cost' and 'ordering cost' with two examples for each in the context of a provision store in Ibadan.
5. A distributor of bottled water sells 4,800 cartons per year. The cost of placing an order is N3,
5
0
0. The annual holding cost per carton is N
7
0. Determine the EOQ for the bottled water.
6. For the bottled water distributor in question 5, calculate the total annual inventory cost if they use the EOQ.
7. A bookstore in Abuja sells 1,500 copies of a popular textbook annually. If the ordering cost is N2,000 per order and the holding cost per book per year is N80, what is the optimal number of textbooks to order at a time?
8. Distinguish between raw materials and finished goods with examples from a shoe-making industry in Abia State.
9. A retailer orders 500 units of an item at a time. The annual demand is 6,000 units. The ordering cost is N1,200, and the holding cost is N30 per unit per