Industry 4.0 · Quiz 2 · Jan 2026

**1. Classify receipt-related inspection:** Inspecting an incoming order is associated with procuring and receiving that order, so (1) maps to (b), ordering cost. **2. Classify loss while stock is held:** A product can become obsolete during storage, so (2) maps to (c), carrying cost. **3. Classify customer dissatisfaction from shortages:** Failing to supply a wanted item can damage goodwill, so (3) maps to (a), shortage cost. **Answer: A — (1): (b), (2): (c), (3): (a).** **Convention note:** The question uses the standard broad inventory-cost classification. Inspection is part of order/receipt cost here; detailed accounting can separately classify quality-appraisal expenses. A: **Correct:** It correctly maps inspection to ordering, obsolescence to carrying and goodwill loss to shortage. B: **Incorrect:** It exchanges ordering-related inspection and inventory obsolescence. C: **Incorrect:** It misclassifies inspection as shortage and goodwill loss as carrying cost.

**1. Classify the product relationship P1:** Tea and coffee can satisfy a similar beverage need, so P1 maps to S2, substitution. **2. Classify P2:** Milk powder can be consumed together with tea, so P2 maps to S1, complementary products. Its unplanned purchase may also involve impulse buying, but the table asks for the product-pair relationship. **3. Explain the biscuit switch P3:** Limited Good-day stock leads Pranjal to buy the available Sun-fist brand. His observed purchase depends on shelf inventory, so P3 maps to S3, inventory-dependent demand. **Answer: B — P1–S2, P2–S1, P3–S3.** A: **Incorrect:** Tea/coffee are not the intended complementary pair, and the other two mappings do not match the scenario. B: **Correct:** It gives P1–S2, P2–S1 and P3–S3. C: **Incorrect:** The biscuit switch is attributed to stock availability rather than an unexplained impulse.

**1. Distinguish cycles from seasons:** Cyclical fluctuations follow expansions and contractions whose duration need not be a fixed calendar interval. Seasonal fluctuations repeat with a known calendar frequency. **2. Classify the examples:** A recession is part of the business cycle. A recurring winter sales effect or year-end tourist peak is seasonal, regardless of whether that particular seasonal direction is typical. **Answer: A — Economic recission.** A: **Correct:** An economic recession is associated with the non-fixed-length business cycle. B: **Incorrect:** A recurring winter pattern is seasonal rather than cyclical; increased fan sales in winter is unusual but the calendar classification is unchanged. C: **Incorrect:** A recurring end-of-year tourist peak has a fixed annual seasonal timing.

**1. Define a stage and its information:** Let week \(t\) be a stage and \(s_t\) be stock available at the beginning of that week. This stock determines how much new material must be ordered to meet demand \(d_t\). **2. Write the transition:** If \(q_t\) units are ordered, \[s_{t+1}=s_t+q_t-d_t,\qquad 0\le q_t\le5,\quad 0\le s_{t+1}\le4.\] The next week’s initial stock is this week’s ending inventory. **3. Connect to backward recursion:** \[V_t(s_t)=\min_{q_t}\left\{4\mathbf{1}_{\{q_t>0\}}+2q_t+0.5s_{t+1}+V_{t+1}(s_{t+1})\right\}.\] The state passed into the week’s decision is beginning inventory. **Answer: C — Beginning inventory with the warehouse in each week.** A: **Incorrect:** The purchase quantity is the decision/action, not the initial state. B: **Incorrect:** Ending inventory is the result of this week’s action and becomes next week’s state. C: **Correct:** Beginning inventory is the information available when the current decision is made.

**1. Identify the controllable choice:** At the beginning of week \(t\), the firm chooses order quantity \(q_t\). Beginning inventory \(s_t\) and demand \(d_t\) constrain that choice. **2. State the feasible action set:** Orders must meet demand and leave feasible inventory: \[\mathcal A_t(s_t)=\{q\in\{0,1,\ldots,5\}:0\le s_t+q-d_t\le4\}.\] **3. Distinguish state and consequence:** \(s_t\) is the state; \(q_t\) is the action; \(s_t+q_t-d_t\) is ending stock. Backward recursion compares the feasible order quantities. **Answer: A — How many units of the raw material needs to be purchased in a week.** A: **Correct:** The action is how many units to purchase in the current week. B: **Incorrect:** Ending inventory follows from beginning stock, the order and demand. C: **Incorrect:** Beginning inventory is the state entering the decision.

**1. Define a shortage cost:** A shortage occurs when inventory cannot satisfy demand. Its cost includes immediate lost margin and longer-term losses caused by disappointing customers. **2. Identify the relevant losses:** Damaged goodwill, losing a customer and missing the profit on an unfulfilled sale all arise directly from a stockout. Returns of already supplied products normally belong to return/quality costs instead. **Answer: A, C, D — A, C and D.** A: **Correct:** Unavailability can damage the seller’s reputation and customer goodwill. B: **Incorrect:** A product return is normally a reverse-logistics or quality cost, not itself a stockout cost. C: **Correct:** A stockout can cause the customer to leave permanently. D: **Correct:** A sale not fulfilled can lose its associated contribution or profit.

**1. Recall the single-period decision:** Choose inventory before uncertain demand for one selling period becomes known. Excess stock and lost sales then create overage and underage consequences. **2. Interpret separability:** In the basic unconstrained model, each product can be optimized independently; no shared budget or capacity couples its decision to others. **3. State the usual loss structure:** The standard expected mismatch cost is \[C_o\,E[(Q-D)^+]+C_u\,E[(D-Q)^+],\] with constant per-unit overage and underage penalties. **Answer: A, C, D — A, C and D.** **Convention note:** “Separable” means independent product decisions here, not physically divisible or necessarily perishable goods. Shared capacity or budget constraints require a coupled multi-product extension. A: **Correct:** Under the basic independent-product interpretation, product objectives are separable and can be solved individually. B: **Incorrect:** The standard model treats demand as uncertain rather than known deterministically. C: **Correct:** Inventory is committed before the period’s uncertain demand is observed. D: **Correct:** Each realized excess or shortage unit incurs a constant marginal penalty in the basic model.

**1. Identify the axes and cost shapes:** A is annual cost and B is order quantity. Ordering cost is \(DS/Q\), decreasing with lot size, so D is ordering cost. Carrying cost is \(HQ/2\), increasing with lot size, so E is carrying cost. **2. Identify the optimum:** C is the sum of ordering and carrying costs, a U-shaped curve. Its minimum occurs at \[Q^*=\sqrt{\frac{2DS}{H}},\] shown by F on the quantity axis. **3. Read the complete mapping:** A=Cost, B=Quantity, C=Total cost, D=Order cost, E=Carrying cost, F=EOQ. **Answer: B — A: Cost; B: Quantity; C: Total cost; D: Order cost; E: Carrying cost; F: EOQ.** A: **Incorrect:** It swaps the decreasing ordering-cost curve and increasing carrying-cost curve. B: **Correct:** It correctly labels both axes, all three curves and the optimum quantity. C: **Incorrect:** It reverses the axes and also swaps the two component-cost curves. D: **Incorrect:** It confuses axes, total cost and the position of the optimal quantity.

**1. Recognize the smoothing update:** Simple exponential smoothing uses \[F_{t+1}=\alpha D_t+(1-\alpha)F_t,\qquad0\le\alpha\le1.\] It combines the latest actual observation with the previous forecast. **2. Interpret the weighting:** Expanding the recurrence gives past observations weights proportional to \(\alpha(1-\alpha)^k\), decreasing geometrically with age. This explains the term “exponential smoothing.” **Answer: A — F(t+1)=αD(t)+(1−α)F(t).** A: **Correct:** It is the standard weighted update using the latest observation and prior forecast. B: **Incorrect:** The displayed mean-minus-lagged-error form is an MA-type error model, not this simple exponential-smoothing update.

**1. Calculate EOQ using estimated parameters:** The table gives \(D_e=1000,S_e=50,H_e=10\). Thus \[Q_e=\sqrt{\frac{2(1000)(50)}{10}}=100\text{ units}.\] **2. Calculate EOQ using actual parameters:** With \(D_a=2000,S_a=25,H_a=20\), \[Q_a=\sqrt{\frac{2(2000)(25)}{20}}=\sqrt{5000}\approx70.7107.\] **3. Use the actual quantity as the error base:** \[\text{Error}\%=\frac{100-70.7107}{70.7107}\times100=41.42\%.\] Enter 41.42; the estimated optimal lot is larger than the actual optimal lot. **Answer: 41.42.**

**1. Identify the three unit values:** Selling price p=₹2, purchase/production cost c=₹1 and salvage value v=₹0.50. The ₹2 is treated as sales revenue per sold diya, not additional net profit. **2. Compute mismatch costs:** \(C_u=p-c=1\) and \(C_o=c-v=0.5\). **3. Calculate the critical ratio:** \[CR=\frac{C_u}{C_u+C_o}=\frac{1}{1+0.5}=\frac23\approx0.67.\] Retain the exact fraction when calculating the optimal quantity. **Answer: 0.67.**

**1. Compute the exact critical ratio:** \(C_u=2-1=1\), \(C_o=1-0.5=0.5\), so \(CR=2/3\). Do not substitute the rounded 0.67 prematurely. **2. Invert the uniform CDF:** For \(D\sim U[800,1000]\), \(F(Q)=(Q-800)/200\). Therefore \[Q^*=800+200\left(\frac23\right)=933.333\ldots.\] **3. Interpret the quantity:** The continuous-model answer is 933.33 units. If only whole diyas may be made, compare the neighboring integers; 933 is optimal under this continuous uniform expected-cost model. **Answer: 933.33.**

**1. Compare mismatch penalties:** If \(C_o<C_u\), then \(CR=C_u/(C_u+C_o)>0.5\). Avoiding an expensive shortage merits stocking above the median. **2. Locate the uniform quantile:** The uniform mean and median are both \((800+1000)/2=900\). A quantile above 0.5 lies to the right of 900; here the ratio 2/3 gives about 933.33. **Answer: C — Towards the right of the mean.** A: **Incorrect:** Below the mean corresponds to a critical ratio below 0.5, when overage is more costly. B: **Incorrect:** The mean is optimal when the two mismatch costs are equal. C: **Correct:** A higher underage cost produces a quantile above the mean/median.

**1. Put all inputs on an annual basis:** \(D=1500(12)=18{,}000\) units/year; ordering cost \(S=150\) rupees/order; holding cost \(H=0.20(27)=5.40\) rupees/unit/year. **2. Apply EOQ:** \[Q^*=\sqrt{\frac{2DS}{H}}=\sqrt{\frac{2(18{,}000)(150)}{5.4}}=\sqrt{1{,}000{,}000}=1000.\] **3. Interpret the minimization:** This balances annual ordering and holding costs. With a fixed unit purchase price and fixed annual demand, purchase expenditure does not vary with lot size. **Answer: 1000.**

**1. Find annual demand and EOQ:** \(D=1500(12)=18{,}000\), \(H=0.20(27)=5.4\), and \(Q^*=\sqrt{2(18{,}000)(150)/5.4}=1000\) units. **2. Divide demand into lots:** \[n=\frac{D}{Q^*}=\frac{18{,}000}{1000}=18\text{ orders/year}.\] Each order supplies the next 1000 units of demand. **Answer: 18.**

**1. Find optimal order frequency:** Annual demand is \(18{,}000\), annual holding cost per unit is \(5.4\), and EOQ is \(\sqrt{2(18{,}000)(150)/5.4}=1000\). Thus the firm places \(18{,}000/1000=18\) orders/year. **2. Multiply by the cost per order:** \[C_{order}=\frac{D}{Q^*}S=18(150)=2700\text{ rupees/year}.\] This excludes material purchase expenditure. **Answer: 2700.**

**1. Determine the replenishment lot:** \(D=18{,}000\) per year and \(H=5.4\) give \(Q^*=\sqrt{2(18{,}000)(150)/5.4}=1000\) units. **2. Describe the EOQ inventory cycle:** Under constant demand and instantaneous replenishment without safety stock, on-hand inventory decreases uniformly from 1000 just after delivery to 0 before the next delivery. **3. Take the average:** \[\bar I=\frac{Q^*+0}{2}=\frac{1000}{2}=500\text{ units}.\] **Answer: 500.**

**1. Compute unit holding cost:** \(H=0.20(27)=5.40\) rupees/unit/year. Annual demand is \(18{,}000\), so \(Q^*=\sqrt{2(18{,}000)(150)/5.4}=1000\). **2. Use average inventory:** With no safety stock, \(\bar I=Q^*/2=500\) units. **3. Calculate annual carrying cost:** \[C_{hold}=H\bar I=5.4(500)=2700\text{ rupees/year}.\] It equals annual ordering cost at the unconstrained EOQ. **Answer: 2700.**