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Difficulty: Very hardSequential Processes and State Transitions

A distribution network manages inventory between two regional facilities, Facility A and Facility B. At the beginning of Stage 1 (k=1k=1), Facility A holds 216216 units of product and Facility B holds 160160 units of product.

In each stage kk (for k=1,2,3k = 1, 2, 3), the inventory levels undergo four sequential state transitions in the exact order listed below:

1. Step 1: Facility A dispatches 12\frac{1}{2} of its current inventory.
2. Step 2: Facility B dispatches 12\frac{1}{2} of its current inventory.
3. Step 3: Facility A transfers 13\frac{1}{3} of its remaining inventory to Facility B.
4. Step 4: Facility B receives 2828 new units from an external manufacturer.

Which of the following statements regarding the inventory levels at the end of Stage 2 and Stage 3 are correct? Select all that apply.

  1. At the end of Stage 2, Facility B holds exactly 112 units.Answer
  2. At the end of Stage 3, the total combined inventory across both facilities is 96 units.Answer
  3. C
    At the end of Stage 2, Facility A holds 36 units.
  4. D
    At the end of Stage 3, Facility B holds 60 units.
  5. The total combined inventory decreases by 40 units from the end of Stage 2 to the end of Stage 3.Answer

Answer

The correct statements are: (1) At the end of Stage 2, Facility B holds exactly 112 units; (2) At the end of Stage 3, the total combined inventory across both facilities is 96 units; and (3) The total combined inventory decreases by 40 units from the end of Stage 2 to the end of Stage 3.
Tracking all 4 transition steps systematically across 3 stages yields the exact state vectors: Stage 1 (72,144)(72, 144), Stage 2 (24,112)(24, 112), and Stage 3 (8,88)(8, 88). Comparing these state vectors confirms that Facility B has 112 units at Stage 2 end, total inventory at Stage 3 end is 8+88=968 + 88 = 96 units, and the change from Stage 2 end to Stage 3 end is 136−96=40136 - 96 = 40 units.

Step-by-Step Solution

1
Calculate state transitions for Stage 1 (k=1k=1)
A1=72A_1 = 72, B1=144B_1 = 144
Start at A0=216,B0=160A_0 = 216, B_0 = 160. Step 1: A→216−108=108A \to 216 - 108 = 108. Step 2: B→160−80=80B \to 160 - 80 = 80. Step 3: AA transfers 13(108)=36\frac{1}{3}(108) = 36 to BB, so A→108−36=72A \to 108 - 36 = 72 and B→80+36=116B \to 80 + 36 = 116. Step 4: BB receives 28, so B→116+28=144B \to 116 + 28 = 144.
2
Calculate state transitions for Stage 2 (k=2k=2)
A2=24A_2 = 24, B2=112B_2 = 112, Total2=136\text{Total}_2 = 136
Start at A1=72,B1=144A_1 = 72, B_1 = 144. Step 1: A→72−36=36A \to 72 - 36 = 36. Step 2: B→144−72=72B \to 144 - 72 = 72. Step 3: AA transfers 13(36)=12\frac{1}{3}(36) = 12 to BB, so A→36−12=24A \to 36 - 12 = 24 and B→72+12=84B \to 72 + 12 = 84. Step 4: BB receives 28, so B→84+28=112B \to 84 + 28 = 112.
3
Calculate state transitions for Stage 3 (k=3k=3)
A3=8A_3 = 8, B3=88B_3 = 88, Total3=96\text{Total}_3 = 96
Start at A2=24,B2=112A_2 = 24, B_2 = 112. Step 1: A→24−12=12A \to 24 - 12 = 12. Step 2: B→112−56=56B \to 112 - 56 = 56. Step 3: AA transfers 13(12)=4\frac{1}{3}(12) = 4 to BB, so A→12−4=8A \to 12 - 4 = 8 and B→56+4=60B \to 56 + 4 = 60. Step 4: BB receives 28, so B→60+28=88B \to 60 + 28 = 88.
4
Evaluate the given statements against calculated stage states
Statements A, B, and E are true; C and D are false.
Facility B holds 112 units at Stage 2 end. Total inventory at Stage 3 end is 8+88=968 + 88 = 96 units. Total inventory change between Stage 2 and Stage 3 is 136−96=40136 - 96 = 40 units.

Key Concept

Sequential Multi-Stage State Transitions
Estimated Time:2m 30s
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Topics

Sequential Processes and State Transitions

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