Production plan and supplier choice at a tailoring unit
A small tailoring unit in Bhubaneswar produces shirts, trousers and jackets. Each shirt uses 2 units of cloth, 1 spool of thread and 1 dozen buttons; each trouser uses 3, 1 and 2; each jacket uses 4, 2 and 3. These recipes form the resource-requirement matrix A, with rows cloth, thread and buttons and columns shirt, trouser and jacket. The current order is 60 shirts, 40 trousers and 20 jackets. The unit holds exactly 320 cloth units, 140 thread spools and 200 button dozens. Supplier P quotes ₹150 per cloth unit, ₹40 per spool and ₹25 per button dozen, and offers 3% off if the bill reaches ₹55,000. Supplier Q quotes ₹140, ₹45 and ₹30, and charges ₹500 delivery. The owner must confirm the order is feasible, choose a supplier, and check whether a changed rush order can be met by a unique product mix.
1. Verify that the current order exactly uses the available stock, then compute the landed cost from each supplier and recommend one.
Read the worked answer
Write the order as x = (60, 40, 20)^T. Then A x gives:
- cloth: 2(60) + 3(40) + 4(20) = 120 + 120 + 80 = 320
- thread: 1(60) + 1(40) + 2(20) = 60 + 40 + 40 = 140
- buttons: 1(60) + 2(40) + 3(20) = 60 + 80 + 60 = 200 So the stock is exactly sufficient.
Supplier P: 320(150) + 140(40) + 200(25) = 48,000 + 5,600 + 5,000 = ₹58,600. The 3% discount is ₹1,758, giving a net cost of ₹56,842.
Supplier Q: 320(140) + 140(45) + 200(30) = 44,800 + 6,300 + 6,000 = ₹57,100; adding ₹500 delivery gives ₹57,600.
Recommend P. It is cheaper by ₹758 after the discount and delivery, although Q is cheaper before those terms. The recommendation depends on P's threshold being met.
2. A rush order requires 260 cloth units, 120 thread spools and 160 button dozens. Using A^(-1) = [[1,1,-2],[1,-2,0],[-1,1,1]], find the product mix and comment on uniqueness.
Read the worked answer
Let b = (260, 120, 160)^T. Then x = A^(-1)b gives:
- x1 = 260 + 120 - 2(160) = 380 - 320 = 60
- x2 = 260 - 2(120) + 0 = 20
- x3 = -260 + 120 + 160 = 20 So the mix is 60 shirts, 20 trousers and 20 jackets.
Check: cloth = 2(60) + 3(20) + 4(20) = 120 + 60 + 80 = 260; thread = 60 + 20 + 40 = 120; buttons = 60 + 40 + 60 = 160. All quantities are non-negative, so the rush order is feasible.
The solution is unique because det A = 2(3-4) - 3(3-2) + 4(2-1) = -2 - 3 + 4 = -1, which is non-zero. A zero determinant would mean either no solution or infinitely many mixes for some resource targets.
3. Supplier P asks what minimum discount rate it must offer to beat Q's landed cost of ₹57,600, given P's list bill is ₹58,600. Compute and interpret.
Read the worked answer
We need 58,600(1 - d) < 57,600. Thus 1 - d < 57,600/58,600 = 0.982935..., so d > 0.017065..., i.e. about 1.71%.
At a 1.70% discount, the net bill is 58,600(0.983) = ₹57,603.80, still above ₹57,600. At 1.71%, it is about ₹57,597.94, which beats Q by a small margin.
Interpretation: P needs only a little more than its current 3% offer to be competitive on this order, but the margin is thin. If Q waived its ₹500 delivery charge, Q's landed cost would be ₹57,100 and P would need a discount above 2.56% to win. The owner should therefore compare final landed costs, not list prices.